Why the Number 1,000,000 Should Be 1,081,080
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- Опубликовано: 5 окт 2024
- What do clocks, calendars, and eggs have to do with the number 1,081,080? Get ready to learn about an exclusive family of special numbers....
Stay tuned for next class when we'll prove things about prime numbers and infinity!
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Combo Class, taught by Domotro, is an unconventional learning experience where anybody (whether they're a fan of normal school or not) can become excited to learn rare things about math, science, language, and more. Also check out the shorter videos on the Combo Class Shorts channel (and TikTok page). Thanks for coming to Combo Class!
DISCLAIMER: The burning clock in the intro (and any other uses of fire, tools, or science experiments in this series) was done in a professional and safe way. Don't try to copy anything you see in this series yourself.
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I love you jack harrlow
I can not tell you how much I love that there is enough overlap between people who watch content like this and those who like rap in general and at least know who jack Harlow is to make this the top comment lmao
@@monhi64 I agree
@@monhi64 Last time I checked rap is the most popular genre in the US and Jack Harlow has made several hit songs
Better Jack Harlow.
@@Omlet221rap is not that popular
Can we appreciate him writing "Highly Composite Numbers" while simultaneously explaining some context verbally?
Had a school acquaintance demonstrate saying a sentence while writing a completely different one and my mind was blown. You don't realize how hard that is to do until you try it.
Like numberphile writing out a 20 digit number.
Yeah right! That was insane
Its really cool I think you could achieve it reasonably with practice
Timestamp?
@@2P9PR 2:54
Admittedly, he pauses to explain some things before writing "Numbers", but he still did it for most of it so it was cool.
The reason for all the duplicated numbers is that 1001 is 11 * 13 * 7 which adds a lot of potential new combinations.
I was searching for exactly this comment.
First thought was it to be some cousin of 11 like 101 or 1001 which it is.
and 10010 is 11*13*7*5*2 which is really good too
You’re incredible Dimitri!! I love this!!
Thanks! More coming soon :)
Your tetration video was the first one of yours that I saw. When you brought up factorial, I immediately thought, is there a higher and lower operation than factorial? I don't think there is. Most of what I found just used sigma.
Either way, I enjoy the detail that you go into. It forces me to wonder what happen if ___. You'll have a Practical Million subscribers in no time!
I just watched your video about primes. Once you said the word "factorial," I knew you had to be answering what google didn't show me in my searches. lol
There's double factorial, counterintuitively named
cool that’s also my first vid of his!
@Chip Wiseman arent these the triangle numbers
@@muskyoxeseah, it should have been called half factorial
I can't help but smile while watching this. Your love of math really speaks in this video. This is the kind of attitude we need for teaching maths!
I think what kills school teachers is that they have to explain the same thing over and over again and they are also constrained on what they must teach. Being on youtube on the other hand gives you much more freedom on what YOU want to present, which also means you are more than likely excited to talk about it
@@MCreeper-eg9xy Oh yea, definitely! This is more of a systematic problem than a problem with individual teachers. I've had lots of inspirational teachers who loved math but were stuck teaching the same dry, boring material year after year. It might have been more accurate to say, "This is the kind of attitude our system needs for teaching maths!"
I originally thought this was a numberphile video because of the enthusiastic looking man with a marker in the thumbnail and a very interesting title that seems to make no sense lol
I like 45045, like an "odd highly composite", has a sweet palindromic binary form, and can just bit shift to get to 720720. (Of course, 7*11*13=1001 is how we get clones)
10:31 that exclamation mark after the 9 threw me theough the loop for a good 20 seconds thinking it meant 9 factorial and I was trying to figure out how that could possibly make sense, you gotta be careful with that especially after just talking about factorials and such in the video
I’m guessing all the “clones” in the later Highly Composite Numbers are because they all have 11 as one of their many factors. With the lower ones, it looks like every highly composite number except 1 is divisible by 2, every highly composite number starting with 6 is divisible by 3, highly composite numbers starting with 60 are divisible by 5, and 840 is where they start being divisible by 7, so it makes sense that 11 joins later on.
