I recently had a professor say medical knowledge took around 50 years to double in the 1950s, and now doubles around every 70 days, I took it at face value at first then I thought about what that actually meant. In 5 year time frame if the rate held, medical knowledge will have grown millions of times larger compared to 5 years prior, and according to that theory in 73 days it would double again. Seems impossible, the larger medical knowledge (whatever that means) the harder it would be to eventually double again due to the vastness. It took 5 years to grow millions of times larger and then that feat would be repeated again in 73 days. Seems utterly impossible if someone takes a moment to think about. I googled looking for a source and I found an article from 2011 that said something similar to what my professor had said however the article provided no citation, no definition of medical knowledge and no explanation how this was estimated, however I found hundreds of articles and blog posts by "experts" stating this estimate as fact and/or citing that article, but not a single article questioning it. This is the best explanation I have found how the author from 2011 potentially arrived at that estimate, however its disheartening to see so many people and "experts" parrot this stat as fact without thinking about it what it means or how it could be possible. Great work on the video, subscribing now.

universities should require an art class for their math students

Fantastic explanation! Very cool. ...11111 (in base 2) + 1 = 0 hence ...11111 has to be equal to -1. I'd like to see the derivation of a/(1-r) in base 2. Great job!

I need 12 adic numbers...

Amazing. Very intriguing - and presented perfectly

it F'*'in great! to finally know what a p-adic number is! damn... Wikipedia was so unintuitive sometimes.

Subbed because of the p-adic videos, but can we get an update on this, please? What a cliffhanger haha

music is a bit offensive.

If I had you as a tutor or friend in undergraduate, I'd have become a math major. Because explanations don't come any clearer than yours.

Mate, a millennium is 1000 years, no-one has been claiming anything with respect to 1000 years, it's only the last 100 years that has been spoken about.

Perfect for the enthusiast amateur maths guy

Very nice explanation

it's accurate, -1 is the right answer, it's a loop '' number infinity is a loop so you start at positive and find yourself at negative -1

It's "Ostrowski" theorem, not " Ostrowki"

im looking at the beautiful foliage beyond the glasses

I need the number of your dentist

This is already a complete proof of Fermat's little theorem because the principle can be seen directly and easily generalized. The more difficult part is to put it into formalized "mathematical speech". So, you only can understand a formal proof in deep if you can also "see" it. Or the other way round: Mathematicians often "see" a proof but struggle to formalize so that others can understand even without "seeing". Ramanujan was such a typical example. Galois to some extent, too or Riemann, who often used physical analogies like streaming water or electric currents to visualize for himself aspects of his complex functions theory, for example. Besides being a mathematical genius he was also an excellent physicist, what most people don't know. That's rare.

What happens to the carry on the far left, of ...111 + 1? How is this 0? Isn't it 1...000?

0:04 "how ig"

This is great and so down to earth! Exciting thank you! Looking forward to other lessons :)

This feels like correlation but not causation. The infinite sum formula you used is obviously not supposed to work when common ratio is greater than one, because a common ratio > 1 will give an extra term in the numerator, -a(r)^n. So the formula for a common ratio greater than one should be, S = {a - a(r)^n} / {1 - r}, as opposed to the one in the video S = {a} / {1 - r} What if it’s just coincidence that the omission of that term coincides with whatever the p-adic value is, at least for the numbers you’ve used so far. I guess what I’m trying to say is omission of the one term in the numerator feels like a calculation error and it’s crazy that a calculation error branches into a whole new realm of numbers.

Wouldn't it be more correct to tslk about "p-adic distance" instead of "p-adic numbers"?

There's a very simple way to count the matching digits, at least for 2-adics: take the number's negative and AND it with the original number. Then you just use a lookup table or tree-like structure to narrow down which power of 2 the result is. Actually, you often won't even need to do this since most computers have a dedicated instruction just for finding the 2-adic valuation of an integer, often by the name of ctz (or "count trailing zeros"). Alternatively, don't even bother with the lookup table and just take the reciprocal immediately, if you can represent it anyways; no need to take the logarithm if you're just going to exponentiate it immediately afterwards. Thinking in terms of computer integers actually helped me understand the p-adic metric, since with only 8 bits, 256 is indistinguishable from 0. In terms of modular arithmetic, you'd say the two numbers are congruent modulo 256. In terms of 2-adic arithmetic, you could say that the difference between 256 and 0 is a rounding error with only 8 known bits. If two numbers can round into each other, then they have to be pretty close. It does give the amusing consequence of flipping the concept of "most/least-significant digit/bit."

