my favorite sequence is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10... its the natural sequence and its perfect. the number in the nth position is n and its the first sequence anyone learns.
@@thomaskaldahl196 The decimal is cool bc you get to know the approximate value of this godly self-knowing number, as opposed to just some fraction whose value you can't tell by looking at it
One of my favorite number sequences is this: 2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, etc. It's all the primes, plus all the values of p^(2^n) where p is prime and n is a positive integer. With these numbers, every other number can be written as a unique product of these, without repeats. For instance 99 = 11 * 9, that's the "factorization" and there's no other way to do it.
Khinchin’s constant is absolutely mind blowing. That any continued fraction expansion of “almost all” numbers gives you Khinchjn’s constant is just jaw-dropping. Question: is the “almost all” numbers all real numbers except the rationals?
It is really incredible and yet ture for ''almost all'' numbers... however it is NOT containing each and every irrational number! For example fi=1.618... or the base of natural exponential e=2.718... are irrational numbers which are not under this rule. The fi's fractional expansion goes this way: [1;1,1,1,...] which is the notation for 1+1/(1+1/(1+1/(1+...))), and with the ''e'' it goes this way: [2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,...] = 2+1/(1+1/(2+1/(1+1/(1+1/(4+...))))). In the first example the geometric mean is constant 1, and in the second it goes to infinity as we deal with more and more terms...
you can show pretty easily that no quadratic irrational number has this property. Since the terms in the continued fraction repeat periodically, it will not converge to an irrational number. Same with numbers where the terms in their cf strictly increase, which diverge to infinity. e is similar to these numbers and doesn't converge to khinchin's constant for similar reasons however most generic irrational numbers do have this property
Its a nice technique that helps with concentration. We are trained to see brown as a constructive material so writing on it makes us think we are doing more than just writing on paper. The tactile sound and feel of the paper also helps with concentration and I honestly think it sounds nice and prefer it over just normal paper or a white board
Thanks for your votes, everyone! Golomb's sequence won the vote, but the only sequence we could fit on the trophy was the Wieferich primes so we said that won instead. Look at the trophy on The Aperiodical, it's magnificent.
When talking about real numbers, "almost all" is typically defined as "all except for a set of (Lebesgue) measure zero". This is the case here as well. The exceptional set here is in fact uncountable! A subset of it is the uncountable set of all reals with only 1 and 2 in their cont. fraction expansions - the geometric mean will be less than (or =) 2, but Khinchin's constant is >2 Another is the uncountable set of reals with numbers >=3 in their expansions - the geo. mean will be >=3, but K0
A086703. The continued fraction of Levy's constant. Levy's is closely related to Khinchin's. This constant also embodies a property of the continued fraction of almost all numbers - and this sequence is itself a continued fraction. We say "almost all" numbers. Just to expand on this, the exceptions are somewhat intriguing. Any number that is a root of a quadratic does not comply. Also, Euler's number e.
Do you know what - I kind of get it and never really mind it.... It is human nature to get a thrill from being first (or among the first) to do or see something...
I had the same problem, and I wasn't using the subtitles. It just sounded like descending to me. I think it's that T at the end of strict being right by the A at the start ascending. It sounds like strict-d-ascending.
Maybe I misunderstood something, but if the first one is just the sequence of digits in a real number, and the integers in the sequence aren't actually used as numbers, it's not really significant as an integer sequence. It's not s very integ sequence at all, let alone the integest.
Could you do a video on the Tree function? I've looked up some things about it but it's over my head without intense explanation. Mostly about TREE(3) and how it compares to grahams number and other big numbers.
Levy's constant applies to itself and "almost all" numbers in the same sense that Khinchine's does. It's a related property of continued fractions. I nominate A087602 (its decimal expansion) and A086703 (its continued fraction expansion) as my favorites.
My favourite: 1, 2, 6, 12, 60, 360 and 2520. The only numbers that have more divisors than every single number apart from itself and up to it's double. These are literally the most divisible numbers can be, seeing as doubling the number adds a new power of two to the factors.
Omygod golombs sequence is absolutely extraordinary. Just explaining how 1 could only occur once and obviously since 2 doesn't occur once because then it would be 1, which it can't be and, therefore MUST be 2 AND occur TWICE is incredible. Golomb's was by FAR the best
More often then not this channel does a good job at explaining the math so that I can understand it and how cool it is even though with my basic knowledge. This is not one of those times. I am sure it is awesome, but it is way over my head.
in a nut shell, a base is how you describe the place holder. we generally use base 10, so each place holder is a power of 10: one's place, 10's place, 100's place...or in other words 10^0 place, 10^1 place, 10^2 place. If I want base 5, each place holder would be a power of 5. 5^0, 5^1, 5^2 ect. so if I want to write the number "six" in base 5, I would write 11. meaning, one set of 5^1=5 and one set of 5^0=1 ==> 5+1=6.
