Transcription
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 p-adic 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... 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 p-adic numbers? These exist for any integer p. I just want to give you an example for p=10. P-adic 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... Now we add one. That's 1000... 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. Write down zero, carry one, and so on. The result is 000... 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. Gabriel's Horn. The surface obtained by rotating the curve 1/x for x larger than one about the x-axis has finite volume but infinite surface area. In other words, you could fill it with paint but could never coat its surface.
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 math problems can be surprisingly hard to solve.
Seven. Metalogical 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 Gödel's theorem.
Six. The 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: U(1), SU(2), and SU(3). 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 10^54. 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 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 * (1 - your 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. 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 sine 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 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 the real or complex numbers we work with are algebraic. This means there are solutions to some polynomial 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, the 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 Bonus 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? Let me know in the comments.
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