What does it feel like to invent math?

Take 1 plus 2 plus 4 plus 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 negative 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,

those 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 ½ plus 1 fourth plus 1 eighth plus 1 sixteenth on and on up to infinity,

whatever that means, equals 1, and imagine that 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 that 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 that 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 ½, ½ plus a fourth, ½ plus a fourth plus an eighth, 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're tempted to say that if these numbers approach anything,

we should be able to write this thing down as a sum that contains the reciprocal 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 the statement off 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 you wrote down, since it kind of feels like nature gave it to you.