Yes for 11 and also 7x11x13 = 1001 might be a factor (ha!) in why the pattern/clone numbers are highly composite.
@@Rack979 Came here to say this.
Good point! I didn’t even consider 1,001, but those numbers look a lot more like multiples of 1,001 than just 11, so you’re probably right!
When you said “for some reason [4324320] is superior” I laughed out loud. Very nice
I'm getting back into all the math and STEM fun that I used to love as a kid, and you're becoming such an inspiration to me on this path! I can tell you're having a blast with it too, a like a new Bill Nye!
There actually exists a duodecimal system with the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, dek, el and do. This would be way more useful to modern-day mathematics, because fractions in this system are more often visually appealing and easier to work with. For example: 1/3 = 0.4, 1/4 = 0.3 and 1/6 = 0.2. These fractions are 0.3333..., 0.25 and 0.1666... in our decimal system, which is way more inconvenient.
There's also base six, officially named senary but also called seximal. It's got a lot of the same benefits of dozenal, while requiring no new symbols and having better representations for a fifth (0.11111...) and a seventh (0.05050505....) than the corresponding ones in dozenal (0.24972497..., 0.186X35186X35...). But it does have a downside of having numbers get long somewhat quickly.
it's far easier to just work with fractions instead of converting them to numerical form
Hey, I love your cheerfulness, and the fact that when you don't know something you nonchalantly admit it.
Great video!
A tip for the videographer (who is doing a great job btw) and for you, would be to make sure the sun is not behind you or in front of the camera. It looks like you lost a fair bit of contrast.
Now, I bought a polarizing filter, and a mattebox to solve this for when I had to put my subjects between me and the sun, and that works really well!
Looking forward seeing more of your stuff!
this channel is something really special. can't wait to see you get bigger! the music feels really nostalgic for some reason, and makes me happy :)
The title had my curiosity. The information/content within had my subscription. Very well done
The only highly composite number is zero, since it has infinite factors. Every larger number has fewer factors.
My favorite quirk of numbers is talking with people who think Base 10 is somehow better than other Bases. Like the only reason Base 10 means Base (9+1) is because we all agree that it does. Every base system would call itself Base 10. Count using 6? That's now Base 10. Counting using 100? 100 is now Base 10. So next time someone says Base [anything with more than 1 digit] poke a little fun at how ambiguous their phrasing is.
2520 is a nice number since it's the smallest number that can be divided by all numbers up to 10
27720 is also nice since it has the same property but goes up to 12
I wonder if there's a name for numbers whose factors can form a sequence from 1 to n. They're very factorial-esque.
The old pre decimal British pound used to have 240 pence to the pound. 240 being one of your numbers and highly divisible.
I sugest the 'Equaly Highy Composite Numbers' this will include all the Highly Composite Numbers and all the numbers that have the same amout of divisible factors than the last Highly Composite Numbers. 3, 8 and 16 will be the first of these, We could also call these 'Strictly Highy Composite Numbers' by just takeing the away the Highly COmposite Numbers from the list
I also though about ' factor dense numbers '
This highly composite numbers' video was a gem! Thank you very much for the mathematical enthusiasm, your channel is the most mathematical creative one I've seen so far. I hope to watch more videos where you show the beauty in math for us starving for it.
The Practical Million makes the whole video worthwhile.
"What do all these items have in common?"
me: they're all measurment tools. Measure time, measure weeks, measure length, measure eggs
"They all have the number 12"
me: oh yeah totally I was gonna say that
You're not that far off though!
we liked using 12s to measure things because it was divisible, so 12 worked it's way into a lot of early measurements
my guess was that they were all related to time
I think it's interesting that most fair dice (aside from the infinite dihedral families) have a number of faces which is a highly composite number. The exceptions being the d8, d20, and d30, and the lack of a d36 existing
8, 20, and 30 are part of the "largely composite" family, where the number of divisors are greater than or equal to, rather than just greater than the numbers less than them that the highly composite numbers abide by.
