e to the pi i equals negative 1
is one of the most famous equations in math, but it's also one of the most
confusing.
Those watching this video likely fall into one of three categories
1) you know what each term means, but the statement as a whole seems nonsensical,
2) you were lucky enough to see what this means and some long formulae explaining why it
works in a calculus class,
but it still feels like black magic, or 3) it's not entirely clear with the
terms themselves are.
Those in this last category might be in the best position to understand the explanation
I'm about to give
since it doesn't require any calculus or advanced math, but will instead require an
open-minded to reframing how we think about numbers.
Once we do this, it will become clear what the question means,
why it's true and most importantly why it makes intuitive sense.
First let's get one thing straight,
what we we write as e to the x is not repeated multiplication
that would only make sense when x is a number that we can count 1, 2, 3 and so on,
and even then you'd have to define the number the number e first. To understand what this
function actually does,
we first need to learn how to think about numbers as actions.
We are first taught to think about numbers as counting things, and addition and
multiplication are thought of with respect counting.
However is more thinking becomes tricky when we talk about fractional amounts,
very tricky when we talk about irrational amounts, and downright nonsensical
when we introduce things like the square root of -1.
Instead we should think of each number as simultaneously being three things
a point on an infinitely extending line, an action which slides that line along itself,
in which case we call it an "adder", and an action which stretches the line
in which case we call it a "multiplier". When you think about numbers as adders,
you could imagine adding it with all numbers as points on the line
all at once. But instead, forget that you already know anything about addition
so that we can reframe how you think about it.
Thing of adders purely as sliding the line with the following rule:
You slide until the point corresponding to zero ends up where the point corresponding
with the adder itself started
When you successively applied two adders, the effect will be the same as just applying some other adder.
This is how we define their sum. Likewise,
forget the you already know anything about multiplication, and think of a
multiplier purely as a way to stretch the line.
Now the rule is to fix zero in place, and bring the point corresponding with
one, to where the point corresponding with the multiplier itself started off,
keeping everything evenly spaced as you do so. Just as with adders
we can now redefine multiplication as the successive application
of two different actions. The life's ambition of e to the x
is to transform adders into multipliers, and to do so is naturally as possible.
For instance, if you take two adders, successfully apply them,
then pump the resulting sum through the function, it is the same as first putting
each adder through the function separately,
then successively applying the two multipliers you get. More succinctly,
e to the x plus y equals e to the x time e to the y.
If e to the x was thought it as repeated multiplication, this property
would be a consequence.
but really it goes the other way around. You should think this property is defining
e to the x, and the fact that the special case and counting numbers has
anything to do with repeated multiplication
is a consequence the property.
Multiple functions satisfy this property,
but when you try to define one explicitly, one stands out as being the most natural,
and we express it with this infinite sum. By the way,
the number e is just to find to be the value of this function at one.
The numbers is not nearly as special as the function as a whole, and the convention to
write this function as e to the x
is a vestige of its relationship with repeated multiplication.
The other less natural function satisfying this property
are the exponentials with different bases. Now the expression "e to the pi i"
at least seems to have some meaning,
but you shouldn't think about this infinite sum when trying to make sense of it.
You only need to think about turning adders into multipliers. You see,
we can also play this game a sliding and stretching in the 2d plane,
and this is what complex numbers are. Each numbers simultaneously a point on
the plane
an adder, which slides the plane so that the point for 0
lands on the point for the number, and multiplayer which fixes zero in place
and brings the point for one to the point for the number while keeping
everything evenly spaced.
This can now include rotating along with some stretching and shrinking.
All the actions of the real numbers still apply, sliding side to side and stretching,
but now we have a whole host of new actions.
For instance, take this point here. We call it "i". As an adder,
it slides the plane up, and as a multiplier, it turns it a quarter of the way around
Since multiplying it by itself gives -1, which is to say
applying this action twice is the same as the action of -1
as a multiplier,
it is the square root of -1. All adding is some combination of sliding sideways
and sliding up or down, and all multiplication is some combination of
stretching and rotating.
Since we already know that e to the x turns slide side to side into stretches,
the most natural thing you might expect his for to turn this new dimension of adders,
slides up and down, directly into the new dimension of multipliers,
rotations. In terms points on the plane, this would mean e to the x takes
points on this vertical line
which correspond to adders that slide the plane up and down, and puts them on the
circle with radius one
which corresponds with the multipliers that rotate the plane. The most natural way
you could imagine doing this
is to wrap the line around the circle without stretching your squishing it
which would mean it takes a length of two pi to go completely around the circle,
since by definition this is the ratio the circumference of a circle to its radius.
This means going up pi would translate to going exactly half way around the circle.
When in doubt, if there's a natural way to do things,
this is exactly what e to the x will do, and this case is no exception.
If you want to see a full justification for why e to the x behaves this way,
see this additional video here. So there you have it,
this function e to the x take the adder pi i to the multiplayer -1.