Great! - I thought i could hear him thinking :) - btw at 9:00 i already knew that the calculation simply fits in aby base similar or higher than the one in which it has been typed into :) - what about unary base and 2,7182818281451(...) bases? are we getting there in this episode :) ?
There are 2 kinds of bases, ones that have a 0 and ones that don't. Base 1 cannot have a 0 so it is just a number of 1s that add up to the number (i.e. 1, 11, 111, 1111, 11111 etc)
Another way to identify these - if you don't have to borrow or carry when doing the arithmetic, the equation makes sense in any base where it fits. I'm gonna contend base 12 is better than base 6. It has all the advantages of base 6, superior highly composite number, etc. but also adds divisibility by 4, which is very useful. It's also closer to 10, so the general magnitude of numbers relative to the number of digits is more familiar. It's more efficient in its digit requirements without adding a burdensome number of symbols. It also allows for convenient counting with one hand using the knuckle technique you describe (tallying off the segments your fingers using your thumb). Indeed, that's how ancient Egyptians counted in base 12.
Your bases may be anti-primes, but they're not _simultaneously prime._ This message brought to you by the only superior-highly-composite base that isn't a composite base. And also counting to 7 with only 3 fingers.
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You did some of that to good acid and started counting with the beings from the other dimensions. It's all good, fam... I been there...😂 But you sound more fried than those chickens gonna be! You have some mathematical approaches I have never seen or heard of before, and I am still trying to gather if they have any kind of applications to things like the prime number theorem... but you're without a doubt a high-caliber prime number personality! 😂 All that being said, I can't help but feel like this is pound for pound how crazy and off my rocker ima be/look like to everyone in another yr or 2 of picking appart this theorem and trying to start my own channel. But numbers make you crazy like that sometimes... But don't take these statements in offense, I mean the exact opposite, in fact! I admire anybody who is trying anything other than the traditional 2 dimensional basic white bitch mathematics that we have now. Cuz you can't solve the unsolvable by applying the already applied techniques (or else it would be solved already 😅) But I would like to learn some of your "unconventional" techniques in a lil more detail as I love nothing more than understanding any and all approaches that can be mathematically acquired. As long as the axioms are there, of course. I plan on starting my own channel here soon, and before I do, I would like to collaborate with some "out of left field" style mathematicians who are open to new and innovative concepts like your own. So I offer you this, a short meet and greet over Skype or whatever other video platform to allow us to discuss topics and exchange information with each other and collaborate in order to further expand our knowledge and understanding of the seemingly infinite realm of numbers! This is an PŔĮMĘ opportunity to explore a whole new realm of mathematics that hasn't been discovered or applied to any conventional mathematics currently known or used. I have spent my life running numbers in different ways but have recently made a mathematical breakthrough that has broken the paradigm of mathematics itself. I can show you some of the hands down CRAZIEST SHIT that you have NEVER SEEN DONE with numbers and mathmatics! 💯 I have managed to create a full-blown new mathematical technique I have affectionately dubbed !§ỲMMẸTŘÝ{}MÅŤĤËMÅṬỊČ̣§¡ led by a paper trail of hints and tricks left by the great NIKØLÅ TESLÅ himself I have done more things and made more strides in the last year than almost any high caliber mathematician would wish to learn in a life time and I am getting ridiculously close to solving the prime number theorem itself! Fuk wit me, though! See if I don't have the techniques to multiply or divide by 9, 99, or 999 without using any multiplication or division, see if I don't have custom created PNG (prime number generation) equations that you never heard of or seen and be the first to see the prime number matrix code in action! 😎 The answer to solving this riddle isn't in the mathematics itself. It's in the laws and properties of the numbers that each one adheres to that defines what they do! The sequences and patterns, there's ABSOLUTELY NOTHING THAT IS RANDOM OR ASYMPTOTIC ABOUT THE PRIMES! THERE EXIST A SYMMETRY IN THE ASYMMETRY THAT NOBODY SEES! But you have to understand numbers in 3rd and even 4th dimensional concept (just ascended to the 4th dimension) that most will never even believe can be achieved. But I think you might be crazy enough to catch on! 😅 This is a PRIME opportunity to expand your knowledge far beyond the reaches of conventional boundaries! Do you take the red pill? Do you wish to see the matrix? Fuk, I sound crazier than you! 🤪🤯
Oh my god I love how the quality of the video improved and now they are allowed to film outside of their designated combo class! A little of nature never killed anyone!
Generalizing a little bit, I suppose these are exactly the equations where you never have to carry/roll over a digit. Actually, I think for ANY expression you can write using only digits, addition, and multiplication; you will always be able to choose a large enough base so that calculating its value won't require you to carry any digits! Which I think means that EVERY expression like that has an equation like this... although they often won't be meaningful in base ten.