I wonder if you were to sort all the natural numbers by this formula (numberOfFactors(N) / N) what that list would look like. I wonder if you could prove what the Nth element was in that list without looking at all of the infinite possible natural numbers first.
Call your formula f(n). There is no Nth member, because for any number n, there are infinitely many m_i where f(m_i) > f(n). As n goes up, there will always be numbers n with larger and larger f(n). For example, if we take numbers 1..120, the leaders are: 120:3 60:14/5 72:65/24 84:8/3 96:21/8 90:13/5 108:70/27 48:31/12 36:91/36 24:5/2. If we go up to 1000: 840:24/7 720:403/120 360:13/4 420:16/5 960:127/40 480:63/20 900:2821/900 540:28/9 600:31/10 240:31/10 504:65/21. If we go up to 10000: 5040:403/105 7560:80/21 9240:288/77 2520:26/7 7920:403/110 8400:1922/525 6720:127/35 9360:217/60 3360:18/5. If we go up to 1081080: 720720:248/55 1081080:640/143 831600:15376/3465 942480:29016/6545 1053360:6448/1463 997920:22/5.
2:08 Yes but then it could get a bit confusing when someone says “three tenths past/till” some hour. Memorizing all the possible fractions would be difficult when there are many ways to divide the hour. If, for example, I saw “14.42” on a clock I might not immediately read it as “three tenths till three.”
I do like using "a third past" or "a third to" though, mostly just to see people's reaction.
That's so cool! So that means Tau=6! degrees which is superior to 6!/2
Yooo this video just blew my mind
Found this channel from a RUclips short you posted, you're doing awesome work.
This feels like a combination of explosions and fire and vsauce 2
I think you may have shown why the metric system is not the most desirable system for many applications.
For me, the biggest pro of SI is the fact all of the units are experimentally derived (though US Customary are legally defined in terms of SI units, and thus physical constants, now). Order of magnitude prefixes are just shorthand for scientific notation, so they aren't really as big of an advantage as is often touted.
Another thing is that scientists have to deal with floating point precision, so in the end base 2 reigns supreme.
Yes! Spread the good word of dozenal/doudecimal!
"Hey, can I please buy exactly 1,260 grains of rice?"
I never knew I needed this but this is actually highly practical.
I can't wait to see how this channel grows. New subscriber!
Great show, always loved 7!
This video is the worthy successor to Numberphile's base 12 video
I really love this channel
I don’t know if it was intentional to shoot into the sun but it totally works for this channel and ONLY this channel lol
720720, 1081080, 1441440, 2162160, those are all video resolutions with the first three digits repeated!
So cool, I wish my teachers made math this interesting when I was in school
12 and 60 are dicisible by 2,3,4,5(for 60) and 6.
5:00 lmao that ugly 12 cracked me up
if you look up what superior highly compossite numbers are. you'll get a very technical definition which doesn't make much sense. so, as i was comparing them i realised shcn are those numbers whose divisors are a highly composite number example, the number of divisors for 12 is 6 which is highly composite. idk if it's an exact rule but it's fairly accurate.
I propose that we call these anti-prime numbers, Amibote Number. Yes, the name is intentionally similar to Amicable Numbers which comes in pairs.
A bit of Classical Latin, "Amabo te" is dirrectly translate as "please", although it is less ambiguous than the English phrase "please". It has additional meaning of pleasing. So, Amibote Numbers are Pleasing Numbers! In the event that you use these numbers, you ("te") will love it ("ami" or genitive "ama") and it is good for you ("bonum").
If anyone cares what the equivalent antiprimes would be when accounting for the same number of divisors:
10 -> 6
100 -> 36
1,000 ->120
10,000 -> 360 (1 less divisor than 10,000)
100,000 -> 1,260
1,000,000 -> 2,520 (1 less divisor than 1,000,000)
I think if you start throwing "largely composite numbers" into the mix I think one could potentially come up with a system that divides much nicer into the next orders of magnitude upward.
