Foundations — Roots of unity — cube roots, nth roots, geometric interpretation

This page assumes nothing. We build every symbol the parent note (the parent topic) leans on, in an order where each idea rests on the one before it.

1. What is a complex number? (the arrow behind $z$)

What does it LOOK like? Draw a flat map. Go $x$ steps right, then $y$ steps up. The dot you land on — or the arrow from the origin to that dot — is the number $z$. This picture-map is the Argand Diagram.

Why does the topic need this? The equation $z^n=1$ has no full answer among ordinary numbers on a line. By upgrading numbers to arrows on a plane, we gain a second freedom — direction — and that is exactly what lets $z^n=1$ have $n$ different solutions instead of just one.

2. Modulus $r$ — how long the arrow is

What does it LOOK like? It is the straight-line distance from the origin to the dot. Look at the right triangle in the figure below: the horizontal leg is $x$, the vertical leg is $y$, and the arrow is the slanted side (the hypotenuse).

Why that formula? This is just Pythagoras — the length of the slanted side of a right triangle is the square root of (base squared + height squared). No new tool needed; the arrow's length is a plain distance.

Why the topic needs it. Roots of unity all sit on the unit circle — the circle of radius $1$. "On the unit circle" is exactly the sentence "$r=1$." Without modulus we cannot even say what circle they live on.

3. Sine and cosine, straight from the unit circle

Before we can talk about direction, we need the two functions that turn an angle into a position. We build them on the circle of radius $1$, which is exactly the circle the roots of unity live on.

What does it LOOK like? In the figure below the moving point rides the unit circle; drop it straight down to read $\cos \theta$ off the horizontal axis, and slide it across to read $\sin \theta$ off the vertical axis.

Why this and not right-triangle-only definitions? A bare right triangle only makes sense for angles between $0$ and $90^{\circ }$. The circle version works for every angle — up, left, down, past a full turn — which is exactly what we need, because roots of unity sit all the way around the circle.

4. Argument $\theta$ — which way the arrow points

Why do we measure angle at all? Because an arrow needs two facts to be pinned down: how long it is (that's $r$) and where it points (that's $\theta$). Together $(r,\theta )$ locate any dot on the plane.

Why anticlockwise, why from 3 o'clock? Pure convention — but a fixed one, so everyone agrees. Anticlockwise is the direction of increasing angle, like a clock running backwards.

Why this matters SO much here. This "$+2\pi k$" freedom is the entire engine of roots of unity. When we later solve for the angle of a root and write $n\theta =0+2\pi k$, the many allowed values of $k$ are what produce the many different roots. If the argument were a single number, $z^n=1$ would have only one answer.

Why the topic needs it. Raising a complex number to a power multiplies its angle. Roots of unity are spaced $\frac{2\pi }{n}$ apart in angle. Every geometric statement in the topic — "equally spaced," "rotate by," "regular polygon" — is a statement about $\theta$.

5. Polar form — writing an arrow as (length, direction)

Where does this come from? From §3, a point at angle $\theta$ on the unit circle is $(\cos \theta ,\sin \theta )$. Stretch that unit arrow to length $r$ and its coordinates scale up to $x=r\cos \theta$ and $y=r\sin \theta$. Substitute both into $z=x+iy$ and factor out $r$: $z=r\cos \theta +ir\sin \theta =r(\cos \theta +i\sin \theta ).$ Nothing invented — same arrow, described by (length, direction) instead of (right, up).

6. Multiplication adds angles — the rotation rule (derived)

Here is the fact that everything else rests on, and we will prove it, not just assert it. Take two arrows of length $1$: one at angle $\alpha$, one at angle $\beta$. Multiply them.