Thank you for sharing this observation. I was super excited about being able to use solutions to multiplication problems I already know to easily construct polynomial expansions, but quickly found it didn't always work and I had no idea why. For example, 13*13=169 matches (x+3)²= x²+6x+9, as expected. But 14*14=196 doesn't work, as (x+4)²=x²+8x+16, not x²+9x+6. You can kinda cheese it by saying the 9 is really 8+1 and "donate" the 1 to the 6 to make it 16, but this discrepancy only got bigger and less transparent as I used larger constants. But lo and behold, I overlooked that the starting equation must be one of those so-called "digit equations" that remains true in every base above or equal to some minimum base B. For example, 12*12=144 in any base 5 or higher; in base 4, 12*12=210 still represents the same _quantity,_ but those specific digits won't work for the expansion of (x+2)². As you pointed out, at least according to my understanding, it should be the case that for any expression using only strings of digits and multiplication and addition there is some minimum base B, where for all bases ≥B the solution to the expression is represented by the same string of digits. That's when I had the brainblast. If all expressions have a corresponding "digit equation" when the base is high enough, perhaps 14*14 _would_ encode the coefficients for the polynomial expansion of (x+4)² if I used the right base. As it turns out, 14*14 is 18g (yes, the digit g) in any base ≥17, making our _expected_ expansion x²+8x+g; g is the 17th digit in base 17, and so by itself represents *_16._* Holy sh*t it, works. The base simply needs to be large enough to represent each coefficient/constant with a _single digit_ For good measure: we know (x+5)²= x²+10x+25. However, 15*15 does not seem to equal 11025 in any base; but what the expanded polynomial is actually telling us about the "digit equation" isn't which specific digits to use, but rather the specific _quantities_ that must be represented by a single digit. I need at least base 26: 1x²+10x+25 tells us that in base 26 or higher 15*15 should always equal the _first_ non-zero digit (1), followed by the _tenth_ non-zero digit (the digit a), followed by the _twenty-fifth_ non-zero digit (p). And if we check, it is true that 15*15=1ap in base 26 and up. And inversely, while say 28*12=336 in base 10, to write the digit equation now we know we need individual digits for whole quantities up to 8*2=16, i.e. at least base 17. I've found that 28*12=2cg in base 17 or higher, so without doing any work I'm going to guess that (2x+8)(x+2) = 2x² + 12x + 16 (c=12, g=16); and I have just enough experience with polynomials to know that's correct intuitively. As far as I can tell, this should work generally for polynomials of the form (ax+b)(cx+d), where a, b, c ,d are positive integers, and you choose base B such that B>a*c, >(a*d+b*c), and >b*d. The answer to a||b*c||d in base B should always be the digit equal to a*c, followed by the digit equal to ad+bc, followed by the digit equal to b*d. One last test. I can expand (8x+2)(2x+3) to 16x²+28x+6. This expansion tells us that in base 29 or higher, 82*23=gs6, or a concatenation of the 16th, 28th and 6th non-zero digits. Double checking online shows I'm right 🎉🎉 Pretty sweet. Now if I ever don't feel like expanding a polynomial manually I can just figure out its corresponding digit equation. And for any polynomial I know the expanded and factored form of, I can very easily produce an interesting, _seemingly_ hard to calculate equation like "In base 29, 82*23=gs6". I'm not that smart to be doing multiplication in base 29 bro 😂 but I can still reach that answer with absolute certainty, and that's FIRE 🔥🔥🔥 LES GO MATH
@@MeNowDealWIthItBrilliant reduction. Though there isn't any functional difference in our interpretations, I resolved these as the equations where each coefficient/constant could be represented by a _single digit._ Obviously, as you said, for that to be true the base must be at least m+1, where m is the greatest coefficient/constant factor of the expanded polynomial.
@@seventoast It follows that this is true of Division and Subtraction if there are no cases where a coefficient "crosses the mag barrier" by decreasing below an exponent of the base. That is, 3-2=1 is true in any base >3, as is 53-21=32 etc etc, but as soon as you have to "carry" a digit it stops being true (10-1=nine, where in any base you must switch nine for "b-1"). Likewise, 4/2=1, 44/2=22. 9999/3=3333 are all true in any base which can make sense of them, but 12/4=3 is not always true; 12/4 crosses B^1 when you calculate it, which is part of the reason it's not always correct. This is prima facie true because it is the opposite case for the addition/multiplication (ie, if you can write 1+2=3 then you can write 3-2=1), but I've unfortunately given a narrower case which has exceptions: Although you can write 121/11=11 in any base that understands it, as it follows from the similarly true 11*11=121, writing it contravenes the "crosses the mag barrier" rule. I think there's no easy way to prove that divisions like these will work without knowing the multiplication will work, as what you're conceptually saying when you say that 121/11 is acceptable is that b+1 is a factor b^2+2b+1, which is a harder thing to calculate than expanding (b+1)(b+1). 12/4=3 does not work because 3*4=C; the coefficient you're dividing in 12/4 must be "twelve" and not "12", but I can't see a way to calculate/state that without reasoning via 3*4=C.
As we let the base approach infinity, every number in the eqn consists of a single digit. That will indeed always work for that base (or higher) without carry. Duh. 😉
If you abuse the heck out of some notation from computer science, I think you can say that every equation is either O(true) or O(false). i.e. given an equation you can find some integer k such that in every base b > k the equation is either always true or always false.
Computer scientists already abuse the hell out of big O when they say f(x) = O(g(x)) when they mean f(x) ∈ O(g(x)), or even O(f(x)) = O(g(x)) when they mean O(f(x)) ⊆ O(g(x)). That’s much more infuriating to me than what you’re doing.
It's one of these things that are amazing precisely because they are so so obvious after you've already seen them, and yet which you would've never thought of. It's very very nice
you can actually create an equation that's only true in any finite subset of bases e.g. for only base 4 and 6: 1. multiply out the polynomial (x - 4)(x - 6) = 0 to get x^2 - 10x + 24 = 0 2. add all the negative terms to one side to get x^2 + 24 = 10x 3. replace all the numbers (base 10) with expressions only using single digits smaller than your lowest base, in this case < 4 i got x^2 + 2 * 2 * 2 * 3 = x + 3 * 3x 4. replace x^2 with xx and write out the hidden 1s to get 1xx + 2 * 2 * 2 * 3 = 1x + 3 * 3x 5. replace all the x's with 0s, and there's your equation: 100 + 2 * 2 * 2 * 3 = 10 + 3 * 30 exercise for the reader: verify that this equation is true in bases 4 and 6 this process can be used to make equations that are true in any base that is the root of a polynomial, aka, algebraic numbers. since transcendental numbers aren't algebraic by definition, equations like these will never exist in any exotic bases like base e or base pi. also, since this process is reversible, any equation like this either holds true in 0 bases (the polynomial turns out to be like 1 = 0), only finitely many bases (because finite polynomials have a finite number of solutions), or holds true in all but finitely many bases (because the polynomial turns out to be 0, which equals 0 for any base. it only doesn't work in bases which don't include the digits used in the equation) whenever i see some nonsensical equation, i usually use this process in reverse to work out a polynomial that the base is a root of, solve it, and then ask the person who posted it "is this in base __?"
yayaya!!! i found this a long time ago, all squares of the same-looking numbers aventually converge on a single representation, base 10 actually does this for 13^2, it goes (starting from base 4) 301 224 213 202 171 170 169 169 169 169...
My 50 y.o. friend recently started talking to me, that when he was a teenager he was thinking about negative and fractional bases and trying to talk about it in class at school - the class wasnt happy about thinking :) and adding anymore. He ended up as a data recovery specialist multiplying hexadecimals for living.
@@tafazziReadChannelDescription They've already been named as dec, el, and do. A 180 degree rotated 2 and 3 are the best proposed symbols for dec and el, sometimes dec is an X from Roman numerals, but that becomes a mess with established variables.