10⁷ -> 7560
10⁸ -> 15120
10⁹ -> 45360
10¹⁰ -> 55440
I shall propose a number system based on 24 called tetraseximal (Sub base of 8 )
10 -24 (24×1)
20-48 (24×2)
100-576 ( 24×24)
1000-1728 ( 72×24)
10000 - 8640 ( 360 x 24 )
Basically this system is more of Roman numeral style where instead of exponents it's more of an addition subtraction thingy hence the lack of uniformity. It goes like this in our base 10 system
24 , 48 , 576 , 1728 ,8640 , 13824 , 46456 ... and then back to 24 ^n to the beyond
Here's a couple of examples
To write 33 (DEC ) in base 24 (TSM).
All you need to do is find if it's close enough to 24 and no more than 8 spaces away from 24 .
Now 33 is 9 spaces away , so this sub base of 8 comes in handy and it's right next to 8×4 so this is how how write 33 in this system
(8×4)+1 =33 ,so everything from one to 0-7 is written as the same and from 8-15 is written with the same numbers from 1 to 7 but with a dash above them and 16 -23 is written with 2 dots
_
So 33 is 1 ×4 +1 = . .
11
Amazing video man
I like your inspiring videos!
13: one shall rise and one shall fall
14: the one that shall fall is you, PRIME!
What a stoatally awesome calendar
Time to crack open that dusty old box of primorials in my attic. I'm not exactly sure what they'll do, but they seem about right.
we love highly composite numbers! such neat and handy lil guys
I'd be curious to see what these numbers look like in a duodecimal system. Whether the patterns start to make more sense...
Why not convert them yourself?
@@michaelcherokee8906 I have, and they do, but not many people watching me.
@@jamescarruthers1967 Even in context, that sentence was nearly unintelligible.
@@michaelcherokee8906 I have converted them myself, and the patterns are quite interesting / seem less "random", but no-one is watching me, I don't have a RUclips channel.
@@michaelcherokee8906 idk what you're on about, the sentence is fine in context, chill out
These videos always amaze me, keep doing these
I recently decided I was going to express time in scores. 4:40 would be 4 'n' 2 score. Though now I'm not sure if you should say "2 score" or "2 scores" but I think the prior sounds better.
In the vein of the practical thousand being highly divisible, but not an HCN, I propose 96 be a practical hundred.
96 or 108 could be good practically hundreds, in slightly different ways. Although I think we can stretch the idea of "hundred" enough that the super-divisible 120 can be the practical hundred, despite being a bit further
Cant wait for this channel to well and truly blow up.
1,081,080 should be a baker's million.
5:21 FEWER numbers, not less :)
6:53 Perfect NUMBER of people
These grammatical rules are outdated, nowadays it is okay to say less on countable stuff.
Less cars
Less problems
Less words
Less challenges
Less fans
I like 720720 = 7! • 11 • 13
Made me see that 1001 is 7•11•13 which explains why the highly composite numbers around a million all look like clones, because 1001 just contains a succession of primes - or I'm looking too much into this
when asking for wrench sizes gonna start saying like 6/12ths
Whew! You got the gift of gab! I loved
your dancing me through all that math; that was fun and instructive! Sign me up for combo class!
This channel makes the math area of my brain spin fast
Why did he have to look in his notes to write down the factorials though?😅
as soon as I looked at the description and title, I thought he was gonna talk about Base 12. But nope, Highly Composite Numbers (anti-primes).
I just can't repeat that 12 eggs falling down moment! 0:49
9:29
Correction:
10,080! is not a highly composite number.
Incredible. Johnny Carson of mathematics!
I had to look twice before I realised that this is not Explosions&Fire
Your "practical million" is pretty close to what one might call a baker's million, if you will. 1000000×13÷12 = 1083333.3333333333.
excellent video. very fun numbers!
If evolution had contrived to give us six fingers on each hand we would likely have adopted base 12 as our everyday number system.
I like to imagine that humanity would be a type 3 civilization by now if that was the case.
Apparently one or more of our ancient civilisations used base 60 for counting, using knuckles rather than fingers. I think it was the baylonians or sumarians but can't remember off the top of my head.
