How to Wrangle Infinity (an intro to p-adic numbers)

if you clicked on this video there's a good chance that at some point you've heard the result that if you add 1 plus 2 plus 3 plus 4 plus 5 on and on up to infinity that in some way that equals negative 1 12. now obviously this result isn't exactly true it depends on what we mean by summing up infinitely many numbers i mean this is an impossible thing to do and so how we define summing affects what the final answer is going to be but it turns out that there's a whole family of infinite sums that nonetheless equal something usually small and often negative and today i'm going to look at a strange one that the sum of all powers of two up to infinity equals negative one before we look at how we encounter such a strange result let's remind ourselves about some infinite sums that we already know how to do so let's take a look at the infinite sum of one plus a half plus a fourth plus an eighth plus a sixteenth and so on adding all of the reciprocals of powers of two now if you don't already know what this equals you can figure it out pretty easily by looking at the number line if we put zero here we'll put one here and two here we can see that adding one is just the same as going from zero to one then adding a half to that is the same thing as going from here to one and a half adding a quarter to this is the same thing as going a quarter the distance here to one and three quarters adding an eighth moves us half the way closer to two adding a sixteenth gets us closer still and for every term we add we're just getting closer and closer to two and so we say that this infinite sum equals two another way we can get this result is by using a formula and the formula is that the sum whatever this infinite series equals is equal to the first term we call that s sub zero in this case it's one divided by one minus the rate and the rate is whatever each next number is being multiplied by so here we see that every new number we're adding is just the previous number but multiplied by a half one times a half is a half a half times a half is a quarter a quarter times a half is an eighth so for this sum our rate r is a half so if we wanted to know what this infinite sum equaled we just plug in the first term one over one minus a half which would give us one over one half which is just two but now what happens if we look at another sum let's look at the sum of one plus two plus four plus eight plus sixteen on and on adding powers of two well if we go look at the formula we'll see that whatever this equals it should be equal to the first term one divided by one minus the rate and in this case every new number is just being multiplied by two right one times two is two two times two is four four times two is eight so the rate is two and this gives us one over one minus two is negative one which equals negative one now obviously this can't be true if i keep adding terms this just blows up to infinity i'm adding bigger and bigger numbers and yet somehow this formula that worked before is now telling us that this answer is going to be negative one now the simple way to fix this problem is just to note that this formula only works when r is less than one so in our previous example when r was a half this formula worked and now that r is greater than one it doesn't work well what if i told you that there's a system of numbers where this really is the right answer there's a system of numbers where if you add one plus two plus four plus eight and so on all the powers of two you really do get closer and closer to negative one the system of numbers is called p attic numbers where the p stands for sum prime so you could have 13 attic numbers or seven attic numbers or two attic numbers and in this world of periodic numbers sizes and scales and distances are all contorted it's like the alice in wonderland of numbers in the two attic system for example four is farther away from five than it is from eight so how do we get periodic numbers well to start let's remind ourselves what happens when we try to do arithmetic with infinity infinity is kind of slippery and what i mean by that is that if we add 1 to infinity well that's still equal to infinity if we multiply infinity by 2 well we still have infinity if we try dividing infinity by 2 we still get infinity so it seems like there's nothing i can do to change the fact that what i have is infinity well this is where periodic numbers come in let's imagine i have some number that starts with a three and then there's another three and another three and another three and another three and we let this number go on to the left for ever for infinity this is an infinitely large number but what happens if i try to multiply it by two well i'll get two times three is six two times three is six and six again and six again obviously this is just an infinite string of sixes so it seems like we multiplied this number by two and definitely got something new but remember this number is infinitely big and so is this one infinity times two is still infinity but now it looks like this is a different infinity it's almost like in this sense we can do arithmetic with infinity but in periodic systems we use the base of whatever p system we're in so if we're in a three attic system we're going to use base three numbers so we might have the number two two two two all the way to infinity going off to the left and we might want to add it to 1. now in a base 10 system when we add 1 to our largest digit which is 9 we roll over we get one zero in a base three system when we add one to our biggest digit which is now two we roll over and again we get one zero i can add these two digits again this is an addition table to get well one zero and then i'll add these to get one zero and this will go on forever i just get an infinite string