Hey guys (said gender neutrally of course). Here are a couple very minor corrections: -Quite a major mistake on my part: when doing the calculations for base -i for 5:47, I forgot to place parenthesis around -i, which lead the results to being equal to -1(i^x) rather than "-i, -1, i, 1..." like it should have been. That's completely my mistake. -I used the wrong spelling of whether at 6:03 -I said at 4:53 that phi was the number that satisfied the equation sqrt(x) = x - 1when the real answer is (sqrt(5) + 3)/2. The correct form of that equation would be that 1/phi = phi - 1. It's a very minor mistake but still important to point out. -At 2:42 I say that base 18,446,744,073,709,551,616 was for a 64 by 64 grid but that base only satisfies all possible combinations of an 8 by 8 grid. A real 64 by 64 grid base would be closer to 1.322112e+123. -The Japanese "100" symbol at 3:07 is slightly malformed, it needs an extra line at the bottom. -I use the term irrational at 4:30 which may have confused some people. The term irrational technically only refers to weather a number can be expressed as a fraction but it is a very useful way of describing a number with infinite, non-repeating digits. -I accidentally refer to what is called a multiplicative system as a bijective system instead. -I, in my foolishness, have made the music too loud. D: I promise there aren't always this many mistakes! Sorry!
Another minor correction: base -𝒾 does not cycle in the same direction as base 𝒾, but in the opposite direction: (-𝒾)² = -1, (-𝒾)³ = 𝒾, (-𝒾)⁴ = 1, and so on. If I may do some armchair inference, I believe you might have forgotten your parentheses when calculating base -𝒾, and instead calculated the negative of base 𝒾, which indeed cycles in the same direction.
Not trying to dogpile you further but: • What you describe as a bijective number system is actually a multiplicative system. • The Japanese/Chinese character for 100 is 百; you missed a line on the bottom. • Honestly I really like the music! I wouldn’t lower it too much I still really like the video though; I’m subscribing for sure. Plus I love your artstyle; reminds me of Typoman.
Why not order the things in this comment so that when watching the video the viewer can easily see when the next correction is? (Though any major mistake can stay at the top or something)
@@randy-x They probably meant using Chinese characters as numerals, not just the Chinese numerals. That being said, that's still not base-∞, "only" base-3500 or base-6500.
*Revelation 3:20* Behold, I stand at the door, and knock: if any man hear my voice, and open the door, I will come in to him, and will sup with him, and he with me. HEY THERE 🤗 JESUS IS CALLING YOU TODAY. Turn away from your sins, confess, forsake them and live the victorious life. God bless. Revelation 22:12-14 And, behold, I come quickly; and my reward is with me, to give every man according as his work shall be. I am Alpha and Omega, the beginning and the end, the first and the last. Blessed are they that do his commandments, that they may have right to the tree of life, and may enter in through the gates into the city.
@@JesusPlsSaveMe *Numbers 25:5-13* So Moses said to Israel’s judges, “Each of you must put to death those of your people who have yoked themselves to the Baal of Peor.” Then an Israelite man brought into the camp a Midianite woman right before the eyes of Moses and the whole assembly of Israel while they were weeping at the entrance to the tent of meeting. When Phinehas son of Eleazar, the son of Aaron, the priest, saw this, he left the assembly, took a spear in his hand and followed the Israelite into the tent. He drove the spear into both of them, right through the Israelite man and into the woman’s stomach. Then the plague against the Israelites was stopped; but those who died in the plague numbered 24,000. The Lord said to Moses, “Phinehas son of Eleazar, the son of Aaron, the priest, has turned my anger away from the Israelites. Since he was as zealous for my honor among them as I am, I did not put an end to them in my zeal. Therefore tell him I am making my covenant of peace with him. He and his descendants will have a covenant of a lasting priesthood, because he was zealous for the honor of his God and made atonement for the Israelites.”
@@JesusPlsSaveMe To Claim Their Bones SkillAttack.png 55 (50+5) Atk Weight ⯀ Amt. x0 [Before Attack] For every 7 Poise.pngPoise on self, Atk Weight +1 (Max +2) [On Use] Until this Skill ends, this unit cannot be Staggered from taking damage [On Use] Deal +2% damage on Critical Hit for every Poise.pngPoise Potency on self (Max 50%) +30% Damage on Critical Hit Deal +0.5% more damage for every 1% missing HP on self (Max 25%) If this Skill targets only a single enemy, deal +100% more damage (In Focused Encounters, a single Part) CoinEffect4.png [On Hit] Inflict 3 Bleed.pngBleed [On Hit] Inflict 5 Paralyze.pngParalyze next turn
Depending on what you think 0^0 is, you can display the number n by writing n times the null symbol, from which there are zero in existence. If you want it more accessible, write | instead, e.g. 4 = ||||, and | has the value 1/0... 🤣
Prof. Arnold Ross: "Think deeply of simple things." ETA: his contention was that in Math, even deep results are near the "surface"--and new discoveries are always possible, even in areas that have been developed for a long time. We, his students, often managed to think simply of deep things. . . .
Base 12 is my favorite base. It isn't inconveniently large, and is divisible by 3 and 4. We used to use it on a large scale as well before the Arabic numerals became the standard, that's where the dozen comes from, and is why eleven and twelve have unique names instead of being lumped in with the 'teens.
Base 12 probably wasn't used on a large scale before Arabic numerals. There are a few words that seem to refer specifically to twenty (like "twenty" itself, "dozen", and "gross") but you could make the same argument for base-20 ("score" and French "vingt"). And when you look at the etymologies of most of those, they clearly derive from base 10: "eleven" and "twelve" derive from Proto-Germanic compounds meaning "one left" and "two left" (as in, after counting to ten), and "dozen" is from Latin "duodecim" meaning" two and ten". The exception is gross meaning 144, which derives from a word meaning "large, coarse, monstrous" and looks to be a case of a word for "a large amount" being repurposed to refer to a specific large amount, similar to how "myriad" can mean either "uncountably many" or "ten thousand". There's some speculation that the Germanic "long hundred" (referring to 120) implies an early use of base-12, but I should point out that 120 isn't really a special number in base-12. It's not a place-value the way hundred is in decimal.
@@gwalla Your nerdery is appreciated. As someone who clearly knows more on this topic than I do, do you know how much truth there is behind the idea of people counting in base 12 by using the thumb to count the three joints on your four fingers of each hand?
@@bow-tiedengineer4453 I mean, you can do that, and it works. As for whether people did it historically, I have no idea. I know in Korea they sometimes use a technique called chisanbop to count up to 100 on two hands, by treating the thumbs kind of like the short rods of an abacus. That was a 20th century invention though. If you hold your hands over a surface you can even count up to 1023 using binary, by touching the surface with different fingers (curling the fingers isn't practical for this).
@@bow-tiedengineer4453 also easier to count on your fingers. 3 bones per finger x 4... Use your 👍 to touch each part of their other fingers to count to 12.
I once read this SCP where an AI was created to devise better compression algorithms for the Foundation's archives, only to remove itself from existence. While the documents in the affected archives had practically disappeared, it was somehow still possible to access them. It turns out the AI figured out how to use nullary: this base can only be understood from a certain frame of reference (Q) which is incompatible with that of normal human thought (K). And a researcher trying to find where the data went ends up becoming nonexistent as well, so yeah we better not let number exist in nonexistence
Ah yea, 6276. They really give Surrealistics a load of wacky anomalies to contain, huh? "The above analysis has been rated helpful by 22% of personnel"
@@umber6937 Not a Surrealistics Dept. skip, but have you sacp slon obr 7978? If you haven’t, it’s this bizarre, mind-warping online cartoon that slon “cult” tlaol tlei plr “cult following”
To throw another work of fiction into the mix, New Phyrexia (MtG) uses hexadecimal as the base for writing numbers. The written system uses a 2x2 grid, rotating the position of a symbol on the grid for each increment and a unique symbol for 1-4, 5-8, 9-12, and 13-16, respectively.
Reminds me of several stories from _Fantasia Mathematica_ (1958): "The No-Sided Professor" by Martin Gardner "The Island of Five Colors" also by Martin Gardner (well, sort of) and "A Subway Named Moebius" by A. J. Deutsch.
4:34 - Common misconception. Rational numbers (including integers) are still rational regardless of whether they have an infinitely long, seemingly random representation. The failing is in the base, not the number. You could say "they appear irrational to people who are familiar with how irrational numbers look in more commonly used bases", but that's about all. They're still rational no matter how they look. Rational means "can be written as the ratio of two integers". This says nothing about how the number or the integers in its ratio happen to look when that writing is done.
A lot of ink has been used to explain "intuitive" concepts in rigorous mathematical detail, precisely because of how often our intuition is wrong in fringe cases. I remember going through my master's level number theory course where they rigorously defined what natural numbers are (using set theory; it's crazy how complicated you have to be with it to fit how they're actually used in the real world while also satisfying everything they need to be mathematically). It's important to remember that literally everything in math is a construct, so in another timeline we could have defined integers (and therefore rational numbers) differently such that the statement would have been correct. But yeah, as things stand the base we choose to represent them in does not change whether a number is rational or not.
@@justin-ju4eo Right. Any integer or rational multiplied by an irrational give an irrational. BUT. It is possible to find pairs of irrationals that give a rational product when multiplied. Then, there are irrationals that can give a rational result when elevated to an irrational power. Just take the natural logarithms of any integer. They are all irrationals. The base is e, and e is an irrational.
- Do you know that you can use different radix for every position? - What a silly idea! Nobody would use such a system! (this comment was written at second 2024 10 11 00 19 41 given place value [31558149.8, 525969.163, 86400, 3600, 60, 1], sidereal year and some timezone)
@@lockaltube The only thing that remains constant, is the modernist interpretation of the idea of constancy, hypothetically existing outside of personal interpretation; I'll be honest, I have no idea what I'm talking about, but you've read this far ☺Edited: to show I edit therefore am not bot, unless...what if I was a robot and didn't know it?
Those are ordinal units as opposed to cardinal units, except for everything smaller than a day. Fun fact: although the day can be treated both as an ordinal unit and a cardinal unit, the *_nychthemeron_* is purely cardinal.
Not even mention of base64 and base85 which are both actually used in computer science... (64 for obvious reasons and 85 is actually just the number of printable ASCII characters, so it uses all of them as symbols for the base)
Hexadecimal (base 16) is much more common, at least by humans who work with computers, than base 64. The main reason for its utility is that each "digit" of Hex can be directly translated into 4 "digits" of binary because 16 is equal to 2^4. Thus you can easily convert between them as needed, taking advantage of using fewer digits when writing long numbers, but being able to break down individual binary bits when needed.