The iffy bit of magic is to put “+” signs and “…” instead of working with a single big map (which we might call a sum) on a series. The thing that makes it work is to define compatible rules between the big map, operations on the series and usual addition and multiplication. That gets a bit more technical though. Well that’s to get the sum to work as a sum. p-adic numbers can just be defined straight up and then the sum properties are theorems.

The key is to think of series of numbers as maps (functions) from the natural numbers into some target like the for example the real numbers (with usual addition and multiplication) If we think of the set of possible series (possible maps from N->R), then we can come up with rules that maps series to numbers (sometimes numbers extended with some extra points like infinity). Some of the ways of assigning a number to series deserve to be called infinite sums. Others may not. Let s and t be a map(function, think series) from N->R. Let Z be a map (functional, think infinite sum) from (N->R)->R. We can add two series to get a new one by adding the corresponding entries at the same place in the series. (s+t) We can append a number from the left to a series by pushing the series back and placing the number at the front. (x@s) We can multiply a series by a number by multiplying each entry by that number. (x*s) Z(x@s) = x+Z(s) Z(s+t) = Z(s)+Z(t) Z(x*s) = x*Z(s) s+o=o+s=s Z(o)=0 Now suppose that for a,r in R and r<1 we have s(n):=a*r^n. Notice s= a@r*s. So: Z(s)=a+r*Z(s). Solving for Z(s) we get: Z(s)=a/(1-r). If r=1/2 then Z(s)=2a. Nowhere did we mention convergence. But if Z is a usual convergent sum which incidentally does satisfy the rules above, then this must be the answer. But the rules could apply to non typically convergent sums, that is if we want to assign a number to this series for values of r where the typical sum of the series diverge but we still want to follow the rules above, then we can do that. Don’t get we wrong, the sum is still divergent if we think of it in the usual sense in analysis!!! But we’ve analytically continued the notion of an infinite sum and this new notion also deserves the name sum. But it is a different thing. Sometimes the notions give the same results sometimes they don’t. That’s perfectly fine if we don’t confuse them. If r=2 (typically divergent sum) then Z(s) = -a. Did we sum up a bunch of positive numbers and get a negative one? No! We extended our rule for assigning numbers to series formally to a larger class of series. The rules obey some rules of finite sums and not others. It obeys some rules of convergent infinite sums but not others.

I would love to know what the 10 adic for -1/12 is now

I wonder how many secrets of quantum and physics are hidden in this numerical encodement

How does....2222 + 1 adds to ....all 0s!? To my mind it's 2223.

so........

Beautiful video brother and u got a new subscriber form INDIA !

this video is amazing

im trying to do research with p-adic numbers, this is super helpful

One of the digits in 3-adic number could be negative, so corresponding to reading variables from the right to the left

Wikipedia disagrees with this. It says that Ostrowski's Theorem says there are only three absolute values on the rationals: the real one, the p-adic one, and the trivial one. An absolute value has the multiplicative property, that is, abs(xy)=abs(x)abs(y). The proof given on the wikipedia page makes use of this propery and it isn't implied by the properties of a metric. Metrics, as defined in the OP video, are still metrics if you arbitrarily permute the inputs.

I love the way you explain things! This is the first of your videos I've come across and I'll be watching all the other ones. I get the feeling that I'll not only learn math from you, but also pick up some tips on clear exposition.

Great job introducing p-adics. Love thinking about numbers that are not on the linear plane. And yes, please do some more videos on p-adics and why they are so useful in solving problems where the Real Numbers seem to wear out.

Trying to create a new distance measure based on surreal birth ordering. The nth surreal number can be ordered by birth between 0 and 2 with: (2n+1)/2^floor(log₂x)-2 Resulting in a linear ordering of the surreal numbers by value. All variable being positive integers. Distances obey: D(a,b) <= D(a,x) + D(x,b) Distance calculation between a and b: D(a,b) = |(2a+1)/2^floor(log₂a)-(2b+1)/2^floor(log₂b)| Example: 3rd is closer to 7th than 8th, because: 3rd surreal = 1 7th surreal = 2 8th surreal = -3 D(3,7) = |(2×3+1)/2^floor(log₂3)-(2×7+1)/2^floor(log₂7)| D(3,7) = |7/2-15/4| = 1/4 D(7,8) = |(2×7+1)/2^floor(log₂7)-(2×8+1)/2^floor(log₂8)| D(7,8) = |15/4-17/8| = 13/8 since 1/4 < 13/8, the distance from 3 to 7 is less than the distance from 7 to 8 Question: Is this the same as 2-adic ordering?

