nth roots of complex numbers — finding all n roots: Foundations

This page builds every symbol the parent note nth roots of complex numbers leans on, starting from a smart 12-year-old who has never seen any of it. Read top to bottom; each block earns the next.

1. The number $i$ — a 90° turn

You already know the number line: numbers stretching left (negative) and right (positive). Now imagine you are allowed to step off that line, straight up. That "up" direction is measured by a brand-new number called $i$ .

Figure — nth roots of complex numbers — finding all n roots

Figure 1 (above). The blue arrow is the ordinary number $+ 1$ pointing right. Multiplying it by $i$ once swings it up to the yellow arrow at $i$ (a 90° turn along the yellow arc). Multiplying by $i$ a second time swings it further to the red arrow at $- 1$ (the red arc, another 90°). Two quarter-turns make a half-turn, so $+ 1$ becomes $- 1$ — a picture of $i^{2} = - 1$ .

Why the topic needs it: without a way to point "off the real line", the equation $z^{2} = i$ or $z^{4} = - 16$ would have no answers at all. $i$ opens up the whole 2-D plane of numbers where roots live.

2. A complex number $z = x + i y$ — a point on a map

Combine a horizontal step $x$ and a vertical step $y$ (measured in units of $i$ ) and you get a complex number: $z = x + i y .$

$x$ is called the real part, written $Re (z)$ — how far right/left.
$y$ is the imaginary part, written $Im (z)$ — how far up/down.

This picture — the map where complex numbers are points — is called the Argand Diagram. The horizontal axis is the real axis, the vertical axis is the imaginary axis.

Why the topic needs it: the parent note says the $n$ roots "form a regular $n$ -gon on a circle." A polygon and a circle are pictures, and this map is the paper we draw them on.

3. The constant $π$ — before we can measure angles

We are about to measure angles, and the number $π$ will appear everywhere. So earn it first.

Why the topic needs it: every full turn will be written as $2 π$ , and the roots formula adds full turns as $2 π k$ . That symbol only makes sense once you know $π$ is this fixed circumference-to-diameter ratio.

4. Modulus $r = ∣ z ∣$ — the length of the arrow

Figure 2 (above). The blue arrow is the complex number $z = 3 + 2 i$ . The yellow segment is its horizontal leg $x = 3$ , the green segment its vertical leg $y = 2$ , and the little white square marks the right angle where they meet. Because the two legs are perpendicular, the blue arrow is the hypotenuse of a right triangle, so its length is $r = x^{2} + y^{2} = 13$ .

Why "opposite/adjacent" and Pythagoras here? The horizontal leg $x$ and vertical leg $y$ meet at a right angle, so the straight-line arrow is the hypotenuse — and Pythagoras is the one tool that turns two perpendicular distances into one straight distance. That is precisely the length we want.

5. Argument $θ = ar g z$ — the direction of the arrow

We measure this angle in radians, not degrees — that is the next symbol to earn.

Figure 3 (above). The dashed white circle has radius $r$ . The blue arrow points at some complex number; the yellow arc measures its argument $θ$ (the angle from the positive real axis). The thick green arc has length exactly equal to the radius — the arc that defines one radian. Sweeping all the way round covers $2 π$ of these, i.e. a full turn $= 2 π = 360°$ .

Why radians and not degrees? Because the roots formula divides the angle by $n$ and adds full turns as $2 π k$ . Using $2 π$ for a full turn keeps every formula clean and consistent. If you ever use degrees you must add $360° k$ instead — never mix the two (a mistake the parent note warns about).

6. sin, cos, and the polar form

To turn a length-and-angle $(r, θ)$ back into horizontal/vertical steps we need trigonometry.

Why these ratios? $cos θ$ measures how much of the arrow points rightward per unit length, $sin θ$ how much points upward. They are exactly the machine that converts a direction into horizontal + vertical components.