@@edwardblair4096 i didn't say they were common i said they are used... Main usage of base 85 is when you want to print out binary data so that it can be then scanned back from paper... Base 64 can also be used for that but also used for hashes and other random string generations... For example RUclips video codes are base64... main advantage of base 64 it uses 4 times fewer characters than base16, which is important if it is actually used as string
I wrote a “big number” program once, which used base (2**64) (meaning every “digit” was a 64-bit digit). So, instead of 0-9, it was 0-18,446,744,073,709,551,615. One of the fun things (?) was figuring out a system for overflow-proof multiplication, while still remaining as performant as possible. You could get some ludicrously big numbers with that thing. For example, it’s much like how the “2” in the tens-position of the decimal number 23 represents (2 * (10 ** 1)), giving us 20, to which the 3 in the ones-position is then added for the final result of 23. But, in this case, [2, 3] in base (2 ** 64) - and I’m representing the digits in an array, to avoid confusion - means we have: • (2 * ((2 ** 64) ** 1) = 36,893,488,147,419,103,232 plus • (3 * ((2 ** 64) ** 0) = 3 for a total of: 36,893,488,147,419,103,235 In decimal, if we have [1, 0, 0], we get 100. In base (2 ** 64), [1, 0, 0] is instead: • (1 * ((2 ** 64) ** 2)) = 340,282,366,920,938,463,463,374,607,431,768,211,456 So, you can see how these numbers quite readily get very out of hand! If you had a 1 in the 64th digit position base 10, and zeroes in the rest, that would be pretty crazy! It would be a 1 followed by 63 0s!: 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000 That’s a lot of number! But, if you did the same thing with base (2 ** 64), you get something a little more insane: 56,616,434,707,392,290,493,830,608,686,343,291,913,914,207,267,329,621,488,609,504,077,443,092,998,260,469,825,507,336,856,810,827,911,261,368,707,758,178,809,587,148,432,632,037,424,161,064,675,500,873,383,487,082,580,173,608,658,838,616,501,128,049,812,714,217,651,328,283,024,830,073,412,384,889,815,432,597,905,927,601,958,951,314,543,937,080,343,085,064,110,993,312,334,723,049,547,437,731,770,779,823,050,760,639,880,425,500,165,204,989,981,997,213,399,871,161,087,661,501,312,220,225,137,504,516,172,823,917,738,211,300,299,807,381,757,818,984,800,273,856,795,211,620,996,748,504,363,142,690,436,276,062,613,301,784,444,812,913,670,500,602,718,367,796,525,145,427,307,300,283,865,951,803,998,808,065,639,029,170,703,022,449,468,190,319,930,507,340,988,677,523,066,084,721,639,425,495,152,552,125,131,611,082,991,384,744,696,522,708,197,306,203,186,569,928,692,274,528,086,558,201,803,207,638,020,888,691,121,265,434,197,293,907,218,384,095,520,560,555,896,947,150,944,133,055,863,397,615,648,411,112,470,754,656,110,748,811,237,653,374,788,477,716,072,552,588,609,616,376,430,894,096,481,364,501,294,655,654,823,142,548,823,259,752,215,716,293,351,999,419,717,562,306,177,064,929,632,050,689,015,014,405,578,694,408,614,353,540,180,649,410,108,905,523,548,191,355,424,055,272,181,142,840,177,487,296,189,764,022,300,385,934,884,483,982,341,901,658,184,426,332,624,776,972,085,905,793,515,459,600,620,068,183,625,181,327,963,636,494,799,582,466,830,531,258,661,325,767,796,742,142,353,530,480,288,174,823,736,291,072,477,683,720,590,971,289,642,787,198,230,081,364,301,868,392,910,675,871,538,608,754,231,607,296 That’s a number 1,214 digits long when represented in base 10!
I saw a few papers on base '2i' and it has some interesting properties. Numbers written in base 2i may look like 'base 4' but their value can be any complex number (positive or negative) and you don't need to use any negative signs. When adding and subtracting such numbers you have to skip a digit when carrying and borrowing and those operations are reversed. For addition you 'borrow' and for subtraction you 'carry'. My favorite property is that if you divide numbers in base 2i then you can divide complex numbers without taking a complex conjugate. Dividing base 2i numbers is tricky so unless you really hate complex conjugates I don't see many reasons to use it.
p-adic is more of an interpretation of an existing base rather than a distinct set of bases. You can write Real decimal numbers like 4561.235..., or you can write p-adic decimal numbers like ...4561.235. Unlike the Reals though, each base actually gives a different number system, so while the 3-adics are just p-adics in ternary, it also has different numbers than the 5-adics, which are p-adics in quinary. Doing math in composite (non prime-power) bases with p-adics though doesn't completely work, hence the "p" in the name standing for "prime". This is different from the Real numbers, where composite bases, and especially superior highly composite number bases, tend to work better. (Binary _cheats_ since 2 is paradoxically both a prime number _and simultaneously_ a superior highly composite number, despite... not being composite. It does however have some weird and unique problems when used for p-adics that pop up in specific situations.)
The Babylonians used to have base 60. They counted each third of the finger excluding thumb and when finishing a 12, they would put a finger up until they have all 5 fingers. 5•12=60
Funny enough, the many symbols problem with larger bases is something people have actually decided to solve: The Argam system bases its symbols for all non-Primes after 9 on their factorization, modifying previous symbols to make new ones (so their symbol for ten, or "dess" is basically an upside down 2, which looks kinda like a 2 and kinda like a 5).
You missed part where negative base can represent all integers(positive and negative). A base with some negative digits can also do that. Like balanced ternary.
If you use something like base -10, you don’t need the negative sign for negative numbers which means you could use a minus sign in front of the positive number. So there are two ways to express a number in this base.
@@TheFrewah No, base -10 doesn't even need the negative sign since a negative number has a even number of digits and a positive number has an odd number of digits. "-10" would just be written as "10", and "10" would be "190" since it's 1(100) + 9(-10) in negadecimal.
@@yeetrepublic9142 Some time since i wrotea program that did the conversion. So if 190 is ten in base 10 then -190 would be -10 in base -10. You could also, more naturally, express it some other way where you would not need the negative sign.
In college, one of my classes gave an assignment to invent a number system other than the commonly used systems. Then, we had to create an addition and subtraction problem using that system and solve. Finally, convert a number in our system to decimal. Most of my classmates went with something small like base3. I did base64 using 0-9, uppercase letters, lowercase letters, and a couple of made-up symbols. The caculations covered an entire page of paper. My professor took one look at it and just said, "I believe you," and gave me an A, lol. Didn't even try to verify it.
Base 16 (Hexadecimal) which uses characters 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e and f: • Neither too little nor too much characters. • Unlike Decimal, doesn't create confusion whether you're referring to the base-10 number system or a number between two integers. • With only four digits, it can represent values as high as (decimal) 65535. • Compatible with 7-segment displays (since letters a-f can be represented using them) • And not to mention its use in Computer Science, in which it has even more advantages in that area, such as: -- • 1 base-16 digit is 4 base-2 digits -- • Can represent any color a screen can process with only 6 digits (8 if counting alpha channel) while Decimal requires 9 (with a lot of leftover numbers). -- • Can represent memory addresses (generally powers of two in amount) and machine code more easily. -- • Can represent one byte in two digits (with no left-over values) In short, Hexadecimal is good.
Binary can be useful for human use in at least one case: counting on your fingers. Think about it, your fingers can either be up or down, representing a 1 or a 0. Counting in binary on your fingers allows you to count up to 1,023. You probably won't need that much, but on just one hand you have up to 31, which is extremely helpful in some cases if you need to do something and count at the same time.
Finger ternary works too -- you just need three positions, such as up, down, and curled for 0,1,2 or T,0,1, etc. Which gets you LOTS more capacity. 243 on each hand -- or +/- 121 with balanced ternary.
This reminds me of 'balanced ternary' which is basically just base 3, but not with symbols {0, 1, 2} but {0, 1, -1}. The -1 is usually written with its own symbol so as to not use two glyphs at once, lets use "T" here. Then a 5, for example, would be written as 1TT in balanced ternary since 9+(-3)+(-1) = 5. This was apparently used in some early computers as a sort of extension to the thinking of binary: Binary makes sense for computers because you can have the electricity [not flowing] (0), or [flowing] (1) Balanced Ternary extented this thought to [not flowing] (0), [flowing one way] (1), and [flowing the other way] (-1) I think this is really cool, but it apparently/evidently never caught on as much as binary did
There are at least 10 types of people in the world: Those that know binary, and those that don't. (And those who know this joke works best in balanced ternary.) Balanced ternary is my favorite base system.
You could also use 0, 1, 2 for balanced ternary and still use negative voltage. The association of 0 with off and 1 with on is also kindof a thing we people made up 😅 (I personally used 0,1,2 because i think the notation is easier, but I did stuff with balanced ternary logic circuits :3)
Fun fact, negative integer bases still allow you to represent every number. Also fun fact, the symbols don't have to be 0-(n-1). I've played with base 10 but where the symbols are -5 through 4, and you can still hit every real number that way, including negatives.
That's damn cool, dude. Maths needs more people like you, who are willing to experiment. Speaking of which, can I ask what your answer is to the question "What number lies between 0 and infinity?"?
Base e is supposed to be the most optimal of all, which is why during the 60's I think the Soviets tried making a base 3 computer since e is closed to 3 as a whole number integer. Unfortunately it failed because making fractional electronic counters is extremely difficult
As a side-note: prime exponent notation is actively used in microtonal music theory to denote fractions (there it is called "Monzo" notation; the individual "digits" can be positive, zero or negative)
When I was in basic training I invented a base 60 system, and I'm actually amazed by how easy it is to use It basically functions like the Mayan system. There's a handful of symbols that you combine to get a value up to 60. A dot is one, a bar is 5, a triangle with it's point up is 15, a triangle pointed down is 30, an equal sign looking thing is 0, and a circle is 60. You combine them, largest to smallest, top to bottom, but you can subtract smaller numbers from bigger ones by writing the smaller number above the larger one. For example, you can write 27 like this ∆ 15 _ +5 _ +5 .. +2 Or you can save a bit of time and space by writing it like this ... -3 V 30 (pretend V is a triangle) This system would probably be very good for teaching math to kids because you can visually see what's happening when you do basic arithmetic.
1:24 base two is not too hard to use for humans. Using the symbols 1 and 0 will make your numbers look very long, but we could use smaller symbols, because we only need do differenciate between 2 symbols instead of 10
In high school, in the early 80's, I had to write a paper for math class. I chose the subject of non-integer bases and irrational bases in particular. I concluded that if you're not going to have a discrete number of symbols for a position, you're essentially just using logs. You only need one position, so you give the value and the position power which indicates whether it's a log, log log, etc. Without personal computers, it's a lot harder to play around with it.
There was an attempt to create a "balanced trinary" base for a computer where each digit or "trit" would have one of three values: Plus, Zero and Minus. If you look at the way negative binary numbers are represented for computers, it is a ugly mess. Balanced Trinary has negative number built in. Zero is represented as "0", one as "+", two as "+ -" (+3 -1), three as "+ 0", four as "+ +" (+3 +1), and five as "+ - -" (+9 -3 -1). Mathematically elegant, but nobody could build a three state junction for the hardware.
Asymmetric Numeral Systems: the base of each digit depends on the frequency of the previous digit. This is actually used in some newer compression algorithms.
A system where each digit uses a mixed radix so instead of powers of a base you have a certain set of multipliers to use for each digit like a system that uses 22 symbols for the first digit 9 symbols for the next digit then 114 symbols then 35 then 178. That way 3.31415 means 3+3/22+1/(9*22)+4/(22*9*114)+1/(22*9*114*35)+5/(22*9*114*35*178) which is about pi
You forgot balanced numeric system! They work like regular natural number bases, but half the symbols are negative. For example, balanced ternary uses digits representing 0, 1 and -1 (usually rendered as T). Here are the numbers from 1 to 27 in that base: 1, 1T, 10, 11, 1TT, 1T0, 1T1, 10T, 100, 101, 11T, 110, 111, 1TTT, 1TT0, 1TT1, 1T0T, 1T00, 1T01, 1T1T, 1T10, 1T11, 10TT, 10T0, 10T1, 100T, 1000
I like balanced ternary for a couple reasons: there is no separate numerical sign for positive or negative--if the first non-zero digit is 1, it's positive, and if it's T it's negative. Also, to approximate a "decimal" ("ternial"?) number in this base, simply truncate it at the desired point.
My favorite underrated numeral base system is Balanced Ternary. You basically use three symbols, such as {+ 0 -}, you can write any integer, positive or negative, without the need for extra symbols, and the addition tables only require 2 carries out of every 9 operations, as opposed to 1 carry out of 4 operations for binary (+ & + = +-, - & - = -+)
This actually cleared something up for me, somehow. I remember reading a bit of math humor many years ago...a list of actual PhD thesis titles, all of which were suitably incomprehensible for non-math nerds. One of them was "...(interesting properties or something)... of base negative three". That always sounded extraordinarily bizarre to me...non-sensical...and therefore funny. But just at the beginning of the section on base -2 in this vid, it all became crystal clear to me how that would work. Thanks! 🤪🤨
Systems with a non-integer base close to 2 (e.g. 1.8) are good for representing numbers in successive approximation ADCs. In this case, the conversion error in the i-th digit can be compensated in the next digit. And the conversion result can be easily converted to binary code.
Issac Asimov made a credible case for using base twelve. We only need two new symbols (call them a,b). Twelve has so many factors, more than any other smaller integer. {2,3,4,6} A lot of arithmetic becomes simple.
And, you can also use a kind of chisanbop to do finger math. I use three fingers for the "ones" and the other two on that hand for the "fours", so you can get 0-11 on each hand.
