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.
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