Foundations — De Moivre's theorem — statement, proof, applications

Before we can believe the theorem, we must own every symbol it uses. This page builds each one from scratch — plain words first, then a picture, then the reason the topic needs it. Read top to bottom: each block leans on the one above.

1. The imaginary unit $i$

The picture. Ordinary numbers live on a horizontal line: $\dots ,-2,-1,0,1,2,\dots$. Multiplying by $-1$ flips you across $0$ — a $180°$ turn. Ask: what single move, done twice, gives that flip? A quarter turn ($90°$). So multiplying by $i$ is a $90°$ rotation, and doing it twice ($i^2$) is the $180°$ flip that lands you on $-1$.

Why the topic needs it. Rotation is the whole story of De Moivre. Since multiplying by $i$ already turns arrows, complex numbers are the natural language for "spinning". Powers of $i$ then cycle: $i^0=1,i^1=i,i^2=-1,i^3=-i,i^4=1,\dots$ That cycle of four is used silently in Example 3 of the parent when $i^3=-i$ appears.

2. A complex number $z=x+iy$

The picture — the Argand diagram. Draw a horizontal axis for $x$ and a vertical axis for $y$. Plot the point $(x,y)$ and draw an arrow from the origin to it. That arrow is $z$. The horizontal axis is called the real axis; the vertical axis is the imaginary axis.

Why the topic needs it. De Moivre is about arrows. The instant you see $z$ as an arrow on this map, "power = stretch and spin" becomes something you can watch rather than compute.

3. Modulus $r=|z|$ — the arrow's length

Why a square root of a sum of squares? Because the arrow is the hypotenuse of a right triangle with horizontal side $x$ and vertical side $y$. Pythagoras: length${}^2=x^2+y^2$. We take  ​ to get the length itself. It is never negative.

Why the topic needs it. The "stretch" half of De Moivre. Under an $n$-th power the length becomes $r^n$ — the modulus multiplies.

4. Argument $\theta =\arg z$ — the arrow's angle

Why we measure from the positive real axis, anticlockwise. It is a shared convention so everyone reads the same angle from the same picture. Anticlockwise is counted positive; clockwise is negative.

How to find it — and why $\tan$. In the right triangle of §3, the vertical side is $y$ (opposite the angle) and horizontal side is $x$ (adjacent). The ratio "opposite over adjacent" is exactly $\tan \theta$, because $\tan$ measures the steepness of the arrow: $\tan \theta =\frac{y}{x}.$ So $\theta =\arctan (y/x)$ only when the arrow points into the right half ($x>0$). If $x<0$ the naive $\arctan$ lands you $180°$ off, because $\tan$ repeats every $180°$; you then add $\pi$ to correct the quadrant. (This quadrant fix is the same one covered in Trigonometric identities.)

The key fact that makes $\theta$ ambiguous — and useful. Spinning a full turn returns you to the same arrow: $\theta \text{and}\theta +2\pi \text{point the same way.}$ This "$+2\pi$ changes nothing" is the reason $n$-th roots come in exactly $n$ copies (parent, Application A).

Why the topic needs it. The "spin" half of De Moivre. Under an $n$-th power the angle becomes $n\theta$ — the argument adds.

5. Polar form and $\mathrm{cis}\theta$

Why $\cos$ for the real part and $\sin$ for the imaginary part. On a unit circle (radius $1$), a point at angle $\theta$ has horizontal coordinate $\cos \theta$ and vertical coordinate $\sin \theta$ — that is the definition of cosine and sine. Multiply the whole circle by $r$ to scale up to length $r$.

Why the topic needs it. De Moivre is stated in this form: $(r\mathrm{cis}\theta {)}^n=r^n\mathrm{cis}(n\theta )$. Without polar form there is no clean "length and angle" to stretch and spin. See Polar form of complex numbers.

6. The multiplication rule: angles add

Why this is true. Expand $(\cos \alpha +i\sin \alpha )(\cos \beta +i\sin \beta )$ and collect real/imaginary parts; the two brackets are exactly the angle-addition formulas for $\cos (\alpha +\beta )$ and $\sin (\alpha +\beta )$ (from Trigonometric identities). The parent proves this in Step 0.

Why the topic needs it. This single fact, applied $n$ times by induction, is De Moivre. Everything downstream is a consequence.

7. Integer exponent $n$

Why the topic needs it. The theorem's scope. Positive $n$ comes from induction; $n=0$ and negative $n$ are patched separately (parent Steps 2–3), together covering all integers.

8. The bridge: Euler's formula

What it says in words. The exponential of an imaginary angle is the unit arrow at that angle. Then $(e^{i\theta }{)}^n=e^{in\theta }$ is just the ordinary rule "exponents multiply", instantly giving $\mathrm{cis}(n\theta )$.

Why the topic needs it. It turns De Moivre into a one-line consequence — but the parent still earns it by induction so you never rely on a fact you haven't proven. See Euler's formula.

Prerequisite map

Equipment checklist

Connections

• 3.5.10 De Moivre's theorem — statement, proof, applications (Hinglish) — parent topic, Hinglish version.
• Argand diagram — the map where every complex number becomes an arrow.
• Polar form of complex numbers — the $r\mathrm{cis}\theta$ representation built here.
• Euler's formula — the exponential bridge $e^{i\theta }=\mathrm{cis}\theta$.
• Trigonometric identities — angle-addition formulas power the multiplication rule.
• Roots of unity — where the "$+2\pi$ changes nothing" fact pays off.
• Binomial theorem — used later to extract multiple-angle identities.