Come to think of it, I have also played with variable number systems -- a fun example is where the first position is base 2, the second base 3, the third base 4, and so forth -- but another one uses bases 4, 6, 8, 10, 10, 12, and 20 -- these are the dice used in D&D, and are mostly based on the Platonic solids (the two 10-sided ones being exceptions) -- and in my experimenting with *this* base, I discovered a couple of fun facts: first, it doesn't matter what order these dice are in, they always produce a valid number -- that changing the order would produce a different number -- but ultimately, the dice set will have the *same* maximum number, no matter what order you choose! Another example of variable-based numbers is how we keep time: base 24 hours, base 60 minutes, base 60 seconds. We can thank the Babylonians for that!
Yeah man, the amount of "whole" numbers you can express depends entirely on what numerals you're using, it'll always be in the "1s" place because all other places are "0s", and it's the only way I can think of to represent 1/0 as anything other than 1/0. Super intriguing, and super untapped.
Base 2 is tedious to use on its own, but fortunately, any base can be represented more compactly by taking a power of that base. Any hexadecimal digit can represent a string of 4 bits, so you see hex everywhere in computing.
Other interesting number systems not mentioned in the video: - Dynamic base number system: A mixed-radix number system who's base or radix values are determined dynamically at runtime, with just-in-time binding. I would elaborate, but naturally its difficult to say much about its properties or behavior before hand. - PRNB (Pseudo-random number base) number system - "Oops! All 5's" number system (This number system only permits one symbol: 5. Fortunately, its also a base-5 number system. Unfortunately, this still doesn't make it a functionally workable number system. Nobody has worked out how to actually DO anything with it) - ROYGBIV number system in base-white (purely theoretical--it is unclear how the operations of addition and multiplication would be performed on colors, or what that would mean) - Dancerithmetic; Uses base-10 numbers with place values, like we are used to. The major difference here is this number system doesn't have any symbols/graphemes, so it cannot be written down. Instead, all symbols have been replaced with large, sweeping and distinct bodily gestures/movements, most of which have been borrowed from Ballet. Therefore arithmetic in this system is largely a performative art. It is possible to divide by zero in this number system, provided you are limber enough.
I feel slightly like Dan Aykroyd talking to Tommy boy when he says “went a little heavy on the pine tree cologne there kid” except I’m thinking about that fricken music playing in the background lol. Little distracting while I’m very interested, but also laying next to my sleeping child. Anyway good video and great content that most don’t think about everyday, but stuff that really keeps things in perspective and I believe to be extremely important. Thanks!
There is also fibinary notation. Each digit is 1 or 0, and the place values of the digits follow the Fibonacci sequence, so 1010 = 1 x 5 + 0 x 3 + 1 X 2 + 0 X 1. Notably, most integers have non-unique representation, e.g. 5 can be either 1000 or 0110. The addition algorithm isn't too hard to derive.
@@urkerab Yes, that is what I consider 'canonical form'. I think this canonical form is unique for each integer. But you need to allow the possibility of consecutive 1s during the addition algorithm. You can tidy them away at the end. Short summary of addition algorithm: to add x and y, initially set s = 0. Shift bits from x or y to s (i.e. in a location where x or y has a 1 but s has a zero, set that bit to 0 in x or y, and 1 in s), and perform substitutions 100 -> 011 or 011 -> 100 to allow more bits to be shifted. All these operations keep the sum x + y + s invariant. Keep going until x and y are both 0.
If I recall, there's a primorial base which uses each primorial instead of powers of a constant number. So the number XYZ in the primorial base would be: X*30 + Y*6 + Z*2 While this looks fairly simple, and maybe kinda cool but quirky, there is a downside. Each digit can take the same number of symbols as that primorial. So the "units" digit can be 0 or 1; the next ("tens") digit can be in the range 0-5; the third ("hundreds") digit can be in the range 0-30 and already we need to invent new symbols. The fourth digit is in the range 0-210, and you can see how this is getting out of hand. The fifth digit handling anything from 0-2310 is just ridiculous. There's a reason this base is nothing more than a fun little mathematical diversion. xD
Factoradic numbers, discovered by Donald Knuth when he was in high school. He noticed that when you "fill up" any number of digits in a normal base system and add one, it "rolls over" to the next digit and everything zeros out, e.g., 999 + 1 = 1000. It also happens to be that factorials have a very similar property, if you add one to 3x3! + 2×2! + 1×1! + 0×0!, you get 4!. This one is only a bit silly, since it has uses in combinatorics.
In the factorial number system the number of symbols grows for each digit. An n digit number in base n uses fewer digits than the factorial number system used for the same value,but a 2 n digit number in base n has more digits than the same value in the factorial number system.
The factoradic base is a mixed-radix number system, so the number of "digits" grows without bound as digits are added to the left. This is why factoradic "digits" are separated by colons, e.g., 3:4:1:0:1:0 represents 3×5! + 4×4! + 1×3! + 0×2! + 1×1! + 0×0!. Digit separators could be omitted when working with factoradic numbers less than 10 digits (as long as the number is noted with a subscript "!" to clarify it's a factoradic). I suppose you could even use the hex digits up to "f" in the same way, but the colons would have to come back if the length of the numbers grew beyond that.
Base 12 is the best base in my opinion, you can count it on just one hand (by counting the segments of your fingers, separated by bone hinges) as well as having 6 divisors, instead of base-10's 4, plus, it wouldn't be all that hard to adjust, plus, making symbols would be easy. we could have something like a backwards 3 for eleven, and an upside down 7 for 12, we even already have separate names for the numbers (coming from a rudimentary base-12 system in proto-germanic)
@@jonasgajdosikas1125That’s not really true. You’re getting it mixed up with counters, which are more like units, or like “head” in “head of cattle”. With only a few exceptions, everything gets counted with the same sino-Japanese numerals
i also think it's strange to call it the Japanese number system when it's really the Chinese number system. that would be like calling the roman numerals the Hollywood system because they are used for the release dates of films in the credits
Gray Code numbering and Zeckendorf numbers are also nice. There are also redundant numbering systems where you use more digits than necessary. Eg you could use 0, 1, 2 for binary numbers. Or you can use digits one to ten for a decimal system, but without a zero. That also works. The same thing to the power of 26 and using letters for the digits is how Excel spreadsheet coordinates work. There's also Skew binary numners. Have a look at Okasaki's book Purely Functional Datastructure for fun with weird numbering systems.
Imaginary/complex bases are really cool, because you can display any complex number as one number instead of 2. You dont even need a negative sign for negatives!
You forgot about balanced bases, where half of your symbols are negative. An example of this is balanced ternary, which is base 3, but the digits are -1,0, and 1
@@tim.martin In a balanced base, every digit can be either positive or negative. In a negative base, every other digit flips between being positive and negative. They are similar, but not identical.
"I prefer base 10", says a computer using binary See, if you call it "base 10" (with the numerals) it only means base ten if... you're already using base ten
True, true. You could say Base Dec though, using the term from Dozenal. And you could write it "Base A" or something similar, however I think most people write "Base 10" out of familiarity more than anything else. If you've never tinkered with number bases, it's not something you've ever thought about.
Hereditary base-n notation (as used in Goodstein's theorem) is really interesting too. You rewrite your base in terms of your base, and it has relations to goodstein's hydra and ordinal number systems.
I LOVE THIS SO MUCH! Next, could you please talk about Base -1 using digits 1 and 0, it's like unary (base 1) but it can also represent negative numbers and it's really interesting, also some algebra becomes more intuitive in Base -1 which i fine fascinating. Also also, you talked about Base 3/2 using the digits 0, 1, and 2, but you failed to mention Base 3/2 using digits 0, H, and 1, where H=1/2. I know it sounds strange and unorthodox, and it leads to this version of Base 3/2 having it's own set of "1.5-integers", but it's actually been written on before in a study (Journal of Integer Sequences, Vol. 23 (2020), Article 20.2.7, Variants of Base 3 Over 2) and it's the system I study, although there are some interesting interactions between the two Base 3/2 systems that you should also check out. I love numeral bases, this is the most well put together video on them on RUclips, and I hope you do more! Thank you for this video!
Thanks very much! I love sharing interesting stuff with people all over the internet. I can't promise another base video but I'll definitely consider it. What I *can* promise is some more weird maths. Also, I'll definitely have to check out that article, sounds like a fun time!
@@RandomAndgit Ahh, I absolutely love nerding out over maths with someone else! Even if you don't make another bases video (although i really really hope you do) I'll be waiting for whatever you have next. Also, it would be super fun if you talked about infinity in the context of hyperreal numbers (specifically the ones where infinity can be represented as 1 followed by infinitely many zeroes, "epsilon" represents an infinitesimal which is related not through division, but actually through a more complex exponential relationship, something like baseless 10 to the power of 1/infinity is epsilon and vice versa, and also infinity is an exact amount, more than infinity-1 and less than infinity+1, which means that a repeating segment of digits can have 1 or more extra digits removed or following it on either side.) If you couldn't already tell, this specific blend of hyperreals is actually my main field of study, and I've been wanting to put together a paper or two on them for ages but all my research is currently tied up in some big legal argument over this other house... it's a whole can of worms for another day. Also, imo limits and their undefined values are corrupting the rest of mathematics. I don't have a problem with the direct study of limits, but I hate when people say stuff like "no, any answer is wrong, it's undefined and should stay like that" it's stifling curiosity and creativity. Like how, no matter how many times it's proved, people can't accept that 0/0=1, 0^0=1, and 0!=1, because the limits of said equations are undefined and give multiple answers. Unless there's limits in the problem, don't assume that the answer will be given by limits. People have said the dumbest things, like infinity lies between infinity and 0, or that infinity+1=infinity. These things break so many mathematical rules for literally no reason, and hyperreals show is that there is no reason to proclaim infinity as NaN and start making up rules purely to confirm that. Hope you have a good day!
@@autumn948 Your passion for maths is palpable and evidently very well informed, have a good day yourself! (Oh, also, I hope that legal issue works out, that sounds very annoying)
@@RandomAndgit Thank you so much! 😊 And yes, I hope to get not just my (ENTIRELY FILLED 😭) notebooks back from that house, but also the rest of my possessions. It's a very sticky situation, but I hope to resolve it soon. Can't wait to see your next video!
Any polynomial can be understood as a number in a variable base x. Then, given a base x, the y-value returns the represented number. Or, given a number y and a polynomial representation, the roots of the polynomial are all the bases for which that expansion represents that number.
thank you so much for this video, i didn't understand a lot of this stuff just by reading wikipedia so a random youtuber making a banger video makes all the difference
Base 3/2 (kind of base 1.5) counts like this: 0, 1, 2, 3=20, 21, 22, 40=210, 211, 212, 213=230=430=2130=2300=4000=2100, ... And so every integer can be represented uniquely base 3/2 using the symbols 0,1,2. In general, if n and m are coprime numbers, such that n>m, you can have an integer system with base n/m using n symbols 0, 1, 2, ... (n-1). All you do is count as normal, but you swap n elements in one column for m in the next higher. In base 10 (10/1) we swap ten digits in one column for an extra 1 in the next, if this overflows we repeat (99+1 = A0=100, because ten, A, overflows). So we base 3/2, just take 3 digirs off one column and add 2 to the next. Keep going until a number is represented with just 0,1 or ,2 digits only
Would you consider uploading a video explaining this (even if it's just you writing on a piece of paper while explaining it)? EDIT: I just found one on RUclips. Also, and I may be completely off the mark, I can't help but wonder whether or not fractional bases could be useful in cryptographic applications since I gather that carrying over to the next column is different from whole-number bases where only one can be carried over to the next column at a time.
There's a version of base 1.5 (base 3/2) where every integer still has a terminating decimal representation! But it uses the digits 0 1 2, just like base 3. It goes like this: the last digit is the remainder when the number is divided by 3, but the digits before that are the base 3/2 representation of *twice* the integer part of the number divided by 3. So it goes: 0, 1, 2, 20, 21, 22, 210, 211, 212, 2100... If you work it out, you'll see that those are actually base 3/2 representations of the integers from 0 to 9. You can use any base p/q for positive integers p>q in this way. The fraction doesn't even have to be in simplest form.