Many societies had a base 6 by 10 system, and even Britain had a 120 system for money for a long time. Ours is really more 10 by 10, because we use the 25 times table for a lot of things; for example, most countries use a quarter of some sort, which doesn't work with base 10, since then it would be a 20 unit coin. 120 is better than 100, because it's divisible by 2, 3, 4, 5, 6, and 8. Sadly, it didn't happen, so we have to deal with the annoying thirds all the time that never fit in well with our numbers.
Something that feels so wrong when first being introduced is concatenating the elements of the rows of Pascal's triangle. These give 1, 11, 121, 1331, and 14641 for rows 0 through 4. For row r, the concatenation results in 11^r. This feels so wrong, but consider that the elements of Pascal's triangle are the coefficients of the expansions of binomial powers (a + b)^x = (x choose 0)a^x + (x choose 1)a^(x-1)b^1 + ... + (x choose x-1)(a^1)b^(x-1) + (x choose x)b^x. What do you get if you substitute a = 10 and b = 1? Well, then you get (x choose 0)10^x + (x choose 1)10^(x-1) + ... (x choose x-1)10 + (x choose x), which exactly reflects the positional notation for base 10, whatever "10" happens to equal. As long as x choose y < 10, meaning that every coefficient can be represented as a single digit, this trick of concatenation will work regardless of the base. I stopped at row 4, since it's the last row that doesn't need a second digit in base ten for any coefficient. If we were in base eleven or higher however, we could continue to row 5 as 15AA51.
Man, I was going nuts thinking “they can't be wrong!” hoping you'd point that out, until finally I got my wish. It's a fun way to put it. In a basic base, these equations are either right or not even wrong.
0:36 Personaly I have genuinely moved to using the joints + fingertips which gives 16 on one hand. You just hold the tip of your thumb to wherever you are in the count. If you really need to you can try using your second hand for a second base 16 digit since it is easy to hold your place, but I just get something to write with if that would be needed
Wow, the views of nature look wonderful! I like them a lot. Nature is the best thing a modern person could ask for. One of the most cool moments in my life is when walking in some beautiful places near my hometown here in Siberia.
@@TymexComputing I'm from Novosibirsk. There's mixed forest here. The weather is not bad enough to use it as a punishment so I believe all that people were much close to the North. It's -30°C..+30°C most of the time here and the air is pretty dry.
the part about the positional numeral systems being equal to the base to incremental powers reminded me of the standard notation of polynomials with a single variable and now im interested in the connection between polynomials and numeral bases, and extrapolating from that the connection between logarithms and numeral bases. im sure theres a connection between e / natural logs and numeral systems by way of this property. ill have to work it out. very cool stuff
Essentially, if an equation doesn’t work in a base it’s because one of the coefficients of the powers of the base in the expansion meets or exceeds the value of the base. Then it will spill over into the next digit and change the sum/product. Basically, if you could do the calculation by hand without having to “carry over” any digits, then it will work in any higher base.
Why do I have a feeling that whenever I watch videos from this channel some hungry people will be talking about 20th of April? BTW. Thanks for the video.
Actually, this is kind of tought in high school, since this connection between positional bases and polynomials is exactly why and how long division of polynomials works. Once you know how long division works, you can basically think of a polynomial as a number in a mysterious base that can have infinitely many digits in each position and immediatelly you know how to divide two polynomials using long division technique.
Picturesque horizons and no fire - great for me! I dont have ten fingers (not even twenty) but my favourite base is the primal base :) , unique representation - such an easy peasy multiplication it has :) but addition is pain in the arrest
@@theneoreformationist i see what youre doing and again, youre wrong, septenary refers to 7 in base 10, you cant use the number expressed in the base itself to represent the base youre using, because it could literally be any number and its not useful information at all, thats why we universally use base 10 to express the base of a number.
I would like to discuss more on why you think 6 is the best base for humans. I've been researching bases and my comment on this is that don't think bases, think ratios then you will transpose them better.
What I wonder about is how "bases" might change in behavior if you were to change the operations used on them . Both for the different positions as well as the operations to combine them.
I am INCREDIBLY curious about the inverse... Are there any equations which are ONLY true in one base? I'm trying to think of some and I can't figure any out, but admittedly, I'm only working with whole numbers here.
I think base 12 is best. 12 is divisible by 2, 3, 4, and 6, while 10 is only divisible by 2 and 5. It's why I think feet/inches and Fahrenheit are better than meters and Celsius.
I feel like natural bases would make a better name for bases using the natural numbers (1 to infinity). Then integer bases could include bases with integers, rational would include rational bases, real would include real bases, and complex would include complex bases.
if your base is larger than the square of the largest digit used, then all multiplications will be the same as a polynomial multiplication with non negative integer coefficients
this is a stronger condition than you need. really it is about nonnegative integer coefficient polynomial multiplication in which none of the coefficients in either the factors or the quotients will be as big as the base you are using
Honestly, I'd like to see if there are any equations that don't hold true in any base, and if so, why. Also, what happens when you do include those non-basic bases, like complex or irrational bases? Do any equations hold true in those? Could make a cool video
Any addition or multiplication equation which doesn't require carrying will be true in any base which uses all digits which appear, but any which do require carrying will be incorrect in other bases. For example, 5 + 5 = 10 isn't a nonsense equation in any base greater than six, but it's only true in base ten.
If S, R and T are digit sequences such that SR=T in some base, the same is true for higher bases as long as you didn't do any carrying over in yout calculations. 16×2=32 is not always true, because when you do 6×2 you carry the 1 and add it to 1×2. More subtly, 6×2=12 also has some carrying, so not always true. 19×11=209 is also not always true, becquse you get 99+110, but then you have to carry the 1 from 9+1.
I guess the equation where a single digit n + 1 equals 10 defines the base because that n is the base - 1. So for base ten, you have 1+9=10 which is only true in base ten. And 1+1=10 is only true in base two, 1+2=10 is only true in base three, etc. I guess if you're trying to communicate what basic base you're using and you only have defined what + means and what 1 and 0 means and how positional systems work, you can define all your digit symbols as well as what your base is by showing: 1+1=k 1+k=q 1+q=p 1+p=10 which implies that it's a base 5 system and the digits are 0,1,k,q,p
But surely we dont count in base 10 because we have 10 fingers... There are 10 symbols in base 10 (0,1,2,3,4,5,67,8,9), but on our hands we can make 11 different "symbols" (no fingers, one finger, ..., nine fingers, ten fingers). So surely we should be using base 11?