I'd love to see these guys do a video on the 216 digit Shemhamphorasch. It's produced using only the 24 digit reduced Fibonacci series and the numerals 1 - 9
Wieferich Primes are hard to explain, but the best I can give it to you is by simply showing it, Wieferich Primes we know of, 1093, so p=1093, 2^(p-1) which is 2^(1092) can be divided by 1093, and come out with an integer, whereas if you tried say p=5, (2^4)/5 isn't an integer. Because you can rewrite the conjecture 2^(p-1) = 1, it needs to come out with an integer, to be a Wieferich Prime, hope that sort of helped with understanding it
Read the Wikipedia article on continued fractions. All rational numbers have terminating (non-infinite) continued fraction representations. Therefore the geometric mean of their terms does not "approach" anything, it just is a fixed value, which will not be the same as Kinschine's constant.
Take pi for example. a0 is the integer part of that, so 3. Now take the reciprocal of the fractional part. a1 is the integer part of that, so a1 = 7, giving a geometric mean so far of 4.58. Take the reciprocal of the fractional part of what you currently have, and a2 is the integer part of that (15), and so on. I'm going to reply to this comment with some actual data on this as it applies to pi.
Here he means 'almost all' in the measure theoretic sense, rather than cardinality. Just as the interval [0,1] contains 'almost none' of the numbers in the Cantor set, despite being an uncountable subset. Essentially if you picked a number at random there is probability 1 that it gives Khinchin's constant and probability 0 that it lies in the Cantor set.
the forth; take any number, write it out, count the letters for it, write that number out, repeat, 4. dunno what you'd call it but i like it ^^ and it works in english, german, dutch and probably some other languages
You've also missed that there are n factors in the product. If they all were equal to x the product would be x^n and the exponents would cancel out. When you take limits you have to take the limit of the entire expression, not just parts. (Furthermore, the limit would be of the type infinity^0 which is undefined)
Andrew Wiles proof of Fermat's Last Theorem takes a genius to even understand it. And he used some math that didn't exist in Fermat's time. It has yet to be truly solved in the way that Fermat first thought of.
I agree, I had to watch it a couple of times, do a google search and work through on a piece of paper myself to understand the first two at all - that's the first time that has happened ever with a video on this channel. Admittedly I'm not the brightest stump in the forest, but I'm no idiot either.
Infinite fraction is a decimal rotation of digits. As the fraction increase the decimals are insignificant and so reduce to k constant. Two most significant and other reduce fast. 3 is the closest. These kind of things are wave guides. Mostly used for encryption FM and AM.
Actually, those are just the powers of 2. A perfect number is a number whose entries in its divisor list -- including 1 but not the number itself -- add up to the number in question. Finding them goes something like this: Iff(sic) 2^(p-1) is prime -- which is only possible, though not guaranteed, when p is prime -- then 2^(p-1)*(2^p-1) is perfect.
how do you define the percentage? if you have a finite set you can just count out the number of elements with your property and compute the percentage. if you have a countable set you can look at all finite subsets and count the elements with your property in each of them and figure out whether the "limit" exists if you make them larger. however on an uncountable set...? compare lesbesgue-measures in any bounded subset? and then figure out whether a limit exists if you make the subsets "larger"?
Imagine the number of the base was the "ten", so, for example, in base 10 you got 123, to calculate that you got 1*10^2 + 2*10^1 + 3*10^0 = 123, you can do the same in other bases like 123 in base 4 is 1*4^2 + 2*4^1 + 3*4^0 in base 10
A continued fraction expansion is basically a decimal turned into a sequence, and any digit is turned into an integer. Let's take pi as an example. 3 is the first digit, so that will be the first number in the expansion sequence: a0. 1 is the first decimal: a1 4 is the second: a2 The sequence would be: {3,1,4,1,5,9, ... } In the above sequence, if I was to take a0 + 1 / a1, it would give me 3.1 If I was to take a0 + 1/ (a1 + 1 / a2), it would give me 3.14 Make sense now?
I agree. Khinchin's constant assumes that the aliens would have even invented continued fractions in the first place. While I do find them interesting, continued fractions are kind of a niche concept in number theory and aren't really important to a lot of mathematics. Then Khinchin's constant is a non-obvious derivation from that which doesn't work for common numbers. Compared to pi which pops up in many different areas of mathematics, Khinchin's constant is not that important.
I called his favorite after he described its self-referential completeness. Ascribing divinity to it -- I tend to think of that as sentimentality, but it also gave me a chuckle. Nothing is as charming (at present) as completeness, eh?