I'm pretty sure phi isn't special when it comes to irrational bases. I have this gut feeling that any root expression for a base can express integers in finitely many digits Like. If I set the base R = sqrt(2) + 1 then I have the relation R^2 = 2R + 1, which manifests in the base as 100 = 021 So a number like 3 would be written as 10.11, and 12 as 200.02 Not too sure since I haven't tested anything from nested roots or cubics or anything higher degree but I feel like it should hold UPDATE: GOT SOME OF THE MATHS WRONG, OOPS... For base Q = sqrt(6) - 1 you instead have 7Q^(-1) = Q + 2, or 007 = 120. So you probably need to work in seven digits. 0-6 is still 0-6 but 7 = 120 now. 14 = 240, 21 = 360, 28 = 1610, 35 = 13030 Funny enough, you need ten symbols for base S = sqrt(8) + 1 because S^2 = 2S + 9 (100 = 029) So 0-9 is still 0-9 but 10 (actual Ten) would now be written as 9.29 and 11 would be... uhhh... huh. no clue, actually
0^0 is indeterminate not undefined... this is because the limits of x^0 and 0^x as x -> 0 are different ! I think something interesting to include when discussing bases are things that are similar to bases but not the same. Those things in specific are Cantor normal form, Conway normal form, and p-adic numbers. Cantor normal form is essentially a way of representing numbers that are in some sense infinite. Here's a brief outline of these three, leaving out quite a fair amount of detail: First, ordinal numbers are a class of numbers where one of ab, a=b is true for all ordinals, and each ordinal has a successor. With ordinal numbers you are allowed to construct what are called limit ordinals. The smallest limit ordinal, ω, is the limit of {0,1,2,3, ... } and is thus larger than all finite numbers. (I'm being a little bit loose with my definitions here to simplify a little bit, but the limit of a set is the union of smaller sets. The empty set, {} is the number zero, the set with the number zero, {0}, is the number one, the set with 0 and 1, {0,1} is the number two, etc.) Cantor normal form is a particular way of writing ordinals (that can in fact write all ordinals) which is a1ω^b1+a2ω^b2+...+akω^bk with a_1,a_2, ..., a_k being natural numbers, b_1>b_2>b_3>...>b_k being ordinal numbers, bk being at least 0. There's another theorem that states that any strictly decreasing chain of ordinals is finite (although it can be arbitrarily large). So, there are only finitely many b_i's. Conway normal form applies to a subject called combinatorial game theory and is slightly harder to explain because of the nature of CGT. However, it constructs something which may initially seem analogous to ordinal numbers called the surreal numbers, but has a nicer seeming arithmetic (you can divide by the surreal number ω, but not by the ordinal number ω). A very rough intuitive way of describing Conway normal form is as a continuous more "arithmetic" (it's commutative, while ordinals are not) analog of Cantor normal form. (If you want a more in-depth definition of surreal numbers, they are essentially games of the form {A|B} where A is the choices for a player (who is called Left) and B is the choices for the other player, where the sets A and B consist of only numbers and for all elements of A, there is no element of B which is less than or equal to it. the game { | }, where neither player has a choice is called 0, and {0 | } is 1 and { | 0} is -1. If you want more info I recommend reading about it, it's a very interesting subject.) p-adic numbers are built on top of analytical tools of calculus. First we need to be able to talk about the distance between two points, which we will call d(a,b). This distance is a nonnegative real number, and we want it to have a few properties. Namely, d(a,a) = 0, d(x,y)=d(y,x), d(x,y)>0 if x and y are distinct, and d(x,z) is at most d(x,y) + d(y,z). If all these properties hold, we call d a metric. The importance of a metric is that it allows the construction of a limit by saying "these values eventually get arbitrarily close" pretty much. This is what it means to say 0.999... = 1, is that any sequence of 0.9,0.99,0.999,... can get arbitrarily close to 1, and is thus the same as 1. We can construct something called the p-adic metric, for any prime number p. this is how it works: |0|_p = 0, | p^n |_p = p^-n, | p^n * (r/s) |_p = | p^n |_p if r and s are coprime to p (n is an integer). With this metric, setting p to be 5 for example, 1+5+5^2+5^3+... converges to a particular point, which can be written as ...1111 in the 5-adic numbers. this is because the distance between each consecutive 1, 11, 111, is decreasing in regards to this metric. This number system seems very opposite our standard intuition in that there can be infinitely many digits to the left of the decimal (radix) point, but only finitely many to the right.
Re base 16: I think it's cool to come up with unique glyphs for 11-15 instead of using letters. There are a couple ideas to play with: 1) make them stylized versions of the letters A-F so they can be read even if you don't know them already. 2) make new digits that look like the existing digits, design-wise.
obscure numbering system: tic-xenotation (TX): - 2 is written as : - (n) is the nth prime number (1-indexed, starting from 2) - every number is written as its factors 2 : 3 (:) (the 2nd prime) 4 :: 5 ((:)) (3rd prime) 6 :(:) 7 (::) (4th prime) 8 ::: 9 (:)(:) 10 :((:)) it's possible to write any integer greater than 2 in this fashion. very impractical since the system is based on factors (even basic addition is impossible without rewriting into a modulus-based system).
I came up with the exact same thing, except instead of using : to denote 2 you just have a blank number represent 1. And if you substitute 1 and 0 for ( and ) you get a very, very cursed form of binary
This video is very entertaining for a math video, i have a critic however. The volume of the music is louder than the voice, maybe you can lower the volume of the music. Other than that it is very good and underrated.
Base 12 is quite interesting: the numbers would be devisible by 2, 4 and 3 (instead of 5 when considering base 10). Which makes it very useful in household use - shame we only have 10 fingers.
The best number systems to use is base (A*2+B*3+C*5+D*7) Where A,B,C, and D are whole numbers >=0 A >= 1 And A >= B >= C >= D Base 2, 6, 8, 12, 16, 64, 60 all meet this requirement. Example, base 12 A=2 and B = 1, C and D = 0. Base 16, A = 4, B,C,D = 0 Base 10 doesn't work though. A = 1, B = 0, C =1, D = 0. Basically the number should be composed entirely of small prime numbers, smaller the better.
There are other options, such as reflected bases--like Gray Code (en.wikipedia.org/wiki/Gray_code) in which adjacent numbers always differ by only one digit change, or, particularly with odd bases, balanced bases--like balanced ternary (en.wikipedia.org/wiki/Balanced_ternary), which needs no separate positive or negative sign, since the digits chose are plus (one), zero and minus (one)--so the largest non-zero digit sets the sign of the number. And it is easy to round numbers: just truncate them when the proper accuracy is reached. Then there is one of my favorite bases--balance reflected ternary. Not very useful at all, but fun to count in. Oh, and can we use continuing fractions for approximations to rational numbers--it's like at each "decimal" we use the optimal base, so we get very good approximations as quickly as possible.
Hey guys (said gender neutrally of course). Here are a couple very minor corrections:
-Quite a major mistake on my part: when doing the calculations for base -i for 5:47, I forgot to place parenthesis around -i, which lead the results to being equal to -1(i^x) rather than "-i, -1, i, 1..." like it should have been. That's completely my mistake.
-I used the wrong spelling of whether at 6:03
-I said at 4:53 that phi was the number that satisfied the equation sqrt(x) = x - 1when the real answer is (sqrt(5) + 3)/2. The correct form of that equation would be that 1/phi = phi - 1. It's a very minor mistake but still important to point out.
-At 2:42 I say that base 18,446,744,073,709,551,616 was for a 64 by 64 grid but that base only satisfies all possible combinations of an 8 by 8 grid. A real 64 by 64 grid base would be closer to 1.322112e+123.
-The Japanese "100" symbol at 3:07 is slightly malformed, it needs an extra line at the bottom.
-I use the term irrational at 4:30 which may have confused some people. The term irrational technically only refers to weather a number can be expressed as a fraction but it is a very useful way of describing a number with infinite, non-repeating digits.
-I accidentally refer to what is called a multiplicative system as a bijective system instead.
-I, in my foolishness, have made the music too loud. D:
I promise there aren't always this many mistakes! Sorry!
Another minor correction: base -𝒾 does not cycle in the same direction as base 𝒾, but in the opposite direction: (-𝒾)² = -1, (-𝒾)³ = 𝒾, (-𝒾)⁴ = 1, and so on.
If I may do some armchair inference, I believe you might have forgotten your parentheses when calculating base -𝒾, and instead calculated the negative of base 𝒾, which indeed cycles in the same direction.
@@TheBasikShow You are precisely right, yes, thank you. That really is quite a comically foolish oversight of mine. Thanks very much.
Not trying to dogpile you further but:
• What you describe as a bijective number system is actually a multiplicative system.
• The Japanese/Chinese character for 100 is 百; you missed a line on the bottom.
• Honestly I really like the music! I wouldn’t lower it too much
I still really like the video though; I’m subscribing for sure. Plus I love your artstyle; reminds me of Typoman.
Why not order the things in this comment so that when watching the video the viewer can easily see when the next correction is? (Though any major mistake can stay at the top or something)
3:43 did you mean integer???
i heard integral hah
In base infinity, every number has its own unique symbol! Sort of like writing numbers with words...
chinese base
@asheep7797 uh no, Chinese is actually base 10 too
@@randy-x They probably meant using Chinese characters as numerals, not just the Chinese numerals. That being said, that's still not base-∞, "only" base-3500 or base-6500.
@@adiaphoros6842I think it would be closer to base-50000
Well, not really like words? Numbers are already written like words. It's just that they're very logical and predictable in how they're spelled
- Let's try to write something in nullary
- So we have zero symbols, so we can't write anything
- Rejects to elaborate further
- Leaves
*Revelation 3:20*
Behold, I stand at the door, and knock: if any man hear my voice, and open the door, I will come in to him, and will sup with him, and he with me.
HEY THERE 🤗 JESUS IS CALLING YOU TODAY. Turn away from your sins, confess, forsake them and live the victorious life. God bless.
Revelation 22:12-14
And, behold, I come quickly; and my reward is with me, to give every man according as his work shall be.
I am Alpha and Omega, the beginning and the end, the first and the last.
Blessed are they that do his commandments, that they may have right to the tree of life, and may enter in through the gates into the city.
@@JesusPlsSaveMe *Numbers 25:5-13*
So Moses said to Israel’s judges, “Each of you must put to death those of your people who have yoked themselves to the Baal of Peor.”
Then an Israelite man brought into the camp a Midianite woman right before the eyes of Moses and the whole assembly of Israel while they were weeping at the entrance to the tent of meeting. When Phinehas son of Eleazar, the son of Aaron, the priest, saw this, he left the assembly, took a spear in his hand and followed the Israelite into the tent. He drove the spear into both of them, right through the Israelite man and into the woman’s stomach. Then the plague against the Israelites was stopped; but those who died in the plague numbered 24,000.
The Lord said to Moses, “Phinehas son of Eleazar, the son of Aaron, the priest, has turned my anger away from the Israelites. Since he was as zealous for my honor among them as I am, I did not put an end to them in my zeal. Therefore tell him I am making my covenant of peace with him. He and his descendants will have a covenant of a lasting priesthood, because he was zealous for the honor of his God and made atonement for the Israelites.”
@@JesusPlsSaveMe
To Claim Their Bones
SkillAttack.png 55 (50+5) Atk Weight ⯀
Amt. x0
[Before Attack] For every 7 Poise.pngPoise on self, Atk Weight +1 (Max +2)
[On Use] Until this Skill ends, this unit cannot be Staggered from taking damage
[On Use] Deal +2% damage on Critical Hit for every Poise.pngPoise Potency on self (Max 50%)
+30% Damage on Critical Hit
Deal +0.5% more damage for every 1% missing HP on self (Max 25%)
If this Skill targets only a single enemy, deal +100% more damage (In Focused Encounters, a single Part)
CoinEffect4.png [On Hit] Inflict 3 Bleed.pngBleed
[On Hit] Inflict 5 Paralyze.pngParalyze next turn
@@JesusPlsSaveMe aight
Depending on what you think 0^0 is, you can display the number n by writing n times the null symbol, from which there are zero in existence. If you want it more accessible, write | instead, e.g. 4 = ||||, and | has the value 1/0... 🤣
i cant believe when we learned expanded form to express numbers in 3rd grade we were peeking into the abyss that is the mathematical bases
That's the good part of mathematics, everything is derivable from everything; and what isn't, you can construct it.
