Q.If $n$ is an integer, then show that $(1 + i)^{2n} + (1 - i)^{2n} = 2^{n+1} \cos\left(\frac{n\pi}{2}\right)$.

The Intuition: Why "Transform" a Number?

You already know that a real number lives on a line — the number line. Adding $+2$ slides you right; multiplying by $-1$ flips you to the opposite side. But what if you want to rotate something? A real number can't do that on its own. Multiplying by $-1$ is a 180° rotation, but what about a 90° rotation? That's where the complex number transform comes in.

Think of a complex number $z = a + bi$ as a point (or an arrow) on a 2D plane. The real part $a$ is the horizontal coordinate, the imaginary part $b$ is the vertical coordinate. Now, when you multiply two complex numbers, something beautiful happens: the lengths multiply, and the angles add.

So a "complex number transform" is simply the act of applying a complex number (as an operator) to another complex number (as a point) — usually by multiplication — to achieve a geometric transformation: rotation, scaling, or both.

The Precise Statement

Let $z = x + yi$ be any complex number (the "point" you want to transform).

Let $w = r(\cos\theta + i\sin\theta)$ be a fixed complex number (the "transformer").

Then the complex number transform of $z$ by $w$ is:

When you multiply this out (using $i^2 = -1$), the result is a new complex number $z'$ whose geometric meaning is:

• Scale the distance of $z$ from the origin by a factor of $r$
• Rotate the point $z$ around the origin by an angle $\theta$ counterclockwise

This is often written using Euler's formula: $e^{i\theta} = \cos\theta + i\sin\theta$, so $w = re^{i\theta}$.

A Concrete Example

Take the point $z = 1 + 0i$ (the number 1 on the real axis).

Let the transformer be $w = i$ (which has $r=1$, $\theta = 90^\circ$).

The point $(1,