What about this sequence? 1, 11, 21, 1211, 111221, 312211, 13112221... It starts with 1 and the next element "describes" the previous, hence the second number in the sequence is "11" meaning "one 1" (describing the previous number). The third number is then "21" meaning "two 1s", and so on. I'm sure there's some very interesting math regarding this sequence..
A similar sequence would be all p^3^n where p is prime and n is a non-negative integer: 2, 3, 5, 7, 8, 11, 13, 17, 19, 23, 27, 29, ... You can get every positive rational number excluding 1 by multiplying or dividing these numbers and factorisations are unique. For example, 28/5 = 7 * 8 / 2 / 5
aha, thanks! looking up "almost all" on wikipedia says that there are "a number of specialised uses" of the term, which continues to confuse ._. definitely not as bad as "mathematical concepts named after leonhard euler" though
Can you do a whole video on Khinchin's constant? Specifically, can you do an example of how a certain number, when you do the continued fraction expansion of it, approaches the constant?
8 years late but essentially if you pick a random number 0-1, It's continued fraction has a 1/1 - 1/2 = 1/2 chance of being 1, 1/2 -1/3 = 1/6 chance of being 2, 1/12 chance of being 3, etc. So the geometric mean is just 1^(1/2) * 2 ^ (1/6) * 3^(1/12)... n^(1/(n(n+1)) which is that constant
No, there is no zeroth place, and each element tells you about the next element, not the previous. The first element will tell you that there are 0 ones. Thus you know that the second element has to be a two, and so on and so forth.
In fact, even more than that, you can just dump zeros wherever you want. Examples: 00000666666777777... is completely valid. So is even 10005555566666 and so on. This sequence only works out if you do it by indexing and not by counting, or you define the range as n >= 1
I guess that would work, but then you could even argue the sequence is valid as a string of infinite zeroes. The most interesting one is the one with positive nonzero integers only because zero is trivial.
The problem with ten is that it uses the factors 2 and 5 when using 2 and 3 would give you just as many divisors, but you would be able to use those factors more frequently as the factors occur in other numbers more frequently. So every benefit you get from base ten you would get from base six, but they would occur more frequently. There are infinitely many numbers you can choose from that have a given number of divisors, but the smaller ones give you the benefits of those number more often.
With the composite numbers vs the primes you'd be comparing two countably infinite sets. You need another way to compare the sets, say natural density in which the primes have density zero. So in that sense, yes the composite numbers are the vast majority. You are correct in that we need to be careful about our definitions and "vast majority" is vague, but the phrase 'almost every' in mathematics, oddly enough, has a very precise definition which has little to do with topological density.
I don't know what this sequence is called, if it even has a name, but I like it 1, 11, 21, 1211, 1231, 131221, 132221, 133221, 232221, 134211, 14131221, 14132241, etc. Wether this sequence has a name or not, try to find out why this is a sequence.
It doesn't work for rational numbers. So x being 3 does not work since a0 would just be 3. Divides into just means it is a factor of. Or in more mathematical terms, n divides x if x is congruent to 0 mod n (leaves no residue once you do the division algorithm.
In that video, James is talking about natural density in the integers which is a different concept than the measure theory definition of 'almost every'. (In fact James shouldn't have used 'almost every' in that video as natural density is not really a measure)
Aleph 0 cannot represent as much as or more than a finite fraction of aleph 1. In this sense you can't look at the number of irrationals in a finite way. But what we're really saying is the density of irrationals over the reals is approaching 100%.
People can rattle off the first few digits of Pi and e, but most people define Pi as the ratio between the diameter of a circle and its circumference...which requires you to measure exactly since in order to calculate one of those things, you need Pi. However, there are other ways to exactly calculate Pi and e via the sum of an infinite series. I would like to see the proof as to why these work
If Khinchin's constant is to be represented by any sequence, surely it should be its continued fraction expansion, not its decimal expansion. That's the deal-breaker for me; I vote Golomb! Because of the phis.
Mathematically, I also think the K constant is amazing. That it should exist is phenomenal. But from a personal point of view, I really like the idea *behind* wild numbers, coming from a novel and being "re-incorporated" into real mathematics...
False, the best name is not Solomon Golomb. It's Nebuchadnezzar, solely because it's pronounced something like "Ne-buh-kuh-ne-zur." ...My Global Studies teacher is quite awesome...
Видео от A.Gorlovich'a You've got it backwards. it's that p² divides 2^(p-1)-1, not divides by it. for three to work, 3 would have to divide by 9, and it doesn't, because 1/3 is not whole.
ah, i see. thanks for clearing it up. it's pretty clear that only irrational numbers result in khinchin's constant. and rationals are dense in the reals. so it's a little hard to wrap your head around, but i never expected for it to be something intuitive, i suppose.