There wouldn't be a zero, we don't acknowledge it these days much, but the jump from "natural numbers" to "abstract numbers" that can't be physically represented in the conventional way (How do you have 0 of something? How do you have -3 of something? etc...) is quite large. All cultures we're aware of had to first define a number system and only then, after indulging in mathematics for a while, would they realise that 0 too is a number - hence why all counting systems, be it our base 10, sumerian base 60, mesoamerican bases 5 and 20 and so on..., seem to be "off by one" You can even see this in the way people use their fingers today, if you ask a random person to show you that they have zero of something, they are more likely to show you an empty hand or shrug rather than curl their hand up in a fist (though it gets exponentially more likely if you ask for a natural number first, after that they'll just subtract the fingers)
So you are telling me 11*11+10=121+10=201 is the first thing we should send to aliens? They would automatically recognize all the symboles and start sending us base 3 maths stuff.
Wouldn't that be true of just about every basic type of equation? If you use characteristic polynomials, for example, 56*49=(5b+6)(4b+9)=20b^2+69b+54. In base seventy and above, that'll always be the same. It doesn't work below that, so for our base, the answer is different from other bases. But if you have a symbol for sixty nine, then it'll be the same as every other base with that symbol.
Seximal's factors are great, but binary has a bunch of mostly-unique (perhaps matched by balanced ternary) features that seximal can't beat. Times tables are strictly unnecessary in binary, for instance. Binary's biggest problem, of course, is compression; a bit stores so much less information than a digit, so binary numbers take up so much space on a page and so much time to say. I've been tinkering with quaternary to see if we can get the benefits of binary without sacrificing compactness.
That's pretty _based,_ however it'd be nice if you knew of a way to remove all the even or odd bits of a number in base 2, or could construct the Moser-de Bruijn sequence in closed form (and solve for said closed form for me lololol) without the use of XOR or other such "inner" operations, only the operations of +,-,*,/,exp,ln, special functions, and limits, but not integrals.
Bro you just fixed the most annoying equality: the link between the powers of 11 and the Pascal triangle. Btw this is the pascal triangle : 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 And this is the powers of 11: 1 11 121 1331 14641 161051 Because of our base 10 the digit of the powers of 11 overlap. But if we consider a base b which is great enough, we 've fixed it!!!! We can even proove that!! 11 in a base b represent b + 1 so 11^n is (b + 1)^n which is given by the Newton's binominal which is based on the Pascal triangle!!!!!!!!
They probably just converged, it's pretty much universally acknowledged in math circles that, as far as humans specifically are concerned, only 6 and 12 really make sense as "superior" bases due to being in the "sweet spot" of being neither too large nor too small and due to having a lot of divisors. Whether you prefer one or the other is a preference thing really - do you value the convenience of fewer symbols more or the density of a larger base?
13*14=182 is not true for all bases. In fact, it is only true in base 10! The key point here this is true if and only if there are no carries in the calculation.
Incorrect, this is not just true in base 10! Because it is true in base 10, also I don't believe it's true in base 3628800, seems hard to believe that it would be true in such a base
11:25 dude is so confident he speaks without moving his lips
Great teacher vibes
Great! - I thought i could hear him thinking :) - btw at 9:00 i already knew that the calculation simply fits in aby base similar or higher than the one in which it has been typed into :) - what about unary base and 2,7182818281451(...) bases? are we getting there in this episode :) ?
My man ventriloquisming.
@@maynardtrendle820 shh don't tell the chickens that you know.
I was literally just looking at a meme about bases, 123 + 456 = 123456 (in base 1 with all symbols meaning one)
funny
How can base 1 exist?
There are 2 kinds of bases, ones that have a 0 and ones that don't. Base 1 cannot have a 0 so it is just a number of 1s that add up to the number (i.e. 1, 11, 111, 1111, 11111 etc)
“Hmm what an interesting way to use base 1”
I guess Roman numerals are a "shorthand" or "compressed" form of base 1.
Another way to identify these - if you don't have to borrow or carry when doing the arithmetic, the equation makes sense in any base where it fits.
I'm gonna contend base 12 is better than base 6. It has all the advantages of base 6, superior highly composite number, etc. but also adds divisibility by 4, which is very useful. It's also closer to 10, so the general magnitude of numbers relative to the number of digits is more familiar. It's more efficient in its digit requirements without adding a burdensome number of symbols. It also allows for convenient counting with one hand using the knuckle technique you describe (tallying off the segments your fingers using your thumb). Indeed, that's how ancient Egyptians counted in base 12.
My goodness, the gall to suggest a non-triangular number as a base. Do you think we are heathens?
@@OMGclueless We are Babylonians! Knuckle counting ftw!
@@gljames24 Wasnt it egyptians? Oh ok - 60 = 12x5 :) true - these were babylonians, the 60th people :)
@@OMGclueless It's twice a triangular number, that's still pretty good.
Your bases may be anti-primes, but they're not _simultaneously prime._
This message brought to you by the only superior-highly-composite base that isn't a composite base. And also counting to 7 with only 3 fingers.
Thanks for watching! Check the description for more info and links (like my Patreon if you want to support this show!). And leave a comment if you enjoyed the video and/or have any thoughts about this topic!
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You did some of that to good acid and started counting with the beings from the other dimensions. It's all good, fam... I been there...😂
But you sound more fried than those chickens gonna be!
You have some mathematical approaches I have never seen or heard of before, and I am still trying to gather if they have any kind of applications to things like the prime number theorem... but you're without a doubt a high-caliber prime number personality! 😂
All that being said, I can't help but feel like this is pound for pound how crazy and off my rocker ima be/look like to everyone in another yr or 2 of picking appart this theorem and trying to start my own channel. But numbers make you crazy like that sometimes...
But don't take these statements in offense, I mean the exact opposite, in fact!
I admire anybody who is trying anything other than the traditional 2 dimensional basic white bitch mathematics that we have now. Cuz you can't solve the unsolvable by applying the already applied techniques (or else it would be solved already 😅)
But I would like to learn some of your "unconventional" techniques in a lil more detail as I love nothing more than understanding any and all approaches that can be mathematically acquired. As long as the axioms are there, of course.