100th like was from me:)
Prof. Arnold Ross: "Think deeply of simple things."
ETA: his contention was that in Math, even deep results are near the "surface"--and new discoveries are always possible, even in areas that have been developed for a long time.
We, his students, often managed to think simply of deep things. . . .
The same system invented for counting sheep, ordinary aritthmetic, winds up in Godel's theorem. More to sheep than meets the eye? Baah humbug!
Math and etymology
Nice
For anyone curious, the background music is the third movement from Beethoven's 14th sonata
Don’t you mean Beethoven’s 0001110th?
@@phoebebaker1575 No, I mean Beethoven's Eth
"Moonlight" for those that prefer base ∞.
@@urkerabMoonlight Sonata 3rd Movement.
K
Base 12 is my favorite base. It isn't inconveniently large, and is divisible by 3 and 4. We used to use it on a large scale as well before the Arabic numerals became the standard, that's where the dozen comes from, and is why eleven and twelve have unique names instead of being lumped in with the 'teens.
Base 12 probably wasn't used on a large scale before Arabic numerals. There are a few words that seem to refer specifically to twenty (like "twenty" itself, "dozen", and "gross") but you could make the same argument for base-20 ("score" and French "vingt"). And when you look at the etymologies of most of those, they clearly derive from base 10: "eleven" and "twelve" derive from Proto-Germanic compounds meaning "one left" and "two left" (as in, after counting to ten), and "dozen" is from Latin "duodecim" meaning" two and ten".
The exception is gross meaning 144, which derives from a word meaning "large, coarse, monstrous" and looks to be a case of a word for "a large amount" being repurposed to refer to a specific large amount, similar to how "myriad" can mean either "uncountably many" or "ten thousand".
There's some speculation that the Germanic "long hundred" (referring to 120) implies an early use of base-12, but I should point out that 120 isn't really a special number in base-12. It's not a place-value the way hundred is in decimal.
@@gwalla Your nerdery is appreciated.
As someone who clearly knows more on this topic than I do, do you know how much truth there is behind the idea of people counting in base 12 by using the thumb to count the three joints on your four fingers of each hand?
@@bow-tiedengineer4453 I mean, you can do that, and it works. As for whether people did it historically, I have no idea. I know in Korea they sometimes use a technique called chisanbop to count up to 100 on two hands, by treating the thumbs kind of like the short rods of an abacus. That was a 20th century invention though. If you hold your hands over a surface you can even count up to 1023 using binary, by touching the surface with different fingers (curling the fingers isn't practical for this).
Absolutely!
USA should convert to the international system and while at it switch to base 12 😁
@@bow-tiedengineer4453 also easier to count on your fingers. 3 bones per finger x 4... Use your 👍 to touch each part of their other fingers to count to 12.
"NOOOO WE NEED TO USE BASE 6 ITS THE FUTURE" - Some mathematician, probably
And then another mathematician says that we should use base 12 instead, starting world war III.
and then some more say base 2
jan misali:
That probable mathematician is jan Misali. See their video "a better way to count".
base 37 🗿
I once read this SCP where an AI was created to devise better compression algorithms for the Foundation's archives, only to remove itself from existence. While the documents in the affected archives had practically disappeared, it was somehow still possible to access them. It turns out the AI figured out how to use nullary: this base can only be understood from a certain frame of reference (Q) which is incompatible with that of normal human thought (K). And a researcher trying to find where the data went ends up becoming nonexistent as well, so yeah we better not let number exist in nonexistence
Ah yea, 6276. They really give Surrealistics a load of wacky anomalies to contain, huh?
"The above analysis has been rated helpful by 22% of personnel"
@@umber6937 Not a Surrealistics Dept. skip, but have you sacp slon obr 7978? If you haven’t, it’s this bizarre, mind-warping online cartoon that slon “cult” tlaol tlei plr “cult following”
it's giving Nan propagation
To throw another work of fiction into the mix, New Phyrexia (MtG) uses hexadecimal as the base for writing numbers. The written system uses a 2x2 grid, rotating the position of a symbol on the grid for each increment and a unique symbol for 1-4, 5-8, 9-12, and 13-16, respectively.
Reminds me of several stories from _Fantasia Mathematica_ (1958):
"The No-Sided Professor" by Martin Gardner
"The Island of Five Colors" also by Martin Gardner (well, sort of)
and "A Subway Named Moebius" by A. J. Deutsch.
4:34 - Common misconception. Rational numbers (including integers) are still rational regardless of whether they have an infinitely long, seemingly random representation. The failing is in the base, not the number. You could say "they appear irrational to people who are familiar with how irrational numbers look in more commonly used bases", but that's about all. They're still rational no matter how they look.
Rational means "can be written as the ratio of two integers". This says nothing about how the number or the integers in its ratio happen to look when that writing is done.
A lot of ink has been used to explain "intuitive" concepts in rigorous mathematical detail, precisely because of how often our intuition is wrong in fringe cases. I remember going through my master's level number theory course where they rigorously defined what natural numbers are (using set theory; it's crazy how complicated you have to be with it to fit how they're actually used in the real world while also satisfying everything they need to be mathematically). It's important to remember that literally everything in math is a construct, so in another timeline we could have defined integers (and therefore rational numbers) differently such that the statement would have been correct. But yeah, as things stand the base we choose to represent them in does not change whether a number is rational or not.
Yeah, infinite digits to the right of the decimal point isn’t the same as irrational, otherwise 1/3 would be irrational in base 10
Just try writing this as a decimal : 14165841/4216516841
It have a period, but it's a very long one.
i get your point, but if it's an integer multiplied by an irrational, then these numbers would have to be irrational, right?
@@justin-ju4eo Right. Any integer or rational multiplied by an irrational give an irrational.
BUT. It is possible to find pairs of irrationals that give a rational product when multiplied. Then, there are irrationals that can give a rational result when elevated to an irrational power. Just take the natural logarithms of any integer. They are all irrationals. The base is e, and e is an irrational.
- Do you know that you can use different radix for every position?
- What a silly idea! Nobody would use such a system!
(this comment was written at second 2024 10 11 00 19 41 given place value [31558149.8, 525969.163, 86400, 3600, 60, 1], sidereal year and some timezone)
_(please don't remind me that every place value in this system is actually a function of N-body problem in distorted spacetime, not a constant 🥲)_
@@lockaltube The only thing that remains constant, is the modernist interpretation of the idea of constancy, hypothetically existing outside of personal interpretation; I'll be honest, I have no idea what I'm talking about, but you've read this far ☺Edited: to show I edit therefore am not bot, unless...what if I was a robot and didn't know it?
@@martian8987is could feasibly have been a quote from the Stanley parable
I like your funny words magic man
Those are ordinal units as opposed to cardinal units, except for everything smaller than a day.
Fun fact: although the day can be treated both as an ordinal unit and a cardinal unit, the *_nychthemeron_* is purely cardinal.
Why use base Infinite when you could just use base -1/12?
because it is easier to use
I love obscure math references!
@@genericgoat you have not seen obscure yet
This ia the tip of the iceberg really. The more obscure the funnier usually :D
😂😂😂
@@genericgoatMay this lowly uncultured peasant conceive the reference?
Not even mention of base64 and base85 which are both actually used in computer science... (64 for obvious reasons and 85 is actually just the number of printable ASCII characters, so it uses all of them as symbols for the base)
Hexadecimal (base 16) is much more common, at least by humans who work with computers, than base 64. The main reason for its utility is that each "digit" of Hex can be directly translated into 4 "digits" of binary because 16 is equal to 2^4. Thus you can easily convert between them as needed, taking advantage of using fewer digits when writing long numbers, but being able to break down individual binary bits when needed.
@@edwardblair4096 i didn't say they were common i said they are used... Main usage of base 85 is when you want to print out binary data so that it can be then scanned back from paper... Base 64 can also be used for that but also used for hashes and other random string generations... For example RUclips video codes are base64... main advantage of base 64 it uses 4 times fewer characters than base16, which is important if it is actually used as string
base 64 does that too
@@Fire_Axus Yes in my reply I did say that
Calling ASCII bade 85 is like saying that english is base 26 because there are 26 letters. You’e not _wrong_ but like…
We all use base 0 far more than we do base 10; whenever you're not writing numbers, you're using base 0
a
"You think you're doing some damage?
2 plus 2 is-... 10
IN BASE 4, I'M FINE"
- GLaDOS
…
2+2 base 3 is 11
:3
@@applememesboom5057 indeed :3
@@applememesboom5057 2 + 2 in base five is..... 4
I wrote a “big number” program once, which used base (2**64) (meaning every “digit” was a 64-bit digit).
So, instead of 0-9, it was
0-18,446,744,073,709,551,615.
One of the fun things (?) was figuring out a system for overflow-proof multiplication, while still remaining as performant as possible. You could get some ludicrously big numbers with that thing.
For example, it’s much like how the “2” in the tens-position of the decimal number 23 represents (2 * (10 ** 1)), giving us 20, to which the 3 in the ones-position is then added for the final result of 23.
But, in this case, [2, 3] in base (2 ** 64) - and I’m representing the digits in an array, to avoid confusion - means we have:
• (2 * ((2 ** 64) ** 1)
= 36,893,488,147,419,103,232
plus
• (3 * ((2 ** 64) ** 0)
= 3
for a total of:
36,893,488,147,419,103,235
In decimal, if we have [1, 0, 0], we get 100. In base (2 ** 64), [1, 0, 0] is instead:
• (1 * ((2 ** 64) ** 2))
= 340,282,366,920,938,463,463,374,607,431,768,211,456
So, you can see how these numbers quite readily get very out of hand!
If you had a 1 in the 64th digit position base 10, and zeroes in the rest, that would be pretty crazy! It would be a 1 followed by 63 0s!:
1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000
That’s a lot of number!
But, if you did the same thing with base (2 ** 64), you get something a little more insane:
56,616,434,707,392,290,493,830,608,686,343,291,913,914,207,267,329,621,488,609,504,077,443,092,998,260,469,825,507,336,856,810,827,911,261,368,707,758,178,809,587,148,432,632,037,424,161,064,675,500,873,383,487,082,580,173,608,658,838,616,501,128,049,812,714,217,651,328,283,024,830,073,412,384,889,815,432,597,905,927,601,958,951,314,543,937,080,343,085,064,110,993,312,334,723,049,547,437,731,770,779,823,050,760,639,880,425,500,165,204,989,981,997,213,399,871,161,087,661,501,312,220,225,137,504,516,172,823,917,738,211,300,299,807,381,757,818,984,800,273,856,795,211,620,996,748,504,363,142,690,436,276,062,613,301,784,444,812,913,670,500,602,718,367,796,525,145,427,307,300,283,865,951,803,998,808,065,639,029,170,703,022,449,468,190,319,930,507,340,988,677,523,066,084,721,639,425,495,152,552,125,131,611,082,991,384,744,696,522,708,197,306,203,186,569,928,692,274,528,086,558,201,803,207,638,020,888,691,121,265,434,197,293,907,218,384,095,520,560,555,896,947,150,944,133,055,863,397,615,648,411,112,470,754,656,110,748,811,237,653,374,788,477,716,072,552,588,609,616,376,430,894,096,481,364,501,294,655,654,823,142,548,823,259,752,215,716,293,351,999,419,717,562,306,177,064,929,632,050,689,015,014,405,578,694,408,614,353,540,180,649,410,108,905,523,548,191,355,424,055,272,181,142,840,177,487,296,189,764,022,300,385,934,884,483,982,341,901,658,184,426,332,624,776,972,085,905,793,515,459,600,620,068,183,625,181,327,963,636,494,799,582,466,830,531,258,661,325,767,796,742,142,353,530,480,288,174,823,736,291,072,477,683,720,590,971,289,642,787,198,230,081,364,301,868,392,910,675,871,538,608,754,231,607,296
That’s a number 1,214 digits long when represented in base 10!
Awe!
Interesting!
The only issue with this is that you've made everything take 8 times as long
I saw a few papers on base '2i' and it has some interesting properties. Numbers written in base 2i may look like 'base 4' but their value can be any complex number (positive or negative) and you don't need to use any negative signs. When adding and subtracting such numbers you have to skip a digit when carrying and borrowing and those operations are reversed. For addition you 'borrow' and for subtraction you 'carry'.