$(\cos \alpha +i\sin \alpha )(\cos \beta +i\sin \beta )$

Step 1 — expand the four products (WHAT). Multiply out like ordinary brackets, using $i^2=-1$ on the last term: $=\cos \alpha \cos \beta +i\cos \alpha \sin \beta +i\sin \alpha \cos \beta +i^2\sin \alpha \sin \beta$ $=(\cos \alpha \cos \beta -\sin \alpha \sin \beta )+i(\sin \alpha \cos \beta +\cos \alpha \sin \beta ).$

Step 2 — recognise the angle-addition identities (WHY). The two bracketed groups are exactly the standard trig sum formulas: $\cos (\alpha +\beta )=\cos \alpha \cos \beta -\sin \alpha \sin \beta ,\sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta .$ So the whole product collapses to $(\cos \alpha +i\sin \alpha )(\cos \beta +i\sin \beta )=\cos (\alpha +\beta )+i\sin (\alpha +\beta ).$

WHAT IT LOOKS LIKE. The result is again a length-$1$ arrow, now at angle $\alpha +\beta$. Multiplying two unit arrows adds their angles — it rotates one by the other's angle.

7. The exponential shortcut $e^{i\theta }$

We first recall the ordinary exponential, then explain how the symbol legally stretches to imaginary powers.

Why extend it to $e^{i\theta }$? Feed the imaginary number $i\theta$ into that same infinite sum, sort the terms into real and imaginary piles, and the real pile is exactly the series for $\cos \theta$ while the imaginary pile is exactly the series for $\sin \theta$. That is Euler's formula:

Why is the exponent law inherited? Because $e^x$ was defined by $e^a{e}^b=e^{a+b}$, and the extension keeps it. Setting $a=i\alpha$, $b=i\beta$: $e^{i\alpha }\cdot e^{i\beta }=e^{i(\alpha +\beta )}.$ This is not a new assumption — it is precisely the "multiplication adds angles" fact we derived in §6, now written in shorthand. Euler's formula is the bridge that turns the trig proof of §6 into a one-line exponent rule.

Why THIS tool and not plain $x+iy$ multiplication? If you multiply $(x_1+i{y}_1)(x_2+i{y}_2)$ the long way you get four products every time. In exponential form the same job is one addition of angles. The whole roots-of-unity story — "multiply angles by $n$," "rotate by $2\pi /n$" — is impossible to say cleanly without this shortcut.

8. De Moivre's Theorem — raising to a power (derived)

Powers are just repeated multiplication, and §6 told us each multiplication adds the angle. So multiplying $e^{i\theta }$ by itself $n$ times adds $\theta$ to itself $n$ times.

Derivation by repeated use of §6: $(\cos \theta +i\sin \theta {)}^2=\cos (2\theta )+i\sin (2\theta )(\text{apply §6 with }\alpha =\beta =\theta ),$ $(\cos \theta +i\sin \theta {)}^3=(\cos 2\theta +i\sin 2\theta \left)\right(\cos \theta +i\sin \theta )=\cos (3\theta )+i\sin (3\theta ),$ and each further multiply by $(\cos \theta +i\sin \theta )$ bumps the angle by one more $\theta$. Doing it $n$ times:

This is the single most important fact behind the topic — see De Moivre's Theorem. The equation $z^n=1$ becomes "find arrows whose length${}^n$ is $1$ and whose angle$\times n$ is a whole number of full turns." Modulus and argument split apart and get solved separately — and the "$+2\pi k$" freedom of the argument (§4) supplies the many roots.

9. Whole-number counting of solutions

Why the topic needs it. $z^n=1$ rearranges to $z^n-1=0$, a degree-$n$ polynomial. This theorem promises there are exactly $n$ solutions — no more, no fewer. It tells us to stop once we have found $n$ of them. See Fundamental Theorem of Algebra.

10. Adding a chain of powers — geometric series

Why is this here? The parent note proves "the roots sum to zero" in one line by plugging ${\omega }^n=1$ into this formula: the top becomes $0$, so the whole sum is $0$. You need to already own the geometric-series formula for that line to make sense. See Geometric Series.

11. Shapes: regular polygons

Why the topic needs it. The punchline of the whole topic is visual: the $n$ roots of unity are the corners of a regular $n$-gon inscribed in the unit circle, one corner nailed at $1$. Knowing what "regular polygon" means turns algebra into a picture. See Regular Polygons.

How these foundations feed the topic

Equipment checklist

Self-test: cover the right side and see if each is solid before opening the parent note.