## Intro [00:00] Mathematics is amazing because it reveals truths, unambiguous, provable truths that defy expectations. Today, I want to tell you my favorite mindblowing math facts. Starting with number 10, the ## 10: P-adic numbers [00:15] piodic numbers. We're used to working with real numbers where we have some digits before the point and then possibly some after the point and all the way to infinitely many digits. In the real numbers, 0.999 and an infinite number of nines is equal to one. But you already knew that. Did you also know though that there's a completely different way to do addition with what's called the paddic numbers? These exist for any integer p. I just want to give you an example for p= 10. Pi numbers have expansions that go to infinitely many digits to the left. So to larger values rather than to the right to smaller values. This leads to the following stunning addition law. Suppose you have 999 and so on all the way to infinity to the left. Now we add one that's 1 0 0 and so on to the left. What do we get? When you add the rightmost 9 + one that gives 10. Write down zero carry one. Add to the next nine that gives 10. Right down zero carry one. and so on. The result is 0 0 0 all the way to infinity to the left which is well zero. This means that this infinite string of nines to the left is actually minus one. I'm not making this up. This is actually how it works. Nine ## 9: Gabriel’s Horn [01:45] Gabriel's horn. The surface obtained by rotating the curve 1 /x for x larger than one about the xaxis has finite volume but infinite surface area. In other words, you could fill it with paint but could never coat its surface. ## 8: The most optimal packing for 17 squares [02:04] Eight. The most optimal packing for 17 squares. Imagine you have a set of n square tiles. What's the smallest larger square that they'll all fit into without overlapping? The answer is obvious. If n is a square number and for most other numbers, the results look reasonably enough. For 17 squares, the best known result is this. This is the best known arrangement. It's not been proved that it's actually the best one. The lesson here is that even simple maths problems can be surprisingly hard to solve. ## 7: Meta-logical contradictions [02:42] Seven, metalological contradictions. This sentence is false is a classical example of a contradiction caused by using a language to make statements about itself. If the sentence is false, then it's true. And if it's true, then it's false. So what is it? Another example is the barber paradox. The barber cuts the hairs of all people who don't cut their own hair. Does he cut his hair or does he not? If he does, he doesn't. If he doesn't, he does. The mathematical version of this is the question of whether the set of all sets that don't contain themselves contains itself. You might have heard of these already, but maybe not this one known as Barry's paradox. What is the smallest positive integer not definable in under 60 letters? The problem is that this phrase itself has 57 letters. So if you could find the number, it wouldn't fulfill its own definition. Does the number exist? These logical problems are all related to Good's theorem. Six. The ## 6: The monster group [03:50] monster group. Mathematicians use groups a lot. These are basically sets with elements that can act on each other and obey certain relations. A typical example is the rotation group in three dimensions whose elements are the rotations in the three directions that you can then combine. These rotation groups exist in any number of dimensions. In fact, most groups exist in these infinite countable series and are reasonably well behaved like the groups in the standard model of particle physics U1, SU2, and SU3. However, besides these infinite series of groups, there are also 26 so-called sporadic simple groups. The largest of them is the monster group. The number of its elements is exactly known and it comes out to be this. In case you don't feel like counting digits, that's approximately 10 to the 54. The stunning thing is that this is provably the largest such group. It's not just the largest known one, it's the largest one, period. That number is a fundamental truth. Five, the logistic ## 5: The logistic map [05:02] map. The logistic map is defined as a sequence of numbers and looks entirely unremarkable. It has only one parameter that I'll write as r. You start with some number between zero and one and then you calculate the next number as r * your starting value times 1 minus the starting value that gives you a new number and then you do it again. For example, if you take r= 3.5 and start with the initial value 0.5, then you get 0.875. Then you do it again and get 0.382. And you do it again and get 0.827 and so on. Looks simple enough. But where does this sequence end? Well, here's the amazing thing. For most values of R, it doesn't end anywhere. You can keep track of the values that the function visits after many steps and plot them as a function of R. At low R, you see a single branch. This is where the sequence settles. It's a fixed point. But when r becomes larger than three that splits into two meaning that the sequence ends up going back and forth between two values. Increase r further and you get more values and then at some point that's approximately r= 3.57 you have the onset of chaos with occasional windows of periodic orbits. The amazing thing here is that such a simple rule can give such a complex result. Four ## 4: Wild Singular Limits [06:33] wild singular limits. A singular limit is a case where the behavior of a sequence suddenly and unpredictably changes. A particularly stunning example is this sequence of integrals over the sign function where you add more factors under the integral. The first gives pi / 2, the second gives pi / 2, the third gives pi / 2. These are exact numbers, not approximations. Yet, when you take 15 factors, it stops working. Three, the ## 3: The Birthday Problem [07:04] birthday paradox. Suppose you're at a party attended by two dozen people. What's the probability that two of them share the same birthday? It's more than 50%. You can calculate the probability of that happening for any number of people, and it makes a surprising jump at about 23. If you have a group of 60 people, the probability that two of them share a birthday is larger than 99%. Two, we don't know most numbers. Most of ## 2: We can’t know most numbers [07:33] the real or complex numbers we work with are algebraic. This means there are solutions to some polomial with rational coefficients. The roots of something basically. However, there are numbers which cannot be written that way. They're called transcendental numbers. The most famous transcendental number is pi. And we've all heard of pi, but it seems rather special. Yet, fact is that almost all real numbers are transcendental. We just can't use them because we can't write them down. Think about it. We can enumerate all possible algorithms to compute numbers. Yet, there are more transcendental numbers than possible algorithms. They're everywhere. and yet in some sense unusable. Bonus fact, you might have heard that every random sequence of digits eventually appears in the digits of pi. But actually, this has never been proved to be true. It's an open question. And finally, the Bonaski ## 1: The Banach–Tarski paradox [08:37] paradox. You can decompose a solid sphere in three-dimensional space into finitely many disjoint pieces and then reassemble them using only rotations and translations into two spheres each the same size as the original. What even is space? How many of those did you know? ## 10:19 Learn Science with Brilliant! [09:01] Let me know in the comments. If this video inspired you to brush up your mathematics knowledge, I recommend you start with Brilliant. All courses on Brilliant have interactive visualizations and come with follow-up questions. What you see here is from their newly updated maths courses, no matter how abstract the topic seems. Brilliant courses have intuitive visualizations that really click into my brain. And Brilliant covers a large variety of topics in science, computer science, and maths from general scientific thinking to dedicated courses, just what I'm interested in. And they're adding new courses each month. I really enjoy the courses on Brilliant, not just because they keep my brain active, but also because it's a great way to systematically build up new knowledge to higher levels. If that sounds like the right thing for you, use my link brilliant.org/ org/zabini to give it a try. First 30 days are free and with this link you'll get 20% off the annual premium subscription. It's a great way to learn more and to support this channel. Thanks for watching. See you tomorrow.