My favorite property is that if you divide numbers in base 2i then you can divide complex numbers without taking a complex conjugate. Dividing base 2i numbers is tricky so unless you really hate complex conjugates I don't see many reasons to use it.
Hard to beat p-adic as a crazy base while still being useful.
Quite so.
p-adic is more of an interpretation of an existing base rather than a distinct set of bases. You can write Real decimal numbers like 4561.235..., or you can write p-adic decimal numbers like ...4561.235. Unlike the Reals though, each base actually gives a different number system, so while the 3-adics are just p-adics in ternary, it also has different numbers than the 5-adics, which are p-adics in quinary. Doing math in composite (non prime-power) bases with p-adics though doesn't completely work, hence the "p" in the name standing for "prime". This is different from the Real numbers, where composite bases, and especially superior highly composite number bases, tend to work better. (Binary _cheats_ since 2 is paradoxically both a prime number _and simultaneously_ a superior highly composite number, despite... not being composite. It does however have some weird and unique problems when used for p-adics that pop up in specific situations.)
💀💀💀💀
I doesn't really matter, from their point of view, all bases are base-10.
7:06 I love the fact that O and Z are technically saying the same thing
lmao average 0 factorial
What’s more, they’re both correct!
r/unexpectedfactorial moment
so you're telling me, that upside-down-u is saying, that 0^0 = undefined factorial?
It's 0.5 end of story
The Babylonians used to have base 60. They counted each third of the finger excluding thumb and when finishing a 12, they would put a finger up until they have all 5 fingers. 5•12=60
Reject base 10. Embrace hexadecimal
hexadecimal cooler 👍
HEX
Nah, embrace Suboptimal (base 17 i think)
Don't forget bintetrabaker's dozenal! (Base 106)
base 24 is also good
There is also a number system based on Fibonacci numbers, it is quite exotic and rare, but no less interesting
Ooh, do elaborate! That sounds very interesting indeed.
oof, ghosted… that’s the worst.
Commenting to get notified if someone answers
There's a Wikipedia article on "Fibonacci coding" that looks relevant, but I don't know if that's the same thing or not.
@@Faroshkassame
Funny enough, the many symbols problem with larger bases is something people have actually decided to solve: The Argam system bases its symbols for all non-Primes after 9 on their factorization, modifying previous symbols to make new ones (so their symbol for ten, or "dess" is basically an upside down 2, which looks kinda like a 2 and kinda like a 5).
You missed part where negative base can represent all integers(positive and negative). A base with some negative digits can also do that. Like balanced ternary.
bro got the blue comment
TTT, TT0, TT1, T0T, T00, T01, T1T, T10, T11, TT, T0, T1, T, 0, 1, 1T, 10, 11, 1TT, 1T0, 1T1, 10T, 100, 101, 11T, 110, 111
(-13 to 13 in balanced ternary)
If you use something like base -10, you don’t need the negative sign for negative numbers which means you could use a minus sign in front of the positive number. So there are two ways to express a number in this base.
@@TheFrewah No, base -10 doesn't even need the negative sign since a negative number has a even number of digits and a positive number has an odd number of digits. "-10" would just be written as "10", and "10" would be "190" since it's 1(100) + 9(-10) in negadecimal.
@@yeetrepublic9142 Some time since i wrotea program that did the conversion. So if 190 is ten in base 10 then -190 would be -10 in base -10. You could also, more naturally, express it some other way where you would not need the negative sign.
In college, one of my classes gave an assignment to invent a number system other than the commonly used systems. Then, we had to create an addition and subtraction problem using that system and solve. Finally, convert a number in our system to decimal. Most of my classmates went with something small like base3. I did base64 using 0-9, uppercase letters, lowercase letters, and a couple of made-up symbols. The caculations covered an entire page of paper. My professor took one look at it and just said, "I believe you," and gave me an A, lol. Didn't even try to verify it.
That sounds like a very cool assignment. Base 64 is a good number to choose as a base too. Nice one!
I'd be tempted to use a random prime. 17, of course.
Base 16 (Hexadecimal) which uses characters 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e and f:
• Neither too little nor too much characters.
• Unlike Decimal, doesn't create confusion whether you're referring to the base-10 number system or a number between two integers.
• With only four digits, it can represent values as high as (decimal) 65535.
• Compatible with 7-segment displays (since letters a-f can be represented using them)
• And not to mention its use in Computer Science, in which it has even more advantages in that area, such as:
-- • 1 base-16 digit is 4 base-2 digits
-- • Can represent any color a screen can process with only 6 digits (8 if counting alpha channel) while Decimal requires 9 (with a lot of leftover numbers).
-- • Can represent memory addresses (generally powers of two in amount) and machine code more easily.
-- • Can represent one byte in two digits (with no left-over values)
In short, Hexadecimal is good.
I made a very good living for 50 years knowing just how good Hexadecimal was. I doubt many days went by that I did not use it.
I’m a big lover of hexadecimal as well. A very useful numbering system that saves memory and can be trimmed to represent decimal.
@@ShaunieDale Me too. I made an oversized 7 segment clock that displays unix time in hexadecimal. It's completely useless.
No, your perspectives are only computer science related and very myopic
@ we talk about things that are relevant to us, what’s wrong with that?
Binary can be useful for human use in at least one case: counting on your fingers.
Think about it, your fingers can either be up or down, representing a 1 or a 0. Counting in binary on your fingers allows you to count up to 1,023. You probably won't need that much, but on just one hand you have up to 31, which is extremely helpful in some cases if you need to do something and count at the same time.
Finger ternary works too -- you just need three positions, such as up, down, and curled for 0,1,2 or T,0,1, etc. Which gets you LOTS more capacity. 243 on each hand -- or +/- 121 with balanced ternary.
Nice, just what I came to add. Learned about binary when I was first learning about computers back in the early 80s.
👍
This reminds me of 'balanced ternary' which is basically just base 3, but not with symbols {0, 1, 2} but {0, 1, -1}. The -1 is usually written with its own symbol so as to not use two glyphs at once, lets use "T" here.
Then a 5, for example, would be written as 1TT in balanced ternary since 9+(-3)+(-1) = 5.
This was apparently used in some early computers as a sort of extension to the thinking of binary:
Binary makes sense for computers because you can have the electricity [not flowing] (0), or [flowing] (1)
Balanced Ternary extented this thought to [not flowing] (0), [flowing one way] (1), and [flowing the other way] (-1)
I think this is really cool, but it apparently/evidently never caught on as much as binary did
There are at least 10 types of people in the world: Those that know binary, and those that don't. (And those who know this joke works best in balanced ternary.)
Balanced ternary is my favorite base system.
@@nikitanugent7165 I actually also like balanced reflected ternary . . .
You could also use 0, 1, 2 for balanced ternary and still use negative voltage. The association of 0 with off and 1 with on is also kindof a thing we people made up 😅 (I personally used 0,1,2 because i think the notation is easier, but I did stuff with balanced ternary logic circuits :3)
@@nikitanugent7165 There are three kinds of mathematicians--those who can count, and those who cannot.
There are two things that easily break maths.
Infinity.
Zero.
And the last one is the king at doing it.
Fun fact, negative integer bases still allow you to represent every number.
Also fun fact, the symbols don't have to be 0-(n-1). I've played with base 10 but where the symbols are -5 through 4, and you can still hit every real number that way, including negatives.
That's damn cool, dude. Maths needs more people like you, who are willing to experiment.
Speaking of which, can I ask what your answer is to the question "What number lies between 0 and infinity?"?
Base e is supposed to be the most optimal of all, which is why during the 60's I think the Soviets tried making a base 3 computer since e is closed to 3 as a whole number integer. Unfortunately it failed because making fractional electronic counters is extremely difficult
4:10 “Decimals work in all base systems as well” is a funny sentence
As a side-note: prime exponent notation is actively used in microtonal music theory to denote fractions (there it is called "Monzo" notation; the individual "digits" can be positive, zero or negative)
1:47 Rainworld refrence... Nice.
I was hoping someone would catch it!
Oh you sneaky devil
no matter where i go, it's always back to rain world
RAIN WORLD MENTIONED🗣🔥
Where’s the reference?
When I was in basic training I invented a base 60 system, and I'm actually amazed by how easy it is to use
It basically functions like the Mayan system. There's a handful of symbols that you combine to get a value up to 60. A dot is one, a bar is 5, a triangle with it's point up is 15, a triangle pointed down is 30, an equal sign looking thing is 0, and a circle is 60.
You combine them, largest to smallest, top to bottom, but you can subtract smaller numbers from bigger ones by writing the smaller number above the larger one. For example, you can write 27 like this
∆ 15
_ +5
_ +5
.. +2
Or you can save a bit of time and space by writing it like this
... -3
V 30 (pretend V is a triangle)
This system would probably be very good for teaching math to kids because you can visually see what's happening when you do basic arithmetic.
Number 0 was the Lemarchand Box of math.
"You open the Box, you must pay the price".
left channel : unknown classical music
right channel : dude yapping about numbers
decelerated learning
Base 16 forever! We'll never get rid of base 10 but if we could start again Base 16 is just lovely.
Base 10i is better! Even though it needs 100 symbols, it can write any number!
1:24 base two is not too hard to use for humans. Using the symbols 1 and 0 will make your numbers look very long, but we could use smaller symbols, because we only need do differenciate between 2 symbols instead of 10
Like Morse code
In high school, in the early 80's, I had to write a paper for math class. I chose the subject of non-integer bases and irrational bases in particular. I concluded that if you're not going to have a discrete number of symbols for a position, you're essentially just using logs. You only need one position, so you give the value and the position power which indicates whether it's a log, log log, etc.
Without personal computers, it's a lot harder to play around with it.
There was an attempt to create a "balanced trinary" base for a computer where each digit or "trit" would have one of three values: Plus, Zero and Minus. If you look at the way negative binary numbers are represented for computers, it is a ugly mess. Balanced Trinary has negative number built in. Zero is represented as "0", one as "+", two as "+ -" (+3 -1), three as "+ 0", four as "+ +" (+3 +1), and five as "+ - -" (+9 -3 -1). Mathematically elegant, but nobody could build a three state junction for the hardware.
Technically, binary has three states, it's just cheating to use the third one. You have plus, minus, and floating.
formulaic base: every time it goes to the next digit place, the base is different according to some formula
Abstract base - each digit is represented through a different numbering system.
Asymmetric Numeral Systems: the base of each digit depends on the frequency of the previous digit.
This is actually used in some newer compression algorithms.
A system where each digit uses a mixed radix so instead of powers of a base you have a certain set of multipliers to use for each digit like a system that uses 22 symbols for the first digit 9 symbols for the next digit then 114 symbols then 35 then 178. That way 3.31415 means 3+3/22+1/(9*22)+4/(22*9*114)+1/(22*9*114*35)+5/(22*9*114*35*178) which is about pi
You should look into the factorial number system
@@Excalibaard How are they interpreted? Through dancing?
You forgot balanced numeric system! They work like regular natural number bases, but half the symbols are negative. For example, balanced ternary uses digits representing 0, 1 and -1 (usually rendered as T). Here are the numbers from 1 to 27 in that base:
1, 1T, 10, 11, 1TT, 1T0, 1T1, 10T, 100, 101, 11T, 110, 111, 1TTT, 1TT0, 1TT1, 1T0T, 1T00, 1T01, 1T1T, 1T10, 1T11, 10TT, 10T0, 10T1, 100T, 1000
I like balanced ternary for a couple reasons: there is no separate numerical sign for positive or negative--if the first non-zero digit is 1, it's positive, and if it's T it's negative. Also, to approximate a "decimal" ("ternial"?) number in this base, simply truncate it at the desired point.
My favorite underrated numeral base system is Balanced Ternary. You basically use three symbols, such as {+ 0 -}, you can write any integer, positive or negative, without the need for extra symbols, and the addition tables only require 2 carries out of every 9 operations, as opposed to 1 carry out of 4 operations for binary (+ & + = +-, - & - = -+)
Also approximations are easy--just truncate the "ternial" at a suitable point.