"Anti Prime" - missed opportunity for a Megatron pun. 😁
This was very cool in a weird way
It would be more compelling to also write them in their most appropriate numeral base.
I think I like "nice thousand" and "nice million" for these.
Actually maybe "neat million" feels just slightly more measurement-like.
About 180, you forgot its importance in geometry with respect to angle measurements of all triangles (vertices, that is).
Every time I see these type of exercises, I want to see them in binary.
0:29
No we didn't decide 12 was important. The Romans did. They probably got it from the Etruscans or the lucanians since they still counted in base 10. The Babylons decided that 60 was an important number that's why we have 60 seconds in a minute and 60 minutes in an hour, and why we have 360 degrees in a circle and calendars used to have 360 days a year before Julius Caesar standardized it.
??? He clearly was talking about the human race when he said "we."
And he was going to say "we decided that 12 eggs [fit good in a carton?]" or something like that, not that 12 was important.
@@mickeyrube6623 that's my point. [We] who made the decision was the Romans and the use of the number 12 in European culture is just a vestige of their empire. Like why breaking mirrors gives 7 years bad luck, or why we call the monthly money we make a salary while if you are paid per week it's called wages, or why people used to measure distance in miles, leagues and feet before the metric system. It's kind of like if you travel to East Asia the number 4 is seen as extremely unlucky. It's because of the influence of the Chinese empire. [We] didn't "decide" that it's just a vestige of history.
@@tnk4me4 ???? Your first comment literally says "No we didn't decide 12 was important. The Romans did."
What do you mean when you said "No?"
@@mickeyrube6623 Dude I'm not even sure about what you're confused about here. My comment was just me voicing that for more than 4 billion people the number 12 isn't as special as it is to western society and that it is only because of Roman imperialism that 12 is important at all.
@@tnk4me4 It was your timestamp. It's at 29 seconds. He doesn't say "we decided" until he starts talking about the carton of eggs, so I thought that was your only problem with the video. He doesn't even finish the sentence, so I thought you were being ridiculous. You should have put the timestamp at 28 secs.
Your argument makes no sense anyways.
When he says "we decided to divide the year into 12 months" it's because we have. We could only mean a few things here.
1. Literally him and at least one other person.
When you say "no, we didn't decide," are you literally saying you and the RUclipsr didn't decide that? I think not. (I hope not!)
2. He means the whole human race. This is true. The entire human race has decided to officially adopt some form of calendar that has 12 months.
If you bring up some tribes in the Amazon or some shit I swear I will find you.
3. He means western civilization, which includes him, his culture, and basically ever one who is watching the vid who speaks English.
2 and 3 are interchangeable.
When he says "decided" he mean accepted as basic reality, or or come to an agreement that that is what we are going to go with.
If a philosophy professor said in a classroom "we decided that murder is immoral" would you say "No, we didn't decide that. The Sumerians decided that."
If you would, then I'm sorry. Either English is not your first language, you are an idiot, or you are being a pendantic troll.
Yes, those Number are so interesting : )
Very fun video
Is there a symbol for lcmUpTo n = lcm [1..n] ? E.g. lcmUpTo 10 = 2³ × 3² × 5 × 7 = 2520?
Is there a way to write an expession for composit number divisible by all numbers smaller than and including "n".
C(1)=1
C(2)=2
C(3)=6
C(4)=12
C(5)=60
C(6)=60
C(7)=420
C(8)=840
C(9)=2520
C(10)=2520
Sugested name for these numbers:
Continuous Composite Numbers.
They are divisible by any number in their *continuous* sequence of positive integer.
While the first in the sequence are Superior Highly Composit Numbers, they start to pick up other composite numbers.
the n'th such number is the least common multiple of 1, 2, 3, ... up to n, so the expression for them is C(n) = lcm(1, 2, 3, ..., n)
@@schweinmachtbree1013 off course, thank you! Apparently a few years since I took Calculus... Now I just feel stupid .
