Powers and roots

Idea

Multiplying complex numbers adds the arguments of the two arrows and multiplies the moduli. Doing “multiply by the same $z$ ” $n$ times adds the argument $n$ times. That is a power. Reversing it divides the circle into $n$ equal roots.

z_{1} z_{2} ⟷ (r e^{i θ})^{n} ⟷ n w

This page does not explain polar form or Euler’s formula again. It lets the same argument rule change roles across three operations: multiply by another number, multiply by itself, and take a root. The separate formulas, and the details of dragging a point, are left to the works below.

Interaction

Multiplication: drag $z_{1}$ and $z_{2}$ , and check that the product’s argument is the sum of the two arguments
Power: switch to $z, z^{2}, \dots, z^{n}$ for one $z$ , and compare that with the argument multiplied by $n$
Roots: watch the $n$ roots divide the circle into equal parts. Raising any one of them to the $n$ th power returns the original $w$
Order n: shown only for a power or for roots. Changing $n$ makes the equal division finer and makes the power turn more sharply

Suggested order: multiplication, then a power, then roots.

What to notice

All three readings do one thing: add arguments and multiply moduli. The difference is whether you add the other number’s argument, or add your own argument $n$ times.
A power applies that same rule $n$ times, and the points step around a spiral (a circle when $∣ z ∣ = 1$ ): each power turns by $ar g z$ and scales by $∣ z ∣$ . A root is the reverse: one rotation is split into $n$ equal parts.
Multiplication, a power, and a root can be run back. Take an $n$ th root and raise it to the $n$ th power, and you return to the start; going the other way, the $n$ th roots of $z^{n}$ give back $z$ only as one of $n$ candidates. The three pictures read the same argument rule forward and back.

Idea

Interaction

What to notice

Further reading

Comments

Idea

Interaction

What to notice

Related

Further reading

Comments