I plan on starting my own channel here soon, and before I do, I would like to collaborate with some "out of left field" style mathematicians who are open to new and innovative concepts like your own. So I offer you this, a short meet and greet over Skype or whatever other video platform to allow us to discuss topics and exchange information with each other and collaborate in order to further expand our knowledge and understanding of the seemingly infinite realm of numbers! This is an PŔĮMĘ opportunity to explore a whole new realm of mathematics that hasn't been discovered or applied to any conventional mathematics currently known or used. I have spent my life running numbers in different ways but have recently made a mathematical breakthrough that has broken the paradigm of mathematics itself. I can show you some of the hands down CRAZIEST SHIT that you have NEVER SEEN DONE with numbers and mathmatics! 💯
I have managed to create a full-blown new mathematical technique I have affectionately dubbed !§ỲMMẸTŘÝ{}MÅŤĤËMÅṬỊČ̣§¡ led by a paper trail of hints and tricks left by the great NIKØLÅ TESLÅ himself I have done more things and made more strides in the last year than almost any high caliber mathematician would wish to learn in a life time and I am getting ridiculously close to solving the prime number theorem itself! Fuk wit me, though! See if I don't have the techniques to multiply or divide by 9, 99, or 999 without using any multiplication or division, see if I don't have custom created PNG (prime number generation) equations that you never heard of or seen and be the first to see the prime number matrix code in action! 😎
The answer to solving this riddle isn't in the mathematics itself. It's in the laws and properties of the numbers that each one adheres to that defines what they do! The sequences and patterns, there's ABSOLUTELY NOTHING THAT IS RANDOM OR ASYMPTOTIC ABOUT THE PRIMES! THERE EXIST A SYMMETRY IN THE ASYMMETRY THAT NOBODY SEES! But you have to understand numbers in 3rd and even 4th dimensional concept (just ascended to the 4th dimension) that most will never even believe can be achieved. But I think you might be crazy enough to catch on! 😅
This is a PRIME opportunity to expand your knowledge far beyond the reaches of conventional boundaries! Do you take the red pill? Do you wish to see the matrix?
Fuk, I sound crazier than you! 🤪🤯
And I already checked my numerical sequences on the online encyclopedia of integers. Nobody to date has submitted these sequences...
Oh my god I love how the quality of the video improved and now they are allowed to film outside of their designated combo class! A little of nature never killed anyone!
Hold on to your hats! Combo Class' mom just let out of the backyard!
… said the man who was bitten by a snake
Actually nature killed me in a horrible accident 10 years ago
Generalizing a little bit, I suppose these are exactly the equations where you never have to carry/roll over a digit.
Actually, I think for ANY expression you can write using only digits, addition, and multiplication; you will always be able to choose a large enough base so that calculating its value won't require you to carry any digits! Which I think means that EVERY expression like that has an equation like this... although they often won't be meaningful in base ten.
These are exactly the equations where the base is greater than the greatest coefficient in the expanded polynomial form.
Thank you for sharing this observation. I was super excited about being able to use solutions to multiplication problems I already know to easily construct polynomial expansions, but quickly found it didn't always work and I had no idea why.
For example, 13*13=169 matches (x+3)²= x²+6x+9, as expected. But 14*14=196 doesn't work, as (x+4)²=x²+8x+16, not x²+9x+6. You can kinda cheese it by saying the 9 is really 8+1 and "donate" the 1 to the 6 to make it 16, but this discrepancy only got bigger and less transparent as I used larger constants.
But lo and behold, I overlooked that the starting equation must be one of those so-called "digit equations" that remains true in every base above or equal to some minimum base B. For example, 12*12=144 in any base 5 or higher; in base 4, 12*12=210 still represents the same _quantity,_ but those specific digits won't work for the expansion of (x+2)².
As you pointed out, at least according to my understanding, it should be the case that for any expression using only strings of digits and multiplication and addition there is some minimum base B, where for all bases ≥B the solution to the expression is represented by the same string of digits.
That's when I had the brainblast. If all expressions have a corresponding "digit equation" when the base is high enough, perhaps 14*14 _would_ encode the coefficients for the polynomial expansion of (x+4)² if I used the right base.
As it turns out, 14*14 is 18g (yes, the digit g) in any base ≥17, making our _expected_ expansion x²+8x+g; g is the 17th digit in base 17, and so by itself represents *_16._* Holy sh*t it, works. The base simply needs to be large enough to represent each coefficient/constant with a _single digit_
For good measure: we know (x+5)²= x²+10x+25. However, 15*15 does not seem to equal 11025 in any base; but what the expanded polynomial is actually telling us about the "digit equation" isn't which specific digits to use, but rather the specific _quantities_ that must be represented by a single digit. I need at least base 26: 1x²+10x+25 tells us that in base 26 or higher 15*15 should always equal the _first_ non-zero digit (1), followed by the _tenth_ non-zero digit (the digit a), followed by the _twenty-fifth_ non-zero digit (p). And if we check, it is true that 15*15=1ap in base 26 and up.
And inversely, while say 28*12=336 in base 10, to write the digit equation now we know we need individual digits for whole quantities up to 8*2=16, i.e. at least base 17. I've found that 28*12=2cg in base 17 or higher, so without doing any work I'm going to guess that (2x+8)(x+2) = 2x² + 12x + 16 (c=12, g=16); and I have just enough experience with polynomials to know that's correct intuitively.
As far as I can tell, this should work generally for polynomials of the form (ax+b)(cx+d), where a, b, c ,d are positive integers, and you choose base B such that B>a*c, >(a*d+b*c), and >b*d. The answer to a||b*c||d in base B should always be the digit equal to a*c, followed by the digit equal to ad+bc, followed by the digit equal to b*d.
One last test. I can expand (8x+2)(2x+3) to 16x²+28x+6. This expansion tells us that in base 29 or higher, 82*23=gs6, or a concatenation of the 16th, 28th and 6th non-zero digits. Double checking online shows I'm right 🎉🎉
Pretty sweet. Now if I ever don't feel like expanding a polynomial manually I can just figure out its corresponding digit equation. And for any polynomial I know the expanded and factored form of, I can very easily produce an interesting, _seemingly_ hard to calculate equation like "In base 29, 82*23=gs6". I'm not that smart to be doing multiplication in base 29 bro 😂 but I can still reach that answer with absolute certainty, and that's FIRE 🔥🔥🔥 LES GO MATH
@@MeNowDealWIthItBrilliant reduction. Though there isn't any functional difference in our interpretations, I resolved these as the equations where each coefficient/constant could be represented by a _single digit._ Obviously, as you said, for that to be true the base must be at least m+1, where m is the greatest coefficient/constant factor of the expanded polynomial.
