Take 1+2+4+8 and continue on and on adding the next power of 2 up to infinity.
This might seem crazy, but there’s a sense in which this infinite sum equals -1.
If you’re like me, this feels strange or obviously false when you first see it
but I promise you, by the end of this video you and I will make it make sense.
To do this, we need to back up, and you and I will walk through what it might feel like
to discover convergent infinite sums, the ones that at least seem to make sense
to define what they really mean, then to discover this crazy equation
and stumble upon new forms of math where it makes sense.
Imagine that you are an early mathematician in the process of discovering that 1/2 + 1/4
+ 1/8 + 1/16, on and on up to infinity, whatever that means, equals 1, and imagine you needed
to define what it means to add infinitely many things for your friends to take you seriously.
What would that feel like? Frankly I have no idea, and I imagine that more than anything
it feels like being wrong or stuck most of the time
but I’ll give my best guess at one way the successful parts of it might go.
One day you are pondering the nature of distances between objects, and how no matter how close
two things are, it seems thast they can always be brought a little bit closer together without touching
Fond of math as you are, you want to capture this paradoxical feeling with numbers
so you imagine placing the two objects on the number line, the first at 0, the second at 1
Then you march the first object towards the second such that with each step
the distance between them is cut in half.
You keep track of the numbers this object touches during its march
writing down 1/2, 1/2 + 1/4, 1/2 + 1/4 + 1/8, and so on.
That is, each number is naturally written as a slightly longer sum with one more power of 2 in it. As such,
you are tempted to say that if these numbers approach anything, we should be able to write that thing as a sum
that contains the reciprocals of every power of 2. On the other hand, we can see geometrically
that these numbers approach 1, so what you want to say is that 1 and some kind of infinite sum
are the same thing.
If your education was too formal, you’d write off this statement as ridiculous.
Clearly you can’t add infinitely many things.
No human, computer, or physical thing ever could perform such a task.
If, however, you approach math with a healthy irreverence, you’ll stand brave in the face
of ridiculousness and try to make sense out of this nonsense that you wrote down, since
it kind of feels like nature gave it to you.
So how exactly do you, dear mathematician, go about defining infinite sums?
Well-practiced in math that you are, you know that finding the right definitions is less
about generating new thoughts than it is about dissecting old thoughts, so you go back to
how you came accross this fuzzy discovery.
At no point did you actually perform infinitely many operations.
You had a list of numbers, a list that could keep going forever if you had the time, and
each number came from a perfectly reasonable finite sum.
You noticed that the numbers in this list approach 1, but what do you mean by “approach”?
It’s not just that the distance between each number and 1 gets smaller, because for
that matter the distance between each number and 2 also gets smaller.
After thinking about it, you realize what makes 1 special is that your numbers can get
arbitrarily close to 1.
Which is to say, no matter how small your desired distance, 1/100th, 1/1,000,000th,
or one over the largest number you can write down, if you go down the list long enough,
the numbers will eventually fall within that tiny tiny distance of 1.
Retrospectively this might seem like the clear way to solidify what you mean by “approach”,
but as a first time endeavor it’s actually incredibly clever.
Now you pull out your pen and scribble down the definition of what it means for an infinite
sum to equal some number, say X.
It means that when you generate a list of numbers by cutting off your sum at finite
points, the numbers in this list approach X, in the sense that no matter how small a
distance you choose, at some point down the list all the numbers start falling within
that distance of X.
In doing this, you just invented some math. But it never felt like you were pulling things
out of thin air. You were just trying to justify what it was the universe gave you in the first
place.
You might wonder if you can find other, more general truths about these infinite sums that
you just invented.
To do so, you look for where you made any arbitrary decisions.
For instance, when you were shrinking the distance between your objects, cutting the
interval into pieces of size ½, ¼, etc., you could have chosen a proportion other than ½.
You could have instead cut your interval into pieces of size 9/10 and 1/10
and then cut that rightmost piece into the same proportions
to get even smaller pieces of size 9/100 and 1/100
then cut that tiny piece of size 1/100 similarly, continuing on and on, you’d see that
9/10 + 9/100 + 9/1000 on and on up to infinity equals 1, a fact more popularly written as
.9 repeating = 1.
To all of your friends that insist that this does not equal 1, and it just approaches it
you can now just smile because you know that with infinite sums, to approach and to equal
mean the same thing.
To be general about it, let’s say that you cut the interval into pieces of size p and
(1-p), where p represents any number between 0 and 1.
Cutting the piece of size p in similar proportions, we now get pieces of size p(1-p) and p^2.
Continuing in this fashion, always cutting up the rightmost piece into those same proportions,
you will find that (1-p)+p(1-p)+p^2(1-p), on and on always adding p to the next power
times (1-p), equals 1.
Dividing both sides by (1-p), we get this nice formula.
In this formula, the universe has offered a weird form of nonsense
Even though the way you discovered it only makes sense for values of p between 0 and 1
the right hand side still makes sense when you replace p with any number, except maybe for 1.
For instance, plugging in -1, the equation reads 1-1+1-1 on and on forever
alternating between the two, equals 1/2
which feels both pretty silly and kind of like the only thing it could be.
Plugging in 2, the equation reads 1 + 2 + 4 + 8 + ... = -1
something which doesn’t even seem reasonable.
