One way to think about the function e^t is

to ask what properties it has. Probably the most important one, from some points of view

the defining property, is that it is its own derivative. Together with the added condition

that inputting zero returns 1, it’s the only function with this property. You can

illustrate what that means with a physical model: If e^t describes your position on the

number line as a function of time, then you start at 1. What this equation says is that

your velocity, the derivative of position, is always equal your position. The farther

away from 0 you are, the faster you move.

So even before knowing how to compute e^t

exactly, going from a specific time to a specific position, this ability to associate each position

with the velocity you must have at that position paints a very strong intuitive picture of

how the function must grow. You know you’ll be accelerating, at an accelerating rate,

with an all-around feeling of things getting out of hand quickly. If we add a constant to this exponent, like

e^{2t}, the chain rule tells us the derivative is now 2 times itself. So at every point on

the number line, rather than attaching a vector corresponding to the number itself, first

double the magnitude, then attach it. Moving so that your position is always e^{2t} is

the same thing as moving in such a way that your velocity is always twice your position.

The implication of that 2 is that our runaway growth feels all the more out of control. If that constant was negative, say -0.5, then

your velocity vector is always -0.5 times your position vector, meaning you flip it

around 180-degrees, and scale its length by a half.

Moving in such a way that your velocity

always matches this flipped and squished copy of the position vector, you’d go the other

direction, slowing down in exponential decay towards 0. What about if the constant was i? If your

position was always e^{i * t}, how would you move as that time t ticks forward? The derivative

of your position would now always be i times itself. Multiplying by i has the effect of

rotating numbers 90-degrees, and as you might expect, things only make sense here if we

start thinking beyond the number line and in the complex plane. So even before you know how to compute e^{it},

you know that for any position this might give for some value of t, the velocity at

that time will be a 90-degree rotation of that position. Drawing this for all possible

positions you might come across, we get a vector field, whereas usual with vector field

we shrink things down to avoid clutter.

At time t=0, e^{it} will be 1. There’s only

one trajectory starting from that position where your velocity is always matching the

vector it’s passing through, a 90-degree rotation of position. It’s when you go around

the unit circle at a speed of 1 unit per second. So after pi seconds, you’ve traced a distance

of pi around; e^{i * pi} = -1. After tau seconds, you’ve gone full circle; e^{i * tau} = 1.

And more generally, e^{i * t} equals a number t radians around this circle. Nevertheless, something might still feel immoral

about putting an imaginary number up in that exponent. And you’d be right to question

that! What we write as e^t is a bit of a notational disaster, giving the number e and the idea

of repeated multiplication much more of an emphasis than they deserve.

But my time is

up, so I’ll spare you my rant until the next video..