I don't understand why Khinichin's constant should be regarded as an integer sequence. We always consider pi or e to be numbers, not integer sequences.
yes, but when you're dealing with infinities they can't be handled in an intuitive manner such as that. i think of conditionally convergent series and grandi's series as immediate examples that infinity does not obey normal laws. doing so would result in contradiction. you could say that composite numbers are the vast majority of whole numbers by the same logic, but that really doesn't work out.
Giving that constant to aliens not only assumes they understand whatever numeric system for transmitting the data we use, but it also assumes that they do math the same way as us, have found the constant, and/or would be able to derive it once we've given it to them. That seems really unlikely.
That makes absolute sense to me now. Thank you for explaining the concept. On another note, would you know why some people argue that base 12 is more intuitive than base 10?
In the way the Fibonacci Sequence has values by summing the previous 2 values, do any constants or behaviors surface by increasing the number to 3 or higher? 1, 1, 1, 3, 5, 9, 17, 31...
I feel cheated by the distinct lack of the Catalan numbers. Beautiful recursion, beautiful closed form, and answer to a massive number of enumerative combinatorics questions how can you not love that.
The important point for a rational is that the top and bottom of the fraction have to be whole numbers. In a circle, at least one of the circumference and the radius is always an irrational number, so it doesn't make pi rational. It's... kind of a circular argument ;)
There must be some other exceptions to Khinchin's Constant than just rational numbers. For instance, the Golden Ratio is an irrational number, and the continued fraction expansion for it is an endless series of 1's as the coefficients, meaning that the geometric mean of the coefficients would be 1 for any number of iterations, rather than converging to the Constant as the number of terms in the expansion approaches infinity. Similarly, any irrational number constructed in the same way so that all the coefficients in the fraction expansion have the same value would have a geometric mean equal to that value instead of the the Constant.
my favorite sequence is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10... its the natural sequence and its perfect. the number in the nth position is n and its the first sequence anyone learns.
@Adi Septiana
1. It was supposed to be sarcasm
2. This sequence is the base for e
It's also the decimal expansion of Champernowne's constant
I agree but in base 12. Sorry
How about that sequence but nth position is -n?
Ah yes, the A000027... My second favorite.
i could tell khinchin's constant was his fave he went on about it much more than the others
also cause its more complicated
thats what i thought too even before seeing any other ones
But why the decimal expansion? Is there anything special about it?
@@thomaskaldahl196 The decimal is cool bc you get to know the approximate value of this godly self-knowing number, as opposed to just some fraction whose value you can't tell by looking at it
@@olivialuv1 But what's significant about base 10 as opposed to binary or some other base?
*Tony Padilla:* "I'm not going to tell you which one I like best"
*also Tony Padilla:* proceeds and starts by talking about his fav
no hints XD
One of my favorite number sequences is this:
2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, etc.
It's all the primes, plus all the values of p^(2^n) where p is prime and n is a positive integer. With these numbers, every other number can be written as a unique product of these, without repeats. For instance 99 = 11 * 9, that's the "factorization" and there's no other way to do it.
Wow!
Same.
It would seem to me that the constants in the continued fraction expansion of Khinchin's constant would be more meaningful than the decimal expansion.
Shout-out to Ireland!
??????????
Khinchin’s constant is absolutely mind blowing. That any continued fraction expansion of “almost all” numbers gives you Khinchjn’s constant is just jaw-dropping. Question: is the “almost all” numbers all real numbers except the rationals?
It is really incredible and yet ture for ''almost all'' numbers... however it is NOT containing each and every irrational number!
For example fi=1.618... or the base of natural exponential e=2.718... are irrational numbers which are not under this rule.
The fi's fractional expansion goes this way:
[1;1,1,1,...] which is the notation for 1+1/(1+1/(1+1/(1+...))),
and with the ''e'' it goes this way:
[2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,...] = 2+1/(1+1/(2+1/(1+1/(1+1/(4+...))))).
In the first example the geometric mean is constant 1, and in the second it goes to infinity as we deal with more and more terms...
you can show pretty easily that no quadratic irrational number has this property. Since the terms in the continued fraction repeat periodically, it will not converge to an irrational number. Same with numbers where the terms in their cf strictly increase, which diverge to infinity. e is similar to these numbers and doesn't converge to khinchin's constant for similar reasons
however most generic irrational numbers do have this property
NERD
In the video description there are links to all the sequences, a chance to vote for a winner and other stuff...