This actually cleared something up for me, somehow. I remember reading a bit of math humor many years ago...a list of actual PhD thesis titles, all of which were suitably incomprehensible for non-math nerds. One of them was "...(interesting properties or something)... of base negative three". That always sounded extraordinarily bizarre to me...non-sensical...and therefore funny. But just at the beginning of the section on base -2 in this vid, it all became crystal clear to me how that would work. Thanks! 🤪🤨
Math, but numbers are getting progressively more cursed
Some might opine that they start cursed.
Systems with a non-integer base close to 2 (e.g. 1.8) are good for representing numbers in successive approximation ADCs. In this case, the conversion error in the i-th digit can be compensated in the next digit. And the conversion result can be easily converted to binary code.
Dividing by zero is allowed in some contexts and typically results in complex infinity regardless of any non-zero imput
Issac Asimov made a credible case for using base twelve. We only need two new symbols (call them a,b). Twelve has so many factors, more than any other smaller integer. {2,3,4,6} A lot of arithmetic becomes simple.
And, you can also use a kind of chisanbop to do finger math. I use three fingers for the "ones" and the other two on that hand for the "fours", so you can get 0-11 on each hand.
Come to think of it, I have also played with variable number systems -- a fun example is where the first position is base 2, the second base 3, the third base 4, and so forth -- but another one uses bases 4, 6, 8, 10, 10, 12, and 20 -- these are the dice used in D&D, and are mostly based on the Platonic solids (the two 10-sided ones being exceptions) -- and in my experimenting with *this* base, I discovered a couple of fun facts: first, it doesn't matter what order these dice are in, they always produce a valid number -- that changing the order would produce a different number -- but ultimately, the dice set will have the *same* maximum number, no matter what order you choose!
Another example of variable-based numbers is how we keep time: base 24 hours, base 60 minutes, base 60 seconds. We can thank the Babylonians for that!
How about lengths? (typically 1/16th inch, inches, feet, yard, furlongs, miles, leagues . . . 16, 12, 3, 220, 8, 3.
Base 0 is just the best though.
Yeah, counting in it is fun: 0, 0, 0, 0, ... oh.
It's the number system that can only represent one number
@@KaneYork and actually 0 since 0^0 is undefined
@@difrractsliver1031 No you're thinking of base negative infinity
Yeah man, the amount of "whole" numbers you can express depends entirely on what numerals you're using, it'll always be in the "1s" place because all other places are "0s", and it's the only way I can think of to represent 1/0 as anything other than 1/0.
Super intriguing, and super untapped.
Base 2 is tedious to use on its own, but fortunately, any base can be represented more compactly by taking a power of that base. Any hexadecimal digit can represent a string of 4 bits, so you see hex everywhere in computing.
0:51 THERE ONCE WAS A MAN NAMED STANLEYYYY
LITERALLY SAME THOUGHT INSTANTLY LMAO
i dont play stanley parable, can someone explain this
@@unnamedscribble-auttp 427 is the number of the office stanley works in
Other interesting number systems not mentioned in the video:
- Dynamic base number system: A mixed-radix number system who's base or radix values are determined dynamically at runtime, with just-in-time binding. I would elaborate, but naturally its difficult to say much about its properties or behavior before hand.
- PRNB (Pseudo-random number base) number system
- "Oops! All 5's" number system (This number system only permits one symbol: 5. Fortunately, its also a base-5 number system. Unfortunately, this still doesn't make it a functionally workable number system. Nobody has worked out how to actually DO anything with it)
- ROYGBIV number system in base-white (purely theoretical--it is unclear how the operations of addition and multiplication would be performed on colors, or what that would mean)
- Dancerithmetic; Uses base-10 numbers with place values, like we are used to. The major difference here is this number system doesn't have any symbols/graphemes, so it cannot be written down. Instead, all symbols have been replaced with large, sweeping and distinct bodily gestures/movements, most of which have been borrowed from Ballet. Therefore arithmetic in this system is largely a performative art. It is possible to divide by zero in this number system, provided you are limber enough.
Base √2 is quite interesting.
Conversion:
1 = 1
2 = 100
3 = 1000 (it's bigger than sqrt(2) cubed)
4 = 10000
5 = 10001
6 = 100000
...
I feel slightly like Dan Aykroyd talking to Tommy boy when he says “went a little heavy on the pine tree cologne there kid” except I’m thinking about that fricken music playing in the background lol. Little distracting while I’m very interested, but also laying next to my sleeping child. Anyway good video and great content that most don’t think about everyday, but stuff that really keeps things in perspective and I believe to be extremely important. Thanks!
Yeah, sorry about the music! I'm glad you liked the video.
There is also fibinary notation. Each digit is 1 or 0, and the place values of the digits follow the Fibonacci sequence, so 1010 = 1 x 5 + 0 x 3 + 1 X 2 + 0 X 1. Notably, most integers have non-unique representation, e.g. 5 can be either 1000 or 0110. The addition algorithm isn't too hard to derive.
You can require that numbers are written without consecutive 1s.
@@urkerab Yes, that is what I consider 'canonical form'. I think this canonical form is unique for each integer. But you need to allow the possibility of consecutive 1s during the addition algorithm. You can tidy them away at the end.
Short summary of addition algorithm: to add x and y, initially set s = 0. Shift bits from x or y to s (i.e. in a location where x or y has a 1 but s has a zero, set that bit to 0 in x or y, and 1 in s), and perform substitutions 100 -> 011 or 011 -> 100 to allow more bits to be shifted. All these operations keep the sum x + y + s invariant. Keep going until x and y are both 0.
If I recall, there's a primorial base which uses each primorial instead of powers of a constant number.
So the number XYZ in the primorial base would be:
X*30 + Y*6 + Z*2
While this looks fairly simple, and maybe kinda cool but quirky, there is a downside. Each digit can take the same number of symbols as that primorial. So the "units" digit can be 0 or 1; the next ("tens") digit can be in the range 0-5; the third ("hundreds") digit can be in the range 0-30 and already we need to invent new symbols. The fourth digit is in the range 0-210, and you can see how this is getting out of hand. The fifth digit handling anything from 0-2310 is just ridiculous. There's a reason this base is nothing more than a fun little mathematical diversion. xD
Factoradic numbers, discovered by Donald Knuth when he was in high school. He noticed that when you "fill up" any number of digits in a normal base system and add one, it "rolls over" to the next digit and everything zeros out, e.g., 999 + 1 = 1000. It also happens to be that factorials have a very similar property, if you add one to 3x3! + 2×2! + 1×1! + 0×0!, you get 4!. This one is only a bit silly, since it has uses in combinatorics.
Ooh, I've not heard of Factoradic numbers but they sound like a good time.
In the factorial number system the number of symbols grows for each digit. An n digit number in base n uses fewer digits than the factorial number system used for the same value,but a 2 n digit number in base n has more digits than the same value in the factorial number system.
Actually 3n digit number in base n takes more digits to write that the number of digits for a factorial number system
20! 10^30
The factoradic base is a mixed-radix number system, so the number of "digits" grows without bound as digits are added to the left. This is why factoradic "digits" are separated by colons, e.g., 3:4:1:0:1:0 represents 3×5! + 4×4! + 1×3! + 0×2! + 1×1! + 0×0!.
Digit separators could be omitted when working with factoradic numbers less than 10 digits (as long as the number is noted with a subscript "!" to clarify it's a factoradic). I suppose you could even use the hex digits up to "f" in the same way, but the colons would have to come back if the length of the numbers grew beyond that.
Base 12 is the best base in my opinion, you can count it on just one hand (by counting the segments of your fingers, separated by bone hinges) as well as having 6 divisors, instead of base-10's 4, plus, it wouldn't be all that hard to adjust, plus, making symbols would be easy. we could have something like a backwards 3 for eleven, and an upside down 7 for 12, we even already have separate names for the numbers (coming from a rudimentary base-12 system in proto-germanic)
3:05 it should be 百 I think. Unless there's something weird going on I don't know
That's quite possible, I'm nowhere close to fluent in Japanese so I'd be willing to bet that I just got it wrong. Sorry about that!
iirc they also have different numbers depending on the type of object you are counting so it could also be that
@@jonasgajdosikas1125That’s not really true. You’re getting it mixed up with counters, which are more like units, or like “head” in “head of cattle”. With only a few exceptions, everything gets counted with the same sino-Japanese numerals
Was definitly not our beloved hyaku.
i also think it's strange to call it the Japanese number system when it's really the Chinese number system. that would be like calling the roman numerals the Hollywood system because they are used for the release dates of films in the credits
Watching this felt like watching a villain describe their evil plans.
I'm so evil. Bear witness to my weird maths >:)
…
Gray Code numbering and Zeckendorf numbers are also nice.
There are also redundant numbering systems where you use more digits than necessary. Eg you could use 0, 1, 2 for binary numbers.
Or you can use digits one to ten for a decimal system, but without a zero. That also works. The same thing to the power of 26 and using letters for the digits is how Excel spreadsheet coordinates work.
There's also Skew binary numners.
Have a look at Okasaki's book Purely Functional Datastructure for fun with weird numbering systems.
Imaginary/complex bases are really cool, because you can display any complex number as one number instead of 2. You dont even need a negative sign for negatives!
You forgot about balanced bases, where half of your symbols are negative. An example of this is balanced ternary, which is base 3, but the digits are -1,0, and 1
Does base -2 not count as a balanced base?
@@tim.martin In a balanced base, every digit can be either positive or negative. In a negative base, every other digit flips between being positive and negative. They are similar, but not identical.
The DDT System of Place-Value Notation.
Oooooooohhh! How about balanced base negative three?
I just realized you can't have a negative number to the power of a non-integer. Don't know how I didn't realize that before.
It depends. You could in the complex numbers. Albeit not uniquely.
you have absolute incredible music taste man
Thanks! Unfortunately I think I might have made it a bit too loud in this one because it's difficult to hear what I'm saying at points.
Intriguing, I’ve always thought that things like base pi seemed very intuitive way to count the intersection points for trigonometric functions
"I prefer base 10", says a computer using binary
See, if you call it "base 10" (with the numerals) it only means base ten if... you're already using base ten
True, true. You could say Base Dec though, using the term from Dozenal. And you could write it "Base A" or something similar, however I think most people write "Base 10" out of familiarity more than anything else. If you've never tinkered with number bases, it's not something you've ever thought about.
All bases then are base 10 (except base 0 and bijective bases...)
But, by convention, since our (human) Math culture uses base ten so widely, we express bases according to base ten. 😝
Hereditary base-n notation (as used in Goodstein's theorem) is really interesting too. You rewrite your base in terms of your base, and it has relations to goodstein's hydra and ordinal number systems.
I LOVE THIS SO MUCH! Next, could you please talk about Base -1 using digits 1 and 0, it's like unary (base 1) but it can also represent negative numbers and it's really interesting, also some algebra becomes more intuitive in Base -1 which i fine fascinating.
Also also, you talked about Base 3/2 using the digits 0, 1, and 2, but you failed to mention Base 3/2 using digits 0, H, and 1, where H=1/2. I know it sounds strange and unorthodox, and it leads to this version of Base 3/2 having it's own set of "1.5-integers", but it's actually been written on before in a study (Journal of Integer Sequences, Vol. 23 (2020), Article 20.2.7, Variants of Base 3 Over 2) and it's the system I study, although there are some interesting interactions between the two Base 3/2 systems that you should also check out.
I love numeral bases, this is the most well put together video on them on RUclips, and I hope you do more! Thank you for this video!
Thanks very much! I love sharing interesting stuff with people all over the internet. I can't promise another base video but I'll definitely consider it. What I *can* promise is some more weird maths. Also, I'll definitely have to check out that article, sounds like a fun time!
@@RandomAndgit Ahh, I absolutely love nerding out over maths with someone else! Even if you don't make another bases video (although i really really hope you do) I'll be waiting for whatever you have next.
Also, it would be super fun if you talked about infinity in the context of hyperreal numbers (specifically the ones where infinity can be represented as 1 followed by infinitely many zeroes, "epsilon" represents an infinitesimal which is related not through division, but actually through a more complex exponential relationship, something like baseless 10 to the power of 1/infinity is epsilon and vice versa, and also infinity is an exact amount, more than infinity-1 and less than infinity+1, which means that a repeating segment of digits can have 1 or more extra digits removed or following it on either side.) If you couldn't already tell, this specific blend of hyperreals is actually my main field of study, and I've been wanting to put together a paper or two on them for ages but all my research is currently tied up in some big legal argument over this other house... it's a whole can of worms for another day.
