What happens if you just keep squaring?

take the number five and square it you get 25 now take 25 and square it you get 625 Square 625 and you get 39,6 125 do you see the pattern 5 squar ends in a five 25 squar ends in 25 and 625 squar ends in 625 so does this pattern continue well let's try squaring 39,6 125 it doesn't quite end in itself but the last five digits match so it extends the pattern by a few places so let's try squaring just that part 9,625 that does end in itself and if we square that whole number it also ends in itself and now we're up to 10 digits and you can keep doing this squaring the part of the answer that matches the previous number and increasing the number of digits they share in common it's as though we are converging on a number but not in the usual sense of convergence this number will have infinite digits and if you square it you'll get back that same number the number is its own Square now I bet you're thinking does it even make sense to talk about numbers that have infinite digits going off to the left of the decimal point I mean isn't that just Infinity in this video my goal is to convince you that such numbers do make sense they just belong to a number system that works very differently from the one we're used to and that allows these numbers to solve problems that are impenetrable using ordinary numbers which is why they are a fundamental tool in Cutting Edge research today in number Theory algebraic geometry and Beyond so let's start by looking at the properties of the number system that includes the number we just found we'll call these 10 adct numbers because they're written in base 10 if you have two 10 addict numbers can you add them together sure you just go digit by digit from the right to the left adding them up as usual addition is not a problem what about multiplication well again you can take any two 10 addict numbers and multiply them out this works because the last digit of the answer depends only on the last digits of the 10 addict numbers and subsequent digits depend only on the numbers to their right so it might take a lot of work ...