Ш vs Щ
@@mr.z111 Прив
This virol ad said something like 95% of UA-cam vid get less than 1000 views and my first reaction was "wow so many UA-cam vids get over 1000 views"
Its a nice technique that helps with concentration. We are trained to see brown as a constructive material so writing on it makes us think we are doing more than just writing on paper. The tactile sound and feel of the paper also helps with concentration and I honestly think it sounds nice and prefer it over just normal paper or a white board
My favorite integer sequence is and will forever be the look and say sequence.
Numberphile had Conway himself talking about the look and say sequence
1 11 21 1211 111221 312211 13112221 1113213211 ...
I love how Tony's collar was popped for most of this.
numbers are just awesome......what a beauty...
for the wild numbers, just add 0.5.
1,11,21,1211,111221,312211, ...
You split it up and describe the previous number, where the next number in the sequence is the description.
See and write 🎉
Thanks for your votes, everyone! Golomb's sequence won the vote, but the only sequence we could fit on the trophy was the Wieferich primes so we said that won instead.
Look at the trophy on The Aperiodical, it's magnificent.
When talking about real numbers, "almost all" is typically defined as "all except for a set of (Lebesgue) measure zero". This is the case here as well. The exceptional set here is in fact uncountable!
A subset of it is the uncountable set of all reals with only 1 and 2 in their cont. fraction expansions - the geometric mean will be less than (or =) 2, but Khinchin's constant is >2
Another is the uncountable set of reals with numbers >=3 in their expansions - the geo. mean will be >=3, but K0
A086703. The continued fraction of Levy's constant. Levy's is closely related to Khinchin's. This constant also embodies a property of the continued fraction of almost all numbers - and this sequence is itself a continued fraction. We say "almost all" numbers. Just to expand on this, the exceptions are somewhat intriguing. Any number that is a root of a quadratic does not comply. Also, Euler's number e.
2:53 But I thought God's Number was 20...
James Grime was in your video on it...
Nice
Do you know what - I kind of get it and never really mind it.... It is human nature to get a thrill from being first (or among the first) to do or see something...
I had the same problem, and I wasn't using the subtitles. It just sounded like descending to me. I think it's that T at the end of strict being right by the A at the start ascending. It sounds like strict-d-ascending.
Maybe I misunderstood something, but if the first one is just the sequence of digits in a real number, and the integers in the sequence aren't actually used as numbers, it's not really significant as an integer sequence. It's not s very integ sequence at all, let alone the integest.
Could you do a video on the Tree function? I've looked up some things about it but it's over my head without intense explanation. Mostly about TREE(3) and how it compares to grahams number and other big numbers.
You were ahead of your time...
4 years ahead of your time.
I knew Khinchin's constant would be his favourite. That sort of kinky stuff makes all mathematicians salivate.
I knew it was the first one, because I knew he would just be able of holding himself in the sequence he liked the most if it was presented first.
Solomon Golomb ! That is a great name.
Levy's constant applies to itself and "almost all" numbers in the same sense that Khinchine's does. It's a related property of continued fractions. I nominate A087602 (its decimal expansion) and A086703 (its continued fraction expansion) as my favorites.
My favourite: 1, 2, 6, 12, 60, 360 and 2520. The only numbers that have more divisors than every single number apart from itself and up to it's double. These are literally the most divisible numbers can be, seeing as doubling the number adds a new power of two to the factors.
is 2520 the maximum how about 5040?
@@skalderman 5040 doesn't work. 7560 has more divisors than it and is less than 10080 (2*5040)
Omygod golombs sequence is absolutely extraordinary. Just explaining how 1 could only occur once and obviously since 2 doesn't occur once because then it would be 1, which it can't be and, therefore MUST be 2 AND occur TWICE is incredible. Golomb's was by FAR the best
More often then not this channel does a good job at explaining the math so that I can understand it and how cool it is even though with my basic knowledge. This is not one of those times. I am sure it is awesome, but it is way over my head.
I already knew about khnichin's constant and love it, but golomb's sequence is definitely my favourite!
in a nut shell, a base is how you describe the place holder. we generally use base 10, so each place holder is a power of 10: one's place, 10's place, 100's place...or in other words 10^0 place, 10^1 place, 10^2 place. If I want base 5, each place holder would be a power of 5. 5^0, 5^1, 5^2 ect. so if I want to write the number "six" in base 5, I would write 11. meaning, one set of 5^1=5 and one set of 5^0=1 ==> 5+1=6.
I'd love to see these guys do a video on the 216 digit Shemhamphorasch.
It's produced using only the 24 digit reduced Fibonacci series and the numerals 1 - 9
Why was 67 twice in the wild numbers, if that was just the list of numbers that would result from the operation being done on any given number?
Golomb's sequence actually kinda blows my mind.