On the one hand, rigor would dictate that you ignore these, since your definition of
infinite sums doesn’t apply in these cases. The list of numbers that you generate by cutting
off the sum at finite points doesn’t approach anything.
But you are a mathematician, not a robot, so you don’t let the fact that something
is nonsensical stop you.
I will leave this sum for another day, so that we can jump directly into this monster.
First, to clean things up, notice what you get when you cut off the sum at finite points:
1, 3, 7, 15, 31. They are all one less than a power of 2.
In general, when you add up the first n powers of 2, you get 2^{n+1} - 1, which this animation
hopefully makes clear.
You decide to humor the universe and pretend that these numbers, all one less than a power
of two, actually do approach -1.
It will prove to be cleaner if we add one to everything and say that the powers of two
approach zero.
Is there any way that this can make sense?
In effect, what you are trying to do is make this formula more general, by saying that
it applies to all numbers, not just those between 0 and 1.
Again, to make things more general you look for any place where you made an arbitrary choice.
Here, that place turns out to be very sneaky.
So sneaky, in fact, that it took mathematicians until the 20th century to find it. It’s
the way that we define distance between two rational numbers.
That is to say, organizing them on a line might not be the only reasonable way to organize
them.
The notion of distance is essentially a function that takes in two numbers and outputs a number
indicating how far apart they are.
You could come up with a completely random notion of distance, where 2 is 7 away from
3, and ½ is 4/5ths away from 100, and all sorts of things, but if you want to actually
use a new distance function the way that you use the familiar distance function, it should
share some of the same properties.
For example, the distance between 2 numbers shouldn’t change if you shift them both
by the same amount. From 0 and 4 should be the same distance away as 1 and 5, or 2 and
6, even if that same distance is something other than four as we're used to.
Keeping things general, the distance between two numbers shouldn't change if you add the
same amount to both of them.
Let's call this property “shift invariance”.
There are other properties that you want your notion of distance to have as well, like the
triangle inequality, but before we start worrying about those, let's start imagining what notion
of distance could possibly make powers of 2 approach 0, and which is shift invariant.
At first you might toil for a while to find a frame of mind where this doesn’t feel
like utter nonsense, but with enough time and a bit of luck you might think to organize
your numbers into rooms, sub-rooms, sub-sub-rooms, and so on.
You think of 0 as being in the same room as all the powers of 2 greater than 1, as being
in the same sub-room as all powers of 2 greater than 2, as being in the same sub-sub-room
as powers of 2 greater than 4, and so on, with infinitely many smaller and smaller rooms.
It's pretty hard to draw infinitely many things, so I'm only going to dwar four different sizes,
but keep in the back of your mind that this process should be able to go on forever.
If we think of every number as lying in a hierarchy of rooms, not just 0, shift-invariance
will tell us where all of numbers must fall. For instance, 1 should be as far away from
3 as 2 is from 0.
Likewise the distance between 0 and 4 should be the same as that between 1 and 5, 2 and
6, and 3 and 7.
Continuing like this, you will see which rooms, sub-rooms, sub-sub-rooms and so on successive
number must fall into.
You can also deduce where negative integers must fall, where for example -1 has to be
in the same room as 1, in the same sub-room as 3, the same sub-sub-room as 7, and so on,
always in smaller and smaller rooms with numbers one less than a power of 2, because 0 is in
smaller and smaller rooms with the powers of 2.
So how do you turn this general idea of closeness based on rooms and sub-rooms into an actual
distance function?
You can’t take this drawing too literally, since it makes 1 look very close to 14, and
0 very far from 13, even though shift-invariance should imply that they are the same distance
away.
Again, in the actual process of discovery you might toil away, scribbling through many
sheets of paper, but if you have the idea that the only thing which should matter in
determining the distance between two objects is the size of the smallest room they share,
you might come up with the following: any numbers lying in different large yellow rooms
are a distance 1 from each other; those which are in the same large room, but not the same
orange sub-room, are a distance ½ from each other; those that are in the same orange sub-room
but not in the same sub-sub-room are a distance ¼ from each other.
And you continue like this, using the reciprocal of larger and larger powers of two to indicate
closeness.
We won’t do it in this video, but see if you can reason about which rooms other rational
numbers like 1/3 and ½ should fall into, and see if you can prove why this notion of
distance satisfies many of the nice properties we expect from a distance function, like the
triangle inequality.
Here I’ll just say that this notion of distance is a perfectly legitimate one, we call it
the “2-adic metric”, and falls into a general family of distance-functions called
the “p-adic metrics”, where p stands for any prime number.
These metrics each give rise to a completely new type of number, neither real nor complex,
and have become a central notion in modern number theory.
Using the 2-adic metric, the fact that the sum of all powers of 2 equals -1 actually
makes sense, because the numbers 1, 3, 7, 15, 31 and so on genuinely approach -1.
This parable does not actually portray the historical trajectory of discoveries, but,
nevertheless, I still think it’s a good illustration of a recurring pattern in the
discovery of math.
First nature hands you something that is ill-defined, or even nonsensical. Then you define new concepts
that make this fuzzy discovery make sense, and these new concepts tend to yield genuinely
useful math and broaden your mind about traditional notions.
So, in answer to the age-old question of whether math is invention or discovery, my personal
belief is that discovery of non-rigorous truths is what leads us to the construction of rigorous
terms that are useful, opening the door for more fuzzy discoveries, continuing the cycle.