Amazing 😃

10.59. Doesnt it converge to 2 to the power of infinity in the 10 digit system?

hmm. Even if a number has an infinite amount of trailing zeroes, it does not mean it is equal to zero. It still may have a non-zero number somewhere infinitely far to the left of the number (if we look initially at the right side of a given number). Otherwise it means this number is equal to infinity and zero at the same time.... very strange. What do I miss here?

Great video!

Nice work here. Very interesting. Thank you for your post. I have always wondered about how those who have advanced this idea define knowledge, who is currently measuring it, and how it is measured. They whole idea rings true for me at some level and I am inclined to accept that knowledge is increasing and doing so at an increasing rate. The exquisite specificity of it, however, presented as though it is the result of a mathematical formula referring to actual data>information>knowledge is misleading and confounding. Of course, while not stated, the knowledge-doubling idea tacitly infers the inclusion of computers in the process which also relates to Moore’s “Law” of the increasing rate of technological progress. Of interest to me when I looked at this a few decades ago was even though one accepts the premise of knowledge doubling, ancillary knowledge of how that knowledge is distributed and becomes known is likewise interesting. This, of course has to do with the evolution of the means of knowledge distribution. Beginning with the origins of language from the grunts and noises shared among proto-humans through knowledge shared as lore passed through oral traditions, to later, created in more durable form as the written word through the printing press, the large-scale distribution of writing though books, journals, magazines and the like to today’s near instantaneous globally shared electronic media knowledge, distribution is part of the formula for the increasing doubling rate of knowledge. With the advent of computer software such as ChatGBT, which do a fairly convincing job of manipulating vast amounts of data to distribution-on-demand as “new” knowledge, the rate of distribution of existing and newly created knowledge seems to go hand-in-hand with the doubling of knowledge per se. According to theories such as the “Tipping Point” as described by Ray Kurzweil, with his “Law of Accelerating Returns” John von Neumann’s “Technological Singularity,” Gordon Moore’s eponymous law on the doubling rate of the number of transistors fitting on an integrated circuit, et al. all point to the obvious idea that the knowledge doubling curve, technological advancement, etc., when projected forward in time eventually reach a point when advances occur at a rate measured in fractions of a second, eventually reach (absurdly?) infinitely small units of time when “something profound” will happen. Knowledge, it seems, cannot be knowledge without there also being a knower of that knowledge. For whom or to what will these increasingly vast amounts of knowledge become known? For humans, the total of knowledge that can be known, or accessed on-demand will appear a vast sea that will outstrip the faculties of the combined power of human brains on the planet. The next step would seem, logically, to be “machines” themselves that will be the “knowing” entities in that future time. Knowing then might resemble “omniscience,” perhaps? Science currently measures time frames in zeptoseconds. If the transformation of data-to-information-to-knowledge and the communication of such occurs in zeptoseconds, then the speed of light would dictate the limit of the size of any physical entity involved in this process. Mathematically, this process would have to be an entity much smaller than a sub-atomic particle. With our current level of understanding of the “doubling” rate of knowledge, we do not know whether that soaring upward curve on a graph is both infinite and inexorable-not only in our conceptual view but, particularly,in our given world of the actual.

Laughing my soul out when you almost wrote 2-dic numbers hahahahaha Excelente video and amazed with p-adic numbers

Good stuff thanks

you REALLY explain well [others LOVE to talk sooo fancy to make sure that nobody understands--- you are VERY different!! keep it UP

Infinite sum of powers of two is the mathematical version of integer overflow.

Thank you very much for this video!!! My partner shared a clip with me from a documentary about “deep history” using the knowledge curve/doubling concept. The “in a world” narrator said that “by 2020[it will double] every 72 hours”. It used dramatic music and stock footage of industry and technology over the last century or so. I was openly skeptical, and began looking into that claim. How do we quantify knowledge? How could we gather data on that knowledge? Your excellent video gives me half of the answer, and you presented it well.

If there were a world medal for best intuitive math explanation in a youtube video, then you would get the gold. ! Please don't stop - that would be a real shame for mathematics.

It's an exponential growth curve, isn't it?

AI doubles every 3 months, do a video on that disaster