Wieferich Primes are hard to explain, but the best I can give it to you is by simply showing it, Wieferich Primes we know of, 1093, so p=1093, 2^(p-1) which is 2^(1092) can be divided by 1093, and come out with an integer, whereas if you tried say p=5, (2^4)/5 isn't an integer. Because you can rewrite the conjecture 2^(p-1) = 1, it needs to come out with an integer, to be a Wieferich Prime, hope that sort of helped with understanding it
Read the Wikipedia article on continued fractions. All rational numbers have terminating (non-infinite) continued fraction representations. Therefore the geometric mean of their terms does not "approach" anything, it just is a fixed value, which will not be the same as Kinschine's constant.
This is the only Numberphile video that went completely over my head.
Take pi for example. a0 is the integer part of that, so 3. Now take the reciprocal of the fractional part. a1 is the integer part of that, so a1 = 7, giving a geometric mean so far of 4.58. Take the reciprocal of the fractional part of what you currently have, and a2 is the integer part of that (15), and so on. I'm going to reply to this comment with some actual data on this as it applies to pi.
Im watching the entire series this summer and i cant stop thinking of this lol
Here he means 'almost all' in the measure theoretic sense, rather than cardinality.
Just as the interval [0,1] contains 'almost none' of the numbers in the Cantor set, despite being an uncountable subset.
Essentially if you picked a number at random there is probability 1 that it gives Khinchin's constant and probability 0 that it lies in the Cantor set.
the forth; take any number, write it out, count the letters for it, write that number out, repeat, 4. dunno what you'd call it but i like it ^^ and it works in english, german, dutch and probably some other languages
You can't beat a bit of numberphile during the school summer holidays , especially in ireland where it always rains !
The largest metronome base n is (n^n-n²+n-1)/(n-1)².
Special case, n=1, the limit as you go to 1 is 0.
9:11 There *_IS_* an equation to that: a(n) = 7; and it rhymes, too 😊.
You've also missed that there are n factors in the product.
If they all were equal to x the product would be x^n and the exponents would cancel out.
When you take limits you have to take the limit of the entire expression, not just parts.
(Furthermore, the limit would be of the type infinity^0 which is undefined)
Thumbs up if you could already feel Tony's excitement when describing the Khintchine's constant :)
Andrew Wiles proof of Fermat's Last Theorem takes a genius to even understand it. And he used some math that didn't exist in Fermat's time. It has yet to be truly solved in the way that Fermat first thought of.
I agree, I had to watch it a couple of times, do a google search and work through on a piece of paper myself to understand the first two at all - that's the first time that has happened ever with a video on this channel. Admittedly I'm not the brightest stump in the forest, but I'm no idiot either.
Sick sequences.
I was thinking a more interesting "all the sevens" would be 7 in each of the bases, but it would just be 111, 21, 13, 12, 11, 10, 7, 7, 7, etc.
What is K. A continuous fraction is supposed to end in a bell curve.
You didn't unpack the formula behind all the 7s: 7 x 1^n, where n in the position in the sequence.
Infinite fraction is a decimal rotation of digits. As the fraction increase the decimals are insignificant and so reduce to k constant. Two most significant and other reduce fast. 3 is the closest. These kind of things are wave guides. Mostly used for encryption FM and AM.
Actually, those are just the powers of 2. A perfect number is a number whose entries in its divisor list -- including 1 but not the number itself -- add up to the number in question. Finding them goes something like this:
Iff(sic) 2^(p-1) is prime -- which is only possible, though not guaranteed, when p is prime -- then 2^(p-1)*(2^p-1) is perfect.
that grows exactly as fast as the busybeaver function.
you could however use f(x) = busybeaver(x) * busybeaver(x)
how do you define the percentage? if you have a finite set you can just count out the number of elements with your property and compute the percentage. if you have a countable set you can look at all finite subsets and count the elements with your property in each of them and figure out whether the "limit" exists if you make them larger. however on an uncountable set...? compare lesbesgue-measures in any bounded subset? and then figure out whether a limit exists if you make the subsets "larger"?
Imagine the number of the base was the "ten", so, for example, in base 10 you got 123, to calculate that you got 1*10^2 + 2*10^1 + 3*10^0 = 123, you can do the same in other bases like 123 in base 4 is 1*4^2 + 2*4^1 + 3*4^0 in base 10
A continued fraction expansion is basically a decimal turned into a sequence, and any digit is turned into an integer.
Let's take pi as an example.
3 is the first digit, so that will be the first number in the expansion sequence: a0.
1 is the first decimal: a1
4 is the second: a2
The sequence would be: {3,1,4,1,5,9, ... }
In the above sequence, if I was to take a0 + 1 / a1, it would give me 3.1
If I was to take a0 + 1/ (a1 + 1 / a2), it would give me 3.14
Make sense now?
