I watched the veritasiunm vid and I was confused like hell when in the start he said after squareing infinity the number squared the original number and as a 13 years old fella I am stupid
In: 0
Normally, when we write a whole number, we use a finite number of digits: eg 1729.
Some numbers have more digits: eg, 23939041729.
We could, if we wanted to, imagine there’s an infinite string of zeroes before the start of the number:
1729 = ….00000000000001729.
23939041729 = …000000000023939041729.
If you want the 10-adic numbers, then you allow other digits, not just 0, infinitely stretching to the left: eg, ….33333330003959, or …..94934720058920000002393933.
I know he said at the start of the video “surely these are just infinity?” but when using p-adic numbers, the answer is “no, don’t think of these as infinity. They aren’t”.
For example, if I take the number ….857142857142857143:
* Well, if we multiply it by 7, then we get …0000000000000001 (as shown in the video).
* But we already know that …00000000000001 is the usual whole number 1.
* If ….857142857142857143 x 7 = 1, then surely ….857142857142857143 = 1/7.
So the 10-adic numbers can include rational numbers like 1/7. And also negative numbers like -1/7.
So they have something in common with our usual number system. On the other hand:
* there are numbers in our normal number system that are missing from the 10-adics: for example, 1/2. It’s not possible to find a string of digits which, when you double it, you get …000000000001. No matter what you pick, the 1’s digit will be even. So the 10-adics only have some fractions, not all of them.
* but there are also numbers the 10-adics have that our normal number system doesn’t have. Using normal numbers, if you want to solve X^2 = X, there are only two solutions: 0 and 1.
* With the 10-adics, however, there are more solutions:
* ….0000000000000 x ….000000000000 = …000000000000 (ie, 0 x 0 = 0)
* ….0000000000001 x ….000000000001 = …000000000001 (ie, 1 x 1 = 1)
* ….259918212890625 x ….259918212890625 = ….259918212890625 (a brand new number that he started the video with)
* …7743740081787109376… x ….7743740081787109376 = …7743740081787109376 (yet another brand new number).
That brand new number isn’t “infinity”. If it was, then every 10-adic would be infinity, but they don’t all give themselves back when you square them. In fact (amongst the 10-adics), it’s just those four.
I’d recommend watching the Veritasium video through a couple of times to try and cement some of its content a bit more, but I can give another explanation here.
The types of numbers he is talking about here are not infinity. They are their own objects in the world of mathematics, with infinitely many digits of increasingly larger size. But they are not the same thing as the value infinity.
When we do a fraction, we often get a value with infinitely many digits AFTER the decimal. For example, 1/7=0.142857142857142… you can see there is a pattern there, where the section “142857” repeats over and over again. Let’s see what happens when we multiply 0.142857142857142… by 7, doing it column by column
0*7 = 0, so we write down 0
0.1*7 = 0.7, so we’re at 0.7
0.04*7 = 0.28, so we add 0.7 and 0.28 and get 0.98
0.002*7 = 0.014, so we add 0.98 and 0.014 and get 0.994
0.0008*7 = 0.0056, so we add 0.98 and 0.014 and get 0.9996
If we keep going, we get 0.999… and the 9s go on forever. We started with 1/7, wrote it out as decimals, multiplied them by 7, and got 0.999… But 1/7 multiplied by 7 must be 1. So 0.9999… IS the same as the number 1, it’s just another way of writing that number.
p-adic numbers are kind of like numbers where all those repeating digits go BEFORE the decimal. As Veritasium shows in the video, there are p-adic numbers that ARE the same as numbers we know. In fact, right [here](https://youtu.be/tRaq4aYPzCc?t=151) he shows a number with infinite digits BEFORE the decimal, that when you multiply it by 7, column by column, just as I did above, you get a 1 preceded by infinite zeroes. So the “p-adic” number* “…142857142857143” is exactly the same as the number we know as 1/7. It can be written 1/7, it can be written 0.142857142857142…, or it can be written …142857142857143. It’s the same number.
The number he shows for n^2=n does not have a neat repeating pattern. He shows quite well in the video how he came up with the number by repeatedly squaring 5 and taking the end of it. That gets him to a p-adic number* that ends …256259918212890625
And so, In a similar “digit-by-digit” multiplication method, if you multiple …256259918212890625 by itself, you’ll find it also ends …256259918212890625. For example, the unit column MUST be a 5 because 5*5 is 25 which ends in 5, and that’s the only contribution to the units column. When you do that kind of calculation for each column in turn – the 10s, then the 100s, then the 1000s… you find it always is the same as the original number.
So his p-adic number …256259918212890625 is its own square.
But I think what you really want to know is, there’s no way for a number to be its own square, unless it’s maybe infinity (or 0 or 1). But this number is not infinity, and it is not a natural number or integer. It’s a new type of number, called a p-adic number. It has its own properties and needs to be handled in its own way, just like how negative numbers were new to people at one point, and fractions were new when they were introduced. It doesn’t sit on the number line you’re used to, the one that goes from -infinity to infinity and you can point at a point and say “this point on the line is 231.5483592465…” – it doesn’t fit on that line at all! But it’s constructed out of numbers on that line using well-defined mathematical operations as shown in the video, so it is **a number**. Just not one on the number line!
* this number is technically not p-adic since it’s in base-10, but it works for the demonstration.
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