@@seventoast It follows that this is true of Division and Subtraction if there are no cases where a coefficient "crosses the mag barrier" by decreasing below an exponent of the base. That is, 3-2=1 is true in any base >3, as is 53-21=32 etc etc, but as soon as you have to "carry" a digit it stops being true (10-1=nine, where in any base you must switch nine for "b-1"). Likewise, 4/2=1, 44/2=22. 9999/3=3333 are all true in any base which can make sense of them, but 12/4=3 is not always true; 12/4 crosses B^1 when you calculate it, which is part of the reason it's not always correct.
This is prima facie true because it is the opposite case for the addition/multiplication (ie, if you can write 1+2=3 then you can write 3-2=1), but I've unfortunately given a narrower case which has exceptions:
Although you can write 121/11=11 in any base that understands it, as it follows from the similarly true 11*11=121, writing it contravenes the "crosses the mag barrier" rule. I think there's no easy way to prove that divisions like these will work without knowing the multiplication will work, as what you're conceptually saying when you say that 121/11 is acceptable is that b+1 is a factor b^2+2b+1, which is a harder thing to calculate than expanding (b+1)(b+1).
12/4=3 does not work because 3*4=C; the coefficient you're dividing in 12/4 must be "twelve" and not "12", but I can't see a way to calculate/state that without reasoning via 3*4=C.
As we let the base approach infinity, every number in the eqn consists of a single digit. That will indeed always work for that base (or higher) without carry.
Duh. 😉
Excited to hear more about Base 6! This multi-base video did a great job explaining the thread that links bases together.
6!?
base 720 is way too big for it to be usable.
jan Misali has some great stuff on seximal (what he calls base 6). It's always fun to see more about it, so I'm looking forward to your video on it
Exactly who I thought of, too!
Seximal base is also perfect when using D6 dices with dots on. And Combo class has a lot of those
there's also "a better way to count" which made a response to the seximal thing saying we should use binary
You can take the square roots of numbers in base 2. And you don't have any multiplication tables, its all algorithm.
@@ArchieHalliwell i love that video, especially since i already had a soft spot for binary prior lol
If you abuse the heck out of some notation from computer science, I think you can say that every equation is either O(true) or O(false). i.e. given an equation you can find some integer k such that in every base b > k the equation is either always true or always false.
Computer scientists already abuse the hell out of big O when they say f(x) = O(g(x)) when they mean f(x) ∈ O(g(x)), or even O(f(x)) = O(g(x)) when they mean O(f(x)) ⊆ O(g(x)). That’s much more infuriating to me than what you’re doing.
Absolutely fascinating.
Thanks for explaining it.
It's one of these things that are amazing precisely because they are so so obvious after you've already seen them, and yet which you would've never thought of. It's very very nice
you can actually create an equation that's only true in any finite subset of bases
e.g. for only base 4 and 6:
1. multiply out the polynomial (x - 4)(x - 6) = 0 to get x^2 - 10x + 24 = 0
2. add all the negative terms to one side to get x^2 + 24 = 10x
3. replace all the numbers (base 10) with expressions only using single digits smaller than your lowest base, in this case < 4
i got x^2 + 2 * 2 * 2 * 3 = x + 3 * 3x
4. replace x^2 with xx and write out the hidden 1s to get 1xx + 2 * 2 * 2 * 3 = 1x + 3 * 3x
5. replace all the x's with 0s, and there's your equation: 100 + 2 * 2 * 2 * 3 = 10 + 3 * 30
exercise for the reader: verify that this equation is true in bases 4 and 6
this process can be used to make equations that are true in any base that is the root of a polynomial, aka, algebraic numbers. since transcendental numbers aren't algebraic by definition, equations like these will never exist in any exotic bases like base e or base pi.
also, since this process is reversible, any equation like this either holds true in 0 bases (the polynomial turns out to be like 1 = 0), only finitely many bases (because finite polynomials have a finite number of solutions), or holds true in all but finitely many bases (because the polynomial turns out to be 0, which equals 0 for any base. it only doesn't work in bases which don't include the digits used in the equation)
whenever i see some nonsensical equation, i usually use this process in reverse to work out a polynomial that the base is a root of, solve it, and then ask the person who posted it "is this in base __?"
So I'm following, this implies that there are some equations that could exist with base e but not some finite base? And visa versa
yayaya!!! i found this a long time ago, all squares of the same-looking numbers aventually converge on a single representation, base 10 actually does this for 13^2, it goes (starting from base 4) 301 224 213 202 171 170 169 169 169 169...
Just watched this with my 11yr old son and he was definitely vibing with the concept of different bases.
My 50 y.o. friend recently started talking to me, that when he was a teenager he was thinking about negative and fractional bases and trying to talk about it in class at school - the class wasnt happy about thinking :) and adding anymore. He ended up as a data recovery specialist multiplying hexadecimals for living.
11 yos who love different bases unite!
I have always wished we as a society ended up with Base 12 instead of Base 10.
For a transition to happen is necessary all of us base twelvist to come up with different digits and names for those new digits
base 2 🙌
@@tafazziReadChannelDescription They've already been named as dec, el, and do. A 180 degree rotated 2 and 3 are the best proposed symbols for dec and el, sometimes dec is an X from Roman numerals, but that becomes a mess with established variables.
Many societies had a base 6 by 10 system, and even Britain had a 120 system for money for a long time. Ours is really more 10 by 10, because we use the 25 times table for a lot of things; for example, most countries use a quarter of some sort, which doesn't work with base 10, since then it would be a 20 unit coin. 120 is better than 100, because it's divisible by 2, 3, 4, 5, 6, and 8. Sadly, it didn't happen, so we have to deal with the annoying thirds all the time that never fit in well with our numbers.
@@litigioussociety4249 if you add 3 digits to base 10 you get base 13
Hey, I think an episode that examines each “angel number” or threepeated digit mathematically would be really cool!
Something that feels so wrong when first being introduced is concatenating the elements of the rows of Pascal's triangle. These give 1, 11, 121, 1331, and 14641 for rows 0 through 4. For row r, the concatenation results in 11^r. This feels so wrong, but consider that the elements of Pascal's triangle are the coefficients of the expansions of binomial powers (a + b)^x = (x choose 0)a^x + (x choose 1)a^(x-1)b^1 + ... + (x choose x-1)(a^1)b^(x-1) + (x choose x)b^x. What do you get if you substitute a = 10 and b = 1? Well, then you get (x choose 0)10^x + (x choose 1)10^(x-1) + ... (x choose x-1)10 + (x choose x), which exactly reflects the positional notation for base 10, whatever "10" happens to equal. As long as x choose y < 10, meaning that every coefficient can be represented as a single digit, this trick of concatenation will work regardless of the base. I stopped at row 4, since it's the last row that doesn't need a second digit in base ten for any coefficient. If we were in base eleven or higher however, we could continue to row 5 as 15AA51.
