e

Imaginary

negative number: change direction on 1D line. (ex: -4 north = go south 4)

imaginary number: rotate on complex plane.

Exponential

step growth = next step, double the current

natural growth = consider each part, grow little by littlehttps://www.youtube.com/watch?v=_-x90wGBD8U&feature=youtu.be

Euler Formula

https://www.youtube.com/watch?v=qpOj98VNJi4

Euler Formula = 2 ways to move on a circle: (1) x, y, (2) radian

exponential growth

• real exponential growth = increase by 1 after an interval

• imaginary exponential growth = rotate 1 after an interval

rotate pi will move to -1.

What is $e^x$?

$e$ = Start at 1, growing continuously to 100% interest after 1 unit of time.

$e^x$= Start at 1, growing continuously to 100% interest after x unit of time.

$3 ^ 4$ = growing from 1 to 3, growth rate = 4x

What is $e ^ i$?

$e ^ i$ = Start at 1, rotating 1 after 1 unit of time.

What is $3 ^ i$?

$e ^ i = cos(1) + i \cdot sin(i) = 0.54 + 0.84 i$

$3 ^ i = 1 \cdot 3 ^ i$

$3 ^ i = (e ^{ln(3)}) ^ i$

$ln(3) ~ 1.1 > 1 = ln(e) = ln(2.718)$

$3 ^ i = cos(ln(3)) + i \cdot sin(ln(3)) = 0.45 + 0.89i$

What is $i ^ i$?

$i ^ i = 1 \cdot i ^ i$

$i = cos(\pi / 2) + i \cdot sin(\pi / 2) = e ^ {i \cdot {\pi / 2}}$

$i ^ i = e ^ {i \cdot i \cdot \pi / 2} = e ^ {\pi / 2} \sim 0.2$

it is shrinking.

Scaling & rotation