Turns, stretches and Euler’s formula.

Step 1 of 5

Angles add

When two complex numbers multiply, the lengths multiply and the angles add:

$r_1(\cos\alpha + i\sin\alpha) \cdot r_2(\cos\beta + i\sin\beta) = r_1 r_2\left(\cos(\alpha + \beta) + i\sin(\alpha + \beta)\right)$$r_1(\cos\alpha + i\sin\alpha) \cdot r_2(\cos\beta + i\sin\beta) =$$r_1 r_2\left(\cos(\alpha + \beta) + i\sin(\alpha + \beta)\right)$⁠

This follows from the formulas for the sine and cosine of a sum of angles. So multiplying by $w$⁠ turns by $w$⁠’s angle and stretches by its length, like the maps in the previous Unit. Here Turn per step is $40^\circ$⁠. Predict the angle after five steps.

Step 2 of 5

Lengths multiply

Stretch per step is $1.08$⁠: each multiplication lengthens the point by $8\%$⁠, like interest in the exponents Unit. The length after $n$⁠ steps is $1.08^n$⁠, and the points climb a widening spiral. Find how many steps double the length.

Step 3 of 5

The rule

Drag Build Euler’s formula a little: the board marks the last step’s turn and stretch. Then use the rule to multiply two numbers given by length and angle.

Step 4 of 5

Spiralling in

When Stretch per step is less than $1$⁠, the points close in on zero in a spiral. The glowing curve through all the points is $w^t$⁠, between the whole steps too. Steps is locked at $12$⁠; set the other two controls so that the last point lands in a given place.

Step 5 of 5

Euler’s formula

The smooth curve $w^t$⁠ is a power with a real exponent, and when $w$⁠ has length $1$⁠ it runs round the unit circle. Leonhard Euler found that it is exactly the exponential function with an imaginary exponent, when the angle is measured in radians:

$e^{i\theta} = \cos\theta + i\sin\theta$$e^{i\theta} =$$\cos\theta + i\sin\theta$⁠

$e^{i\theta}$⁠ is the point on the unit circle at angle $\theta$⁠. Here Turn per step is $180^\circ$⁠, $\pi$⁠ radians. Predict $e^{i\pi}$⁠, then watch: the board will drag Build Euler’s formula all the way.