@@thorbjrnhellehaven5766 You shouldn't feel stupid! When you were thinking of an "expression" for C(n) you were probably thinking of some formula for C(n) involving +, −, ×, ÷, powers, roots, maybe logarithms and exponentials and so on, but it is all too easy to forget - and most of us do forget - that non-smooth functions are perfectly valid functions too, even though we don't have buttons on our calculators for them. for example:
- |n| and sign(n);
- min(a,b), max(a,b);
- gcd(a,b), lcm(a,b);
- σ_0(n), the number of divisors of n;
- σ_1(n), the sum of the divisors of n;
- v_p(n), the exponent of the prime p in the prime factorisation of n;
- ω(n) and Ω(n), the number of prime factors of n not counting or counting multiplicity, respectively;
- φ(n), the number of positive integers ≤ n that are coprime to n;
- π(n), the number of primes ≤ n;
- p(n), the number of ways of ways of partitioning n, i.e. writing n as a sum of positive integers (ignoring the order of the summands);
- r_2(n), the number of ways of writing n as a sum of two squares (of integers, and not ignoring the order of the summands);
- r_k(n), the number of ways of writing n as a sum of k squares (of integers, not ignoring order)
(I had to look on wikipedia towards the end there xD)
Your sequence C(n) is in the online encyclopedia of integer sequences (OEIS) and has garnered quite a lot of interest. For example some of the simpler-to-state things about it are that 2^n ≤ C(n) ≤ 4^n, so its growth rate is exponential, and the value of C(n) only changes at prime powers (values n = p^k for a prime p and a natural number k): from the values you calculated you can see that it changes value at 2, 3, 4=2^2, 5, 7, 8=2^3, and 9=3^2, and that it doesn't change at the non-prime-powers 6=2·3 and 10=2·5. It follows that there are arbitrarily long streaks where C(n) does not change value (as there are arbitrarily long streaks of non-prime-powers) and that the longest streak with all different values is the initial streak C(n) = 1, 2, 6, 12, 60 (as the longest streak of prime powers is 1, 2, 3, 4, 5) :D
@@schweinmachtbree1013 don't worry, it was more like feeling stupidily embarrassed with a heavy face palm to myself, then quickly getting over it and moving on , 🙂 again thank you 👍
Surprised you didn’t mention that 11 and 12 get special names. And are not called oneteen and twoteen.
this is so cool! i have to know why the copies are there though…
10:00 long thousand is 2×2×2×2×3×5×5 but 1260 is only 2×2×3×3×5×7 so I think long thousand has more divisibility(?) than 1260
Losing two 2s and one 5 for one 3 and one 7 is bad deal (7 can do well with weeks sure but how about weekdays it's 5)
It helps a lot that 1260 has one more unique prime divisor (4 instead of 3). The number of divisors of 1200= a^4 x b^1 x c^2 , add 1 to each exponent, resulting in 5 x 2 x 3 divisors = 30. 1260 = a^2 x b^2 x c^1 x d^1 gives 3 x 3 x 2 x 2 = 36 divisors. 1200 could never be highly composite because it has more factor 5 than factor 3. After exchanging a 5 by a 3 you get the number 720 with also 30 divisors.
Highly Composite Numers... you could just multiply primes, right?
6:44 Plato was not worrying about dividing up votes
Explain
@@honeytubs Plato was not a democracy enjoyer
The best number is 0, because it can be divided by anything, except zero
hi! i think you wrote a number towards the end of the list wrong, because if as you said all of these are divisible by 9, 17297290 should not be on the list. very interesting video regardless, thank you for making this.
edit: okay i checked the list and turns out you only have a digit wrong, it should be 17297280.
Thanks for noticing that. I must have written a digit wrong by accident. I’ll add a correction in the video description
So... is 12 now a "long ten"?
12 already has a nickname ("dozen") but sure, I like "long ten" too lol
This is cool!
That was awesome
I would live easily with my internet access of 1 PMb/s - practical million per second :)
The 1680x1050 was a thing like half HD.
Fractions are useful! Have people forgotten? Yes...thank you electronic calculators.
"1080×1001", I thought.
Also, 720720 is a nice one.
Unscrambling the egg might be a bigger problem than "continuity"... Just saying.