WAKE UP!
-ITS DA 1ST OF ZA MONTH!-
COMBO CLASS UPLOADED
Let's gooooo, new combo class video!!
"Babe, wake up, that guy who like setting math on fire just uploaded"
Great videos and I even understand some parts of some of them. Glad I'm not in charge of the risk assessment.
Man, I was going nuts thinking “they can't be wrong!” hoping you'd point that out, until finally I got my wish. It's a fun way to put it. In a basic base, these equations are either right or not even wrong.
It's fuckin' wild that this show works
ps that's a compliment
0:36 Personaly I have genuinely moved to using the joints + fingertips which gives 16 on one hand. You just hold the tip of your thumb to wherever you are in the count. If you really need to you can try using your second hand for a second base 16 digit since it is easy to hold your place, but I just get something to write with if that would be needed
This is one of the best math channels on RUclips
This is such a different take on such rudimentary things like basic arithmetic. I also feel a good bit of mind blown.
Wow, the views of nature look wonderful! I like them a lot. Nature is the best thing a modern person could ask for.
One of the most cool moments in my life is when walking in some beautiful places near my hometown here in Siberia.
Wow - you're from siberia - but from tajga? or tundra? Are there any polish "repatrians" that were sent there hundred of years ago?
@@TymexComputing I'm from Novosibirsk. There's mixed forest here. The weather is not bad enough to use it as a punishment so I believe all that people were much close to the North. It's -30°C..+30°C most of the time here and the air is pretty dry.
You deserve way more subscribers.
Loving the quirkiness, and the nature in and of the videos!
You're really doing a great job, brother!🌞
Thank you again sir.
the part about the positional numeral systems being equal to the base to incremental powers reminded me of the standard notation of polynomials with a single variable and now im interested in the connection between polynomials and numeral bases, and extrapolating from that the connection between logarithms and numeral bases. im sure theres a connection between e / natural logs and numeral systems by way of this property. ill have to work it out. very cool stuff
0:45 I prefer to use the thumb as an opperator, but its quite fun to see how numbers wrap around your knuckles in base 10
My favorite is 11² = 121 in any base greater than base 2.
My favoroute is not mentioned here - its 0+0=0 - you dont even need a base for that equation :)
Essentially, if an equation doesn’t work in a base it’s because one of the coefficients of the powers of the base in the expansion meets or exceeds the value of the base. Then it will spill over into the next digit and change the sum/product.
Basically, if you could do the calculation by hand without having to “carry over” any digits, then it will work in any higher base.
Very cool vid in many ways! These thought experiments are a challenge to the common mindset and education. Love it!
Why do I have a feeling that whenever I watch videos from this channel some hungry people will be talking about 20th of April?
BTW. Thanks for the video.
Actually, this is kind of tought in high school, since this connection between positional bases and polynomials is exactly why and how long division of polynomials works. Once you know how long division works, you can basically think of a polynomial as a number in a mysterious base that can have infinitely many digits in each position and immediatelly you know how to divide two polynomials using long division technique.
Tell your cats theyre the stars of the show! They are so cute and fluffy 😻😻😻
That's why it's valid to think of floor(log_n(x)) as The number of digits x would have in base n
New Combo Class vid!!... woohoo, it's the same feeling as before taco-night or something.. some nice feast - starting vid now
Picturesque horizons and no fire - great for me!
I dont have ten fingers (not even twenty) but my favourite base is the primal base :) , unique representation - such an easy peasy multiplication it has :) but addition is pain in the arrest
This video changed how I do mental math
I love it when we get to talk about my favorite base, base 10
base 10 your favorite base?! so basic and non quirky, mine is base 7 (septenary)
@@guillermo3412 septenary is base 10
@@theneoreformationist septenary literally means base 7 which means you’re utterly wrong.
@@guillermo3412 If you were using septenary, the symbol "7" is meaningless. The correct name for the base number is 10.
@@theneoreformationist i see what youre doing and again, youre wrong, septenary refers to 7 in base 10, you cant use the number expressed in the base itself to represent the base youre using, because it could literally be any number and its not useful information at all, thats why we universally use base 10 to express the base of a number.
finally, i'm on time for class!
I would like to discuss more on why you think 6 is the best base for humans. I've been researching bases and my comment on this is that don't think bases, think ratios then you will transpose them better.
What I wonder about is how "bases" might change in behavior if you were to change the operations used on them . Both for the different positions as well as the operations to combine them.
Chicken
Chicken! :D
This is pretty amazing
I am INCREDIBLY curious about the inverse... Are there any equations which are ONLY true in one base? I'm trying to think of some and I can't figure any out, but admittedly, I'm only working with whole numbers here.
this is so cool! i wonder if there’s a relation between polynomial division as well
I think base 12 is best. 12 is divisible by 2, 3, 4, and 6, while 10 is only divisible by 2 and 5. It's why I think feet/inches and Fahrenheit are better than meters and Celsius.
Watching any video on this channel equals to touching grass at least twice
I feel like natural bases would make a better name for bases using the natural numbers (1 to infinity). Then integer bases could include bases with integers, rational would include rational bases, real would include real bases, and complex would include complex bases.
if your base is larger than the square of the largest digit used, then all multiplications will be the same as a polynomial multiplication with non negative integer coefficients
so if you are looking at decimal, any multiplication involving numbers whose digits are only 0,1,2,3 will work in all higher bases
this is a stronger condition than you need. really it is about nonnegative integer coefficient polynomial multiplication in which none of the coefficients in either the factors or the quotients will be as big as the base you are using
Honestly, I'd like to see if there are any equations that don't hold true in any base, and if so, why. Also, what happens when you do include those non-basic bases, like complex or irrational bases? Do any equations hold true in those? Could make a cool video
Your videos are so cool
Oh shit, this guy again.
i just realized
if you have a base n system, and write base n in that number system, it'll always be base 10
jan misali baseless base names moment
In base 3 for 12x12 is 5x5 which is 25 in base 10 or 221 in base 3
I feel like there ought to be opportunity for proof by induction here to find other types of equations that Just Work.