Also, imo limits and their undefined values are corrupting the rest of mathematics. I don't have a problem with the direct study of limits, but I hate when people say stuff like "no, any answer is wrong, it's undefined and should stay like that" it's stifling curiosity and creativity. Like how, no matter how many times it's proved, people can't accept that 0/0=1, 0^0=1, and 0!=1, because the limits of said equations are undefined and give multiple answers. Unless there's limits in the problem, don't assume that the answer will be given by limits. People have said the dumbest things, like infinity lies between infinity and 0, or that infinity+1=infinity. These things break so many mathematical rules for literally no reason, and hyperreals show is that there is no reason to proclaim infinity as NaN and start making up rules purely to confirm that.
Hope you have a good day!
@@autumn948 Your passion for maths is palpable and evidently very well informed, have a good day yourself! (Oh, also, I hope that legal issue works out, that sounds very annoying)
@@RandomAndgit Thank you so much! 😊
And yes, I hope to get not just my (ENTIRELY FILLED 😭) notebooks back from that house, but also the rest of my possessions. It's a very sticky situation, but I hope to resolve it soon.
Can't wait to see your next video!
Props on your style. The music makes this way more enjoyable than many other math channels.
Why did it take my informatics prof 2 lectures what took you 4 minutes?
This was sillily brilliant, subbed!
Any polynomial can be understood as a number in a variable base x. Then, given a base x, the y-value returns the represented number. Or, given a number y and a polynomial representation, the roots of the polynomial are all the bases for which that expansion represents that number.
This person is really changing the scene of math explainers.
I like ur FG vids
@@julesharris6383 Thanks!
thank you so much for this video, i didn't understand a lot of this stuff just by reading wikipedia so a random youtuber making a banger video makes all the difference
I'm super glad I could help! Do take everything with a grain of salt though because it's been simplified quite a bit.
In every position based systen the base number is always written as 10. Right?
That's exactly right!
www.youtube.com/@RandomAndgit @z3my4l And what about IMposition-based systems? (Initially meant as a pun, but now treat it SERIOUSLY, please.)
Inlove how phi in your art style looks
Phi being silly
Also loved the video and i could actually kind of understand it
I'm glad to hear it! Number systems are a pretty complex topic so I'm pleased the video isn't totally incomprehensible.
Base 3/2 (kind of base 1.5) counts like this:
0, 1, 2, 3=20, 21, 22, 40=210, 211, 212, 213=230=430=2130=2300=4000=2100, ...
And so every integer can be represented uniquely base 3/2 using the symbols 0,1,2.
In general, if n and m are coprime numbers, such that n>m, you can have an integer system with base n/m using n symbols 0, 1, 2, ... (n-1).
All you do is count as normal, but you swap n elements in one column for m in the next higher. In base 10 (10/1) we swap ten digits in one column for an extra 1 in the next, if this overflows we repeat (99+1 = A0=100, because ten, A, overflows).
So we base 3/2, just take 3 digirs off one column and add 2 to the next. Keep going until a number is represented with just 0,1 or ,2 digits only
A very well informed and interesting comment. Excellent job.
Would you consider uploading a video explaining this (even if it's just you writing on a piece of paper while explaining it)? EDIT: I just found one on RUclips.
Also, and I may be completely off the mark, I can't help but wonder whether or not fractional bases could be useful in cryptographic applications since I gather that carrying over to the next column is different from whole-number bases where only one can be carried over to the next column at a time.
There's a version of base 1.5 (base 3/2) where every integer still has a terminating decimal representation! But it uses the digits 0 1 2, just like base 3. It goes like this: the last digit is the remainder when the number is divided by 3, but the digits before that are the base 3/2 representation of *twice* the integer part of the number divided by 3.
So it goes: 0, 1, 2, 20, 21, 22, 210, 211, 212, 2100... If you work it out, you'll see that those are actually base 3/2 representations of the integers from 0 to 9.
You can use any base p/q for positive integers p>q in this way. The fraction doesn't even have to be in simplest form.
I'm pretty sure phi isn't special when it comes to irrational bases. I have this gut feeling that any root expression for a base can express integers in finitely many digits
Like. If I set the base R = sqrt(2) + 1 then I have the relation R^2 = 2R + 1, which manifests in the base as 100 = 021
So a number like 3 would be written as 10.11, and 12 as 200.02
Not too sure since I haven't tested anything from nested roots or cubics or anything higher degree but I feel like it should hold
UPDATE: GOT SOME OF THE MATHS WRONG, OOPS...
For base Q = sqrt(6) - 1 you instead have 7Q^(-1) = Q + 2, or 007 = 120. So you probably need to work in seven digits.
0-6 is still 0-6 but 7 = 120 now. 14 = 240, 21 = 360, 28 = 1610, 35 = 13030
Funny enough, you need ten symbols for base S = sqrt(8) + 1 because S^2 = 2S + 9 (100 = 029)
So 0-9 is still 0-9 but 10 (actual Ten) would now be written as 9.29
and 11 would be... uhhh... huh. no clue, actually
0^0 is indeterminate not undefined... this is because the limits of x^0 and 0^x as x -> 0 are different ! I think something interesting to include when discussing bases are things that are similar to bases but not the same. Those things in specific are Cantor normal form, Conway normal form, and p-adic numbers. Cantor normal form is essentially a way of representing numbers that are in some sense infinite. Here's a brief outline of these three, leaving out quite a fair amount of detail:
First, ordinal numbers are a class of numbers where one of ab, a=b is true for all ordinals, and each ordinal has a successor. With ordinal numbers you are allowed to construct what are called limit ordinals. The smallest limit ordinal, ω, is the limit of {0,1,2,3, ... } and is thus larger than all finite numbers. (I'm being a little bit loose with my definitions here to simplify a little bit, but the limit of a set is the union of smaller sets. The empty set, {} is the number zero, the set with the number zero, {0}, is the number one, the set with 0 and 1, {0,1} is the number two, etc.) Cantor normal form is a particular way of writing ordinals (that can in fact write all ordinals) which is a1ω^b1+a2ω^b2+...+akω^bk with a_1,a_2, ..., a_k being natural numbers, b_1>b_2>b_3>...>b_k being ordinal numbers, bk being at least 0. There's another theorem that states that any strictly decreasing chain of ordinals is finite (although it can be arbitrarily large). So, there are only finitely many b_i's.
Conway normal form applies to a subject called combinatorial game theory and is slightly harder to explain because of the nature of CGT. However, it constructs something which may initially seem analogous to ordinal numbers called the surreal numbers, but has a nicer seeming arithmetic (you can divide by the surreal number ω, but not by the ordinal number ω). A very rough intuitive way of describing Conway normal form is as a continuous more "arithmetic" (it's commutative, while ordinals are not) analog of Cantor normal form.
(If you want a more in-depth definition of surreal numbers, they are essentially games of the form {A|B} where A is the choices for a player (who is called Left) and B is the choices for the other player, where the sets A and B consist of only numbers and for all elements of A, there is no element of B which is less than or equal to it. the game { | }, where neither player has a choice is called 0, and {0 | } is 1 and { | 0} is -1. If you want more info I recommend reading about it, it's a very interesting subject.)
p-adic numbers are built on top of analytical tools of calculus. First we need to be able to talk about the distance between two points, which we will call d(a,b). This distance is a nonnegative real number, and we want it to have a few properties. Namely, d(a,a) = 0, d(x,y)=d(y,x), d(x,y)>0 if x and y are distinct, and d(x,z) is at most d(x,y) + d(y,z). If all these properties hold, we call d a metric. The importance of a metric is that it allows the construction of a limit by saying "these values eventually get arbitrarily close" pretty much. This is what it means to say 0.999... = 1, is that any sequence of 0.9,0.99,0.999,... can get arbitrarily close to 1, and is thus the same as 1. We can construct something called the p-adic metric, for any prime number p. this is how it works: |0|_p = 0, | p^n |_p = p^-n, | p^n * (r/s) |_p = | p^n |_p if r and s are coprime to p (n is an integer). With this metric, setting p to be 5 for example, 1+5+5^2+5^3+... converges to a particular point, which can be written as ...1111 in the 5-adic numbers. this is because the distance between each consecutive 1, 11, 111, is decreasing in regards to this metric. This number system seems very opposite our standard intuition in that there can be infinitely many digits to the left of the decimal (radix) point, but only finitely many to the right.
base infinity: every number will have their own symbol
Base panic, where numbers are defined by infinitely increasing panic
He he he!
(possibly hysterical laughter)
I really enjoyed this video! I definitely noticed some similarity with TodePond's style, but also it's definitely unique in many more ways, subscribed
Re base 16: I think it's cool to come up with unique glyphs for 11-15 instead of using letters. There are a couple ideas to play with:
1) make them stylized versions of the letters A-F so they can be read even if you don't know them already.
2) make new digits that look like the existing digits, design-wise.
obscure numbering system: tic-xenotation (TX):
- 2 is written as :
- (n) is the nth prime number (1-indexed, starting from 2)
- every number is written as its factors
2 :
3 (:) (the 2nd prime)
4 ::
5 ((:)) (3rd prime)
6 :(:)
7 (::) (4th prime)
8 :::
9 (:)(:)
10 :((:))
it's possible to write any integer greater than 2 in this fashion. very impractical since the system is based on factors (even basic addition is impossible without rewriting into a modulus-based system).
It appears that at least 8 came down the wire wrong. Did you just invent this btw?
I came up with the exact same thing, except instead of using : to denote 2 you just have a blank number represent 1. And if you substitute 1 and 0 for ( and ) you get a very, very cursed form of binary
@@DanDart it was created by an experimental philosophy collective called the "cybernetic culture research unit". massive rabbit hole
The most natural number system is Base 4x10^80 where every digit is a unique particle in the 10 (universe).
This video is very entertaining for a math video, i have a critic however. The volume of the music is louder than the voice, maybe you can lower the volume of the music. Other than that it is very good and underrated.
Base 12 is quite interesting: the numbers would be devisible by 2, 4 and 3 (instead of 5 when considering base 10).
Which makes it very useful in household use - shame we only have 10 fingers.
This is just silly, but infinitely interesting.
The best number systems to use is base
(A*2+B*3+C*5+D*7)
Where A,B,C, and D are whole numbers >=0
A >= 1
And A >= B >= C >= D
Base 2, 6, 8, 12, 16, 64, 60 all meet this requirement.
Example, base 12 A=2 and B = 1, C and D = 0. Base 16, A = 4, B,C,D = 0
Base 10 doesn't work though. A = 1, B = 0, C =1, D = 0.
Basically the number should be composed entirely of small prime numbers, smaller the better.
6:28
Base-0 doesn't exist. There are no digits. There is no math happening. Nothing.
It's very user-friendly for innumerates and dyscalculators.
If we get lost in a forest or stranded on an island, the best base is a stick. We just need to cross out IIII to count the days lol
Base-32768 number systems are quite cool.
Base-32768 is a very powerful number base in computers because it can store 15 bits of data in a 16bit UTF16 codepoint, for an efficiency of 94%.
Wait till you hear about base 9007199254740992.
If we had 8 fingers instead of 10 - and, as such, used a base-8 system - converting between the "standard" base and binary would be so much easier
This is the Kzin advantage in math!
Lets try base 1
-there is only 1 symbol
-but there has to be a null symbol
-so this symbol is the null symbol
-so we get only 0
There are other options, such as reflected bases--like Gray Code (en.wikipedia.org/wiki/Gray_code) in which adjacent numbers always differ by only one digit change, or, particularly with odd bases, balanced bases--like balanced ternary (en.wikipedia.org/wiki/Balanced_ternary), which needs no separate positive or negative sign, since the digits chose are plus (one), zero and minus (one)--so the largest non-zero digit sets the sign of the number. And it is easy to round numbers: just truncate them when the proper accuracy is reached.
Then there is one of my favorite bases--balance reflected ternary. Not very useful at all, but fun to count in.
Oh, and can we use continuing fractions for approximations to rational numbers--it's like at each "decimal" we use the optimal base, so we get very good approximations as quickly as possible.
4:53, isn't 1/phi equal to phi-1? the sqrt of phi is. 1.272...
Oh, yes you're completely right. My mistake.