of zeros so it seems like if i add an infinite string of twos to one that i get zero and what we actually just did was learn something really important about periodic numbers is that we can have negative numbers without actually having a negative sign anywhere in the 3 attic system this number is acting like negative 1. and now i briefly want to talk about the notion of convergence because it's important in periodic numbers when we were looking at the infinite sum of one plus a half plus a quarter plus an eighth and so on that infinite sum was converging on two the more terms we added the closer our final answer got to two so let's think about the sum of three plus thirty plus 300 plus 3000 and so on adding another zero each time it seems like this number is just gonna blow up to infinity but for every new term we add let's see what we get we start with 3 and if i add the next term i'll get 33. if i add the next term i'll get 333. if i add the next term i'll get 3333 and eventually i've just got what i had before this infinite string of threes it might feel to you that in some way this infinite sum was converging on this number even though this number was infinitely big if you were discovering all of this stuff on your own for the first time you might posit that what allowed us to multiply this infinitely large number by 2 to actually get something meaningful was that this number was an infinite sum that converged it was this infinite sum that converged to this so now that we've seen some things where we were able to do math with seemingly infinite numbers let's try to do some math with the sum we started with 1 plus 2 plus 4 plus 8 and so on remember we were able to do math with the 3 plus 30 plus 300 because that infinite sum seemed to converge on some infinitely large number as we were adding terms so let's start adding together these terms and see if anything converges well we start off with just one then when we add the first two terms we get three if we add the next we get seven if we add the next one we get 15 and then adding 16 to this we would get 31. continuing this we would get 63 and 127 255 this doesn't seem to be converging on anything maybe it would if we added more and more terms but unfortunately it just doesn't this number never settles on something the way the other sum settled on an infinite string of threes so let's remind ourselves real quick what we mean by writing numbers in different base systems if i have four hundred and in base 10 well what that means is that i have 4 times 10 squared right that's 400 plus 1 times ten to the one right that's just ten plus seven times ten to the zero this is just a fancy way of saying one four hundred plus ten plus 7 is just 417 and this is how we do this no matter what base system we're in so if i was in a base 3 system and i wrote down the number 201 well this wouldn't be the 201 we're familiar with what this is really is 2 times 3 squared plus 0 times three to the one plus one times three to the zero so what the two zero and one represent is what we're multiplying each of these powers of three by as the powers of three go down and now we finally get to tie everything we've learned together we saw earlier that we weren't able to get one plus two plus 4 plus 8 all the powers of 2 to converge on something the way we were able to get all of those 3 times 10 to the somethings to converge on an infinite string of 3s but maybe that's just because we're looking at it from the wrong angle we're looking at this infinite sum in base 10 maybe we should be looking at it in base 2 or a 2 attic attic system so if i've got 1 plus 2 plus 4 plus 8 well in base 2 what i really have is 1 times 2 to the 0 plus 1 times 2 to the 1 plus 1 times 2 squared plus 1 times 2 cubed and so on and so in base 2 this infinite sum can be represented as one one one one one and now doing the convention where the powers are getting bigger as we go to the left like they normally are this would just be an infinite string of ones just like our infinite string of threes and now for the final piece to the puzzle let's see what happens if we add this infinite string of ones that we know that this sequence equals to one well in base two remember the biggest digit that we have is one and so we roll over one plus one in base two is one zero one plus one is one zero one plus 1 again is 1 0 and 1 0 so this infinite string of ones is nothing more than negative 1. so long as we're looking in a two attic system so just to tie some things in full circle when we were looking at that formula earlier that was giving us negative one that wasn't an accident the derivation of that formula works for values of r that number in the denominator greater than one if you're in different number systems where these infinite sums actually converge so it's not just a coincidence it's just that that formula will only work if you're in the right number system where the sum is converging okay so if it feels like i glossed over too many details way too fast it's because i did there's really a lot going on here and i wanted to just get us to a cool result without going into too much detail that would burden the whole video but maybe i'll make a video about some of those details later on so let me know if you guys want me to make another video talking about some of the more fine details of piatic numbers and thanks for watching [Music] [Music] you

Channel: SuperScript

Views: 4,292

Rating: 4.889328 out of 5

Keywords: math, education, number, primes, p-adic, infinity

Id: o02uipdcT7Y

Length: 15min 42sec (942 seconds)

Published: Sun Jan 31 2021

Reddit Comments

[u/excarnateSojourner]

Reminded me of Numberphile's video on 1 + 2 + 3 ... = - 1/12, but I don't recall them really explaining how this result can seem to be correct and also preposterous simultaneously. Well done!