I agree. Khinchin's constant assumes that the aliens would have even invented continued fractions in the first place. While I do find them interesting, continued fractions are kind of a niche concept in number theory and aren't really important to a lot of mathematics. Then Khinchin's constant is a non-obvious derivation from that which doesn't work for common numbers. Compared to pi which pops up in many different areas of mathematics, Khinchin's constant is not that important.
I called his favorite after he described its self-referential completeness. Ascribing divinity to it -- I tend to think of that as sentimentality, but it also gave me a chuckle. Nothing is as charming (at present) as completeness, eh?
I noticed what Tony's favourite sequence was from how he spoke of it. =)
Could you do a video on A027746? It's a list of n by prime factors.
What about this sequence?
1, 11, 21, 1211, 111221, 312211, 13112221...
It starts with 1 and the next element "describes" the previous, hence the second number in the sequence is "11" meaning "one 1" (describing the previous number). The third number is then "21" meaning "two 1s", and so on. I'm sure there's some very interesting math regarding this sequence..
Kurtis Fraser
13112221,413213,21122314,31321314,31123314,13123314,13123314,13123314,loop.
Conway did some work on this sequence, I think numberphile even has a video about it
A similar sequence would be all p^3^n where p is prime and n is a non-negative integer:
2, 3, 5, 7, 8, 11, 13, 17, 19, 23, 27, 29, ...
You can get every positive rational number excluding 1 by multiplying or dividing these numbers and factorisations are unique. For example, 28/5 = 7 * 8 / 2 / 5
he wrote the general form of a continued fraction expansion.
a0, a1, a2 are the digits of that expansion.
aha, thanks! looking up "almost all" on wikipedia says that there are "a number of specialised uses" of the term, which continues to confuse ._.
definitely not as bad as "mathematical concepts named after leonhard euler" though
Can you do a whole video on Khinchin's constant? Specifically, can you do an example of how a certain number, when you do the continued fraction expansion of it, approaches the constant?
8 years late but essentially if you pick a random number 0-1, It's continued fraction has a 1/1 - 1/2 = 1/2 chance of being 1, 1/2 -1/3 = 1/6 chance of being 2, 1/12 chance of being 3, etc.
So the geometric mean is just 1^(1/2) * 2 ^ (1/6) * 3^(1/12)... n^(1/(n(n+1)) which is that constant
Couldn't you start Golomb's sequence with a 0? 1 appears 0 times. it would be: 0, 2, 2, 3, 3, etc.
0 would appear one time, so the sequence would be 0, 1, 2, 2, 3, 3...
No, there is no zeroth place, and each element tells you about the next element, not the previous. The first element will tell you that there are 0 ones. Thus you know that the second element has to be a two, and so on and so forth.
In fact, even more than that, you can just dump zeros wherever you want. Examples: 00000666666777777... is completely valid. So is even 10005555566666 and so on. This sequence only works out if you do it by indexing and not by counting, or you define the range as n >= 1
I guess that would work, but then you could even argue the sequence is valid as a string of infinite zeroes. The most interesting one is the one with positive nonzero integers only because zero is trivial.
Or, you could just define the sequence as one that every single number in it is referenced to in the sequence. That would also work.
The problem with ten is that it uses the factors 2 and 5 when using 2 and 3 would give you just as many divisors, but you would be able to use those factors more frequently as the factors occur in other numbers more frequently.
So every benefit you get from base ten you would get from base six, but they would occur more frequently.
There are infinitely many numbers you can choose from that have a given number of divisors, but the smaller ones give you the benefits of those number more often.
With the composite numbers vs the primes you'd be comparing two countably infinite sets. You need another way to compare the sets, say natural density in which the primes have density zero. So in that sense, yes the composite numbers are the vast majority.
You are correct in that we need to be careful about our definitions and "vast majority" is vague, but the phrase 'almost every' in mathematics, oddly enough, has a very precise definition which has little to do with topological density.
8:17 how to get 194 in this sequence?
5"3+3*5+5"2+2*5+5"1+5+4=189. No 194
I got, I got
125*1+25*2+5*3+1*4=194 😂😂🎉🎉
I think the interesting thing about Khichin's number is that it neatly avoids the rationals.
Yes, of course... and there is even more interesting about it, especially if we start it with pi.
I don't know what this sequence is called, if it even has a name, but I like it
1, 11, 21, 1211, 1231, 131221, 132221, 133221, 232221, 134211, 14131221, 14132241, etc. Wether this sequence has a name or not, try to find out why this is a sequence.
Which is why they should do a video about it. Every one knows about pi too and they have lots of videos about it.
for golombs sequence you could have begun it 211
Edit: I looked it up and the sequence is non descending so 211 wouldnt work
Yes, your first guess is correct, "almost all" in this case means there are "countably infinite" exceptions.