Definition cos and sin beyond the right triangle (all four quadrants)

A right triangle only makes sense for angles between $0$ and $\frac{π}{2}$ . To handle any angle we use the definition that always holds: $cos θ = \frac{x}{r}, sin θ = \frac{y}{r},$ where $(x, y)$ is the tip of the arrow of length $r$ at angle $θ$ — and now $x$ or $y$ is allowed to be negative. The sign of $cos$ follows the sign of $x$ ; the sign of $sin$ follows the sign of $y$ :

Quadrant I ( $0 < θ < \frac{π}{2}$ ): $x > 0, y > 0$ → $cos > 0, sin > 0$ .
Quadrant II ( $\frac{π}{2} < θ < π$ ): $x < 0, y > 0$ → $cos < 0, sin > 0$ .
Quadrant III ( $π < θ < \frac{3 π}{2}$ ): $x < 0, y < 0$ → $cos < 0, sin < 0$ .
Quadrant IV ( $\frac{3 π}{2} < θ < 2 π$ ): $x > 0, y < 0$ → $cos > 0, sin < 0$ .

This is why a root sitting in the second or third quadrant (as in the parent note's cube roots of $8$ ) still has a perfectly good $cos$ and $sin$ — they just carry the correct signs.

Substituting into $z = x + i y$ :

The shorthand $cis θ$ means exactly $cos θ + i sin θ$ , so $z = r cis θ$ .

7. Powers, $z^{n}$ , and De Moivre's Theorem

The notation $z^{n}$ means " $z$ multiplied by itself $n$ times", where $n$ is a whole number (the index or exponent). Applying the multiply-lengths-add-angles rule $n$ times gives:

This is the engine of the whole topic; its own page is De Moivre's Theorem.

Why the topic needs it: an $n$ -th root is any $z$ with $z^{n} = w$ . De Moivre tells us what $z^{n}$ does to length and angle, so we can run it backwards to recover $z$ .

8. $e$ , $e^{i θ}$ , and Euler's Formula (the compact spelling)

The parent note also writes roots as $e^{2 π ik / n}$ . Here $e \approx 2.718$ is a fixed constant, and the surprising fact you may take as a definition for now is:

See Euler's Formula. It is only shorthand — every result on this topic can be done with $cos$ and $sin$ alone.

9. The integer counter $k$ and "modulo $2 π$ "

The symbol $k$ in the formula is just a counter that runs through whole numbers $k = 0, 1, 2, \dots, n - 1$ . It exists because a single arrow can be labelled by many angles that all point the same way: $θ, θ + 2 π, θ + 4 π, \dots (each one more full turn) .$ We say angles are equal "modulo $2 π$ " — meaning up to whole turns. This is the single fact that produces $n$ different roots instead of one, and the Roots of Unity are the cleanest example (all with $r = 1$ , $θ = 0$ ).

i

i^{2} = - 1

; multiplying by

i

rotates 90° counter-clockwise.

What are the real and imaginary parts of

z = x + i y

Re (z) = x

(rightward step),

Im (z) = y

(upward step).

What is the constant

π

The circumference-to-diameter ratio of any circle,

\approx 3.14159

; the circumference is

2 π r

What does the modulus

∣ z ∣

measure and how do you compute it?

The arrow's length;

∣ z ∣ = x^{2} + y^{2} \geq 0

What does

ar g z = θ

measure, and when is it undefined?

The angle of the arrow from the positive real axis; undefined at

z = 0

What principal-value range does this page use for

Arg z

(- π, π]

How many radians in a full turn?

2 π

(equals

360°

How are

cos θ

and

sin θ

defined for angles beyond a right triangle?

cos θ = x / r

sin θ = y / r

, with

x, y

(and their signs) taken from the arrow's tip in any quadrant.

Write polar form and what

cis θ

means.

z = r (cos θ + i sin θ)

;

cis θ = cos θ + i sin θ

State De Moivre's Theorem.

(r cis θ)^{n} = r^{n} cis (n θ)

— power the length, multiply the angle.

What does Euler's formula say?

e^{i θ} = cos θ + i sin θ

, the unit arrow at angle

θ

Why does the counter

k

exist in the roots formula?

Angles repeat every

2 π

; adding

2 π k

before dividing gives the distinct roots.

Why must

z^{n} = w

have exactly

n

solutions?

The Fundamental Theorem of Algebra: a degree-

n

polynomial has

n

complex roots.

Foundations — nth roots of complex numbers — finding all n roots