This makes me wonder if there are any equations which are true in a number of bases other than 0, 1, or infinite
Any addition or multiplication equation which doesn't require carrying will be true in any base which uses all digits which appear, but any which do require carrying will be incorrect in other bases.
For example, 5 + 5 = 10 isn't a nonsense equation in any base greater than six, but it's only true in base ten.
If S, R and T are digit sequences such that SR=T in some base, the same is true for higher bases as long as you didn't do any carrying over in yout calculations.
16×2=32 is not always true, because when you do 6×2 you carry the 1 and add it to 1×2.
More subtly, 6×2=12 also has some carrying, so not always true.
19×11=209 is also not always true, becquse you get 99+110, but then you have to carry the 1 from 9+1.
Watch out for ticks my guy
I guess the equation where a single digit n + 1 equals 10 defines the base because that n is the base - 1. So for base ten, you have 1+9=10 which is only true in base ten. And 1+1=10 is only true in base two, 1+2=10 is only true in base three, etc.
I guess if you're trying to communicate what basic base you're using and you only have defined what + means and what 1 and 0 means and how positional systems work, you can define all your digit symbols as well as what your base is by showing:
1+1=k
1+k=q
1+q=p
1+p=10
which implies that it's a base 5 system and the digits are 0,1,k,q,p
i wanna hear more about what these continuous spectrums are/would be
1+1 = 2 in every integer base above base 3
My tech friend can easily count up to 8191 on one hand, over 64m on both hands, very crazy, especially since he can run calculations with it.
Are you acknowledging base 64000000
Where are you? Looks like a nice place.
oh god he's free
But surely we dont count in base 10 because we have 10 fingers... There are 10 symbols in base 10 (0,1,2,3,4,5,67,8,9), but on our hands we can make 11 different "symbols" (no fingers, one finger, ..., nine fingers, ten fingers). So surely we should be using base 11?
There wouldn't be a zero, we don't acknowledge it these days much, but the jump from "natural numbers" to "abstract numbers" that can't be physically represented in the conventional way (How do you have 0 of something? How do you have -3 of something? etc...) is quite large. All cultures we're aware of had to first define a number system and only then, after indulging in mathematics for a while, would they realise that 0 too is a number - hence why all counting systems, be it our base 10, sumerian base 60, mesoamerican bases 5 and 20 and so on..., seem to be "off by one"
You can even see this in the way people use their fingers today, if you ask a random person to show you that they have zero of something, they are more likely to show you an empty hand or shrug rather than curl their hand up in a fist (though it gets exponentially more likely if you ask for a natural number first, after that they'll just subtract the fingers)
@@asd-wd5bj it's quite easy to have zero of something. I have zero maidens
The settings 😂
I want to live wherever those chickens are.
#metoo !
good choice..
By increasing the number of multiplications one by one, we can see how many numbers aliens use.
Can't we use it to make some big number multiplication faster ?
ah, so you watched that jan misali video too
7:50 why does it look so post apocalyptic
givin Lil B a run for their money for Based God of the Bay
Have a good 0x6650-0000 !
Car butt scratches= like
Once I invented a base 40, 40 digits. Just rotate each digits, add them some accents or double line ( example 1 with accent ecc. ) 🤣
I feel like those polynomials may have something to do with modern factorization methods
So you are telling me 11*11+10=121+10=201 is the first thing we should send to aliens? They would automatically recognize all the symboles and start sending us base 3 maths stuff.
Wouldn't that be true of just about every basic type of equation? If you use characteristic polynomials, for example, 56*49=(5b+6)(4b+9)=20b^2+69b+54.
In base seventy and above, that'll always be the same. It doesn't work below that, so for our base, the answer is different from other bases. But if you have a symbol for sixty nine, then it'll be the same as every other base with that symbol.
i love my algorithm
💗 + 💘 = 🥰
Huh I never noticed how nicely symmetrical squaring 111 is, works all the way up to 1111111111. Sadly loses the symmetry after that.
Seximal's factors are great, but binary has a bunch of mostly-unique (perhaps matched by balanced ternary) features that seximal can't beat. Times tables are strictly unnecessary in binary, for instance. Binary's biggest problem, of course, is compression; a bit stores so much less information than a digit, so binary numbers take up so much space on a page and so much time to say.
I've been tinkering with quaternary to see if we can get the benefits of binary without sacrificing compactness.
binary digits only take 1 pixel (lit or not) on the screen :)
By that logic, each decimal digit could be represented by a different color.
@@MeNowDealWIthIt decimal, hexadecimal, 2^24-mal and there are even displays with 10bit Lookup tables for each colour :)
That's pretty _based,_ however it'd be nice if you knew of a way to remove all the even or odd bits of a number in base 2, or could construct the Moser-de Bruijn sequence in closed form (and solve for said closed form for me lololol) without the use of XOR or other such "inner" operations, only the operations of +,-,*,/,exp,ln, special functions, and limits, but not integrals.
Bro you just fixed the most annoying equality: the link between the powers of 11 and the Pascal triangle.
Btw this is the pascal triangle :
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
And this is the powers of 11:
1
11
121
1331
14641
161051
Because of our base 10 the digit of the powers of 11 overlap.
But if we consider a base b which is great enough, we 've fixed it!!!!
We can even proove that!!
11 in a base b represent b + 1
so 11^n is (b + 1)^n which is given by the Newton's binominal which is based on the Pascal triangle!!!!!!!!
Erratum : The n line of Pascal triangle correspond to the n-1 power of n
Dimitri ZIN-URU
Makes me wonder what graphs stay the same in a triangular coordinate system, probably none tho.
1-2=-1
19:23 were you in any way influenced by jan Misali in that opinion or did you two just coincidentally come to the same conclusion?
They probably just converged, it's pretty much universally acknowledged in math circles that, as far as humans specifically are concerned, only 6 and 12 really make sense as "superior" bases due to being in the "sweet spot" of being neither too large nor too small and due to having a lot of divisors. Whether you prefer one or the other is a preference thing really - do you value the convenience of fewer symbols more or the density of a larger base?
very col
Base 2 is better than base 6
Source: "the best way to count" by the best way to count
That's not a joke btw... Dude created a channel just to settle this debate
It seems like these equations will hold until you get a carry.
13*14=182 is not true for all bases. In fact, it is only true in base 10! The key point here this is true if and only if there are no carries in the calculation.
Incorrect, this is not just true in base 10! Because it is true in base 10, also I don't believe it's true in base 3628800, seems hard to believe that it would be true in such a base