Number that knows its a number. Brilliant.
@Numberphile
look out for a paper with a conjecture on Pi and the "All the Seven's" coming to a computer near you.
It doesn't work for rational numbers. So x being 3 does not work since a0 would just be 3.
Divides into just means it is a factor of. Or in more mathematical terms, n divides x if x is congruent to 0 mod n (leaves no residue once you do the division algorithm.
In that video, James is talking about natural density in the integers which is a different concept than the measure theory definition of 'almost every'.
(In fact James shouldn't have used 'almost every' in that video as natural density is not really a measure)
Can there be a whole video on Fermat's Last Theorem?
Aleph 0 cannot represent as much as or more than a finite fraction of aleph 1. In this sense you can't look at the number of irrationals in a finite way. But what we're really saying is the density of irrationals over the reals is approaching 100%.
People can rattle off the first few digits of Pi and e, but most people define Pi as the ratio between the diameter of a circle and its circumference...which requires you to measure exactly since in order to calculate one of those things, you need Pi. However, there are other ways to exactly calculate Pi and e via the sum of an infinite series. I would like to see the proof as to why these work
If Khinchin's constant is to be represented by any sequence, surely it should be its continued fraction expansion, not its decimal expansion. That's the deal-breaker for me; I vote Golomb! Because of the phis.
Mathematically, I also think the K constant is amazing. That it should exist is phenomenal. But from a personal point of view, I really like the idea *behind* wild numbers, coming from a novel and being "re-incorporated" into real mathematics...
False, the best name is not Solomon Golomb.
It's Nebuchadnezzar, solely because it's pronounced something like "Ne-buh-kuh-ne-zur."
...My Global Studies teacher is quite awesome...
Joe GameWaffle global studies 🤣 ought to be usa uni
Yep because most of other world learn how it's pronounced from watching the first Matrix movie in original.
The best name might well be Arup Gupta - at least according to Click and clack, the Tappet brothers...from Car Talk. ;-)
Hes in the bible
Walt F. Those end credits were always hillarious.
Isn't 3 a Wieferich prime?
Because 3^2=9 and 9 divides by 3 (2^3-1)-1
Видео от A.Gorlovich'a You've got it backwards. it's that p² divides 2^(p-1)-1, not divides by it. for three to work, 3 would have to divide by 9, and it doesn't, because 1/3 is not whole.
ah, i see. thanks for clearing it up.
it's pretty clear that only irrational numbers result in khinchin's constant. and rationals are dense in the reals. so it's a little hard to wrap your head around, but i never expected for it to be something intuitive, i suppose.
I don't understand why Khinichin's constant should be regarded as an integer sequence. We always consider pi or e to be numbers, not integer sequences.
yes, but when you're dealing with infinities they can't be handled in an intuitive manner such as that. i think of conditionally convergent series and grandi's series as immediate examples that infinity does not obey normal laws. doing so would result in contradiction.
you could say that composite numbers are the vast majority of whole numbers by the same logic, but that really doesn't work out.
They've used it for a while (since the beginning I think). It's provided by Brady, the person who runs the channel and films the videos.
Giving that constant to aliens not only assumes they understand whatever numeric system for transmitting the data we use, but it also assumes that they do math the same way as us, have found the constant, and/or would be able to derive it once we've given it to them. That seems really unlikely.
That makes absolute sense to me now. Thank you for explaining the concept.
On another note, would you know why some people argue that base 12 is more intuitive than base 10?
12 has more factors
In the way the Fibonacci Sequence has values by summing the previous 2 values, do any constants or behaviors surface by increasing the number to 3 or higher? 1, 1, 1, 3, 5, 9, 17, 31...
I feel cheated by the distinct lack of the Catalan numbers. Beautiful recursion, beautiful closed form, and answer to a massive number of enumerative combinatorics questions how can you not love that.
The important point for a rational is that the top and bottom of the fraction have to be whole numbers. In a circle, at least one of the circumference and the radius is always an irrational number, so it doesn't make pi rational. It's... kind of a circular argument ;)
that was i was trying to say...
i think it is called a "measure-zero set" (or its complement set)
thx
7:19 WTH is "base 1"? In the only thing that resembles a "base 1" (unary numbers), 1s are used, not 0s.
There must be some other exceptions to Khinchin's Constant than just rational numbers.
For instance, the Golden Ratio is an irrational number, and the continued fraction expansion for it is an endless series of 1's as the coefficients, meaning that the geometric mean of the coefficients would be 1 for any number of iterations, rather than converging to the Constant as the number of terms in the expansion approaches infinity.
Similarly, any irrational number constructed in the same way so that all the coefficients in the fraction expansion have the same value would have a geometric mean equal to that value instead of the the Constant.