- And again, going to Perplexity, "Transcendental numbers are 'real' or 'complex' numbers; they are not the root of any nonzero polynomial with integer or rational coefficients. This means they cannot be expressed as solutions to algebraic equations with integer coefficients, setting them apart from algebraic numbers."- So some of the famous transcendental numbers would include the number pi, you know, the 3.14159265 and so on.
The average of two fractions is another fraction. And so, sometimes people, it seems to be a different character than the integers , which are discretely ordered, right? From any integer , there's a next one and a previous one, and so on. But that's not true in the rationalnumbers. And yet, the rational numbers are also still only a countable infinity. And the way to see that is actually
But what about the integers ? Integers stretch off to infinity in both the positive and negative directions.Well Cantor realized he could just pick zero as the starting point and from there his ordering went one, negative one, two,
of Hilbert's Hotel becomes kind of clear. There are many other ways to talk about it too. For example, let's think about, say, the integer lattice, the grid of points that you get by taking pairs of natural numbers, say, so the upper right quadrant of the integer lattice, yeah? So there's the, you know, row zero, row one, row two and so on, column zero, column one, column two and so on,
And we cook this method up so that if our knots were the same we would have assigned them the same integer . This integer it's really a property of the knot. So that's great, because it lets me say, well, I got five and you got seven.
It was not the exact form of the conjecture which now bears his name, but something related. "It seems that every integer greater than 2 can be written as the sum of three primes." Euler was intrigued by the proposition, but he thought the idea could be refined further.
And when people want to study sliceness, they still use invariance, but they come up with some invariant. Maybe it's still integer valued, but it has this property that when you compute this integer for a slice knot, you're supposed to get zero. And if you compute it for a non-slice knot, you get, I don't know, not zero.
We don't want any dominoes overlapping anything else. So this is another integer programming problem. It seems very similar to the linear assignment problem.
So to kind of think in terms of how it was like programming this machine, think in terms of restricting yourself to 20 integer variables. If you can't solve atomic bomb problems with 20 integer variables, you're not good enough to program the ENIAC.
having finished what they were doing with that first pulse. So small integer coefficients in your equations don't actually have to go through the multiplier.
In this case, six, you always know which number comes before and which comes next. But what about the integers ? Integers stretch off to infinity in both the positive and negative directions.
Nelson has not seen these numbers. I have 30 integers . I'm reading out to him.
real number system. But then also, we have the algebraic numbers like the square root of 2 or the cube root of 5 and so on. Numbers that solve an algebraic equation over the integers , those are known as algebraic numbers. It was an open question for a long time whether that was all of the real numbers or whether there would exist numbers that are the transcendental numbers. The transcendental numbers are real numbers that are not algebraic.
I've got three patterns here. The coordinates are integers or ??
And as a way to introduce you, in 2014, you sparked a massive controversy on the internet with your video where you show what happens when you add up all the positive integers at once. Can you share a bit about that video and the reaction?
So for example, when people say, well, the sum obviously just goes up to infinity, and it's clearly true that if you take any finite-- take the first billion integers and add them together, you're going to get a big number. If you take the first trillion, you're going to get an even bigger number, and so on.
So the regular particle has some phenomena of their conductance, like the resistance or conductance we usually measure can be quantized in units of 0, 1, 2, 3, and so on. So they behave like integers in a quantization step. So we once had a Eureka moment that, if the Majorana particle is in some sense half of a regular particle, then
So the idea here is that I can add up simple harmonic periodic waves that we just spoke about that I can understand purely in terms of integers . The minute I specify an integer , I know something about the wave.
All of them are related by a pattern called the laws of physics. Just like the integers 0, 1, 2, 3, minus 1, minus 2, minus 3, the number two happens after the number one and before the number three, but we don't say that the number one is the cause of number two, or two is the cause of three.
So there's a power operator-- which you see is a very similar symbol to the mathematical exponentiation, it just has a dieresis above it-- We have 1 bit integers , 8 bit, 16, 32 bit integers in the implementation.
We can say, for example, 6. You know that's an even integer . So can we write it as the sum of two primes?
And then we have 25 constraints that say that all of the xij's are binary, 0 or 1. This is what's called an integer programming problem, discrete linear optimization problem. Some IPs are very hard.
But most of the time when we solve this, it takes very little time for a good optimization solver. Often the relaxation yields an integer value solution, as is guaranteed in the linear assignment problem. If that's not the case, usually a very small branching boundary is needed.
And what you are going to do is pick a positive integer . The person who submits the lowest integer which no one else has chosen will be one winner. And then, since we need two, the person who picks the second lowest integer that no one has chosen.
So we once had a Eureka moment that, if the Majorana particle is in some sense half of a regular particle, then they should display some plateau at half integer steps. Namely, at 1/2, 3/2, and so on and so forth.
And that's good enough. You end up with a positive integer .
And that's good enough. And it's a weirdly arbitrary looking integer .
purely in terms of integers . The minute I specify an integer , I know something about the wave. So I can think of waves, these harmonic waves, as an alphabet.
And then there are results that seem impossible. Like that the sum of all positive integers up to infinity is negative 1 12th. How did he come up with this?
rational and irrational numbers. We've spoken about the rational numbers, and the rational numbers, by the way, are by definition, the numbers that can be represented as a fraction of two integers . - That's right. So with the real numbers, we have the algebraic numbers. We have, of course, all the rational numbers. The integers and the rationals are all part of the
And if you invert the integers to 17, it's like we were making a movie about current events.
And to me, speaks to what I find fascinating about real math. The way that seemingly simple properties of integers , addition, and so on yields this wonderful rich complexity. And so, here to introduce that, Avner and Rob.
It's not easy. I'll give you a little more homework, which is much more interesting. The sum of the first n integers is n times n plus 1 over 2. So show directly that if you add up the first n integers and square it, you get the sum of the first n cubes.
So there's a power operator-- which you see is a very similar symbol to the mathematical exponentiation, it just has a dieresis above it-- And the nice thing about them being integers is that they are in the domain of all the mathematical functions.
Do you think that other sort of integers like format or display or the ways in which people interact with novels can be as form-bending as like pieces of a novel itself?
So those dots, if you think about them as dots, are really the same as the train cars if you think about each column of... in that integer lattice, it's a countable infinity. It's like one train car and then there's the next train car next to it, and then the next column next to that, the next train car.
How long did it take for you to factorize an integer 100 digits long?
to 20 integer variables. If you can't solve atomic bomb problems with 20 integer variables, you're not good enough to program the ENIAC. OK. That's a little overstated, but you kind of get the idea.
OK. So this was a Riddler last summer. And what you are going to do is pick a positive integer . The person who submits the lowest integer which no one else has chosen will be one winner.
So it means if you allow i to become kind of integer , so integer plus integer times i are another new integer ,
And it begins actually with Pythagoras, who was the one that came up with the Pythagorean scale by basically looking at this thing called a mono chord, one string instrument and actually moving a bridge in these integer relations of the length of the string and generating the scale. And Pythagoras really believed in this idea of this harmony of the spheres.
I would find the piece of paper with the lowest integer choice.
There's no loose color or charge or non-integer .
It's the upper band of a single byte signed integer .
Well Cantor realized he could just pick zero as the starting point and from there his ordering went one, negative one, two, negative two, ranking the integers by their absolute value, their distance from zero. It doesn't matter if you put the positives first or the negatives first, as long as you are consistent.
Ordering them this way is actually what allows us to map the integers to the natural numbers and see that both sets are the same size, but there are other ways we could well order the integers . We could start with zero and then have one, two, three, all the way to positive infinity and then negative one, negative two, negative three,
Yeah, so I think when you think of it that way, I think we presented it in a way that was meant to be for a lay audience. And the result that the some of the integers , we can debate about whether it's equal to minus 1/12 or whether it should be assigned a value minus 1/12. But that result is a robust resource which is used extensively, for example, in string theory and in theoretical physics in general,
That's the Poincare recurrence. infinite sums like our famous sum of the integers , which kind of brings us back full circle, right?
The sum of the first n integers is n times n plus 1 over 2. So show directly that if you add up the first n integers and square it, you get the sum of the first n cubes. I'm actually going to give you a little bit more of an assignment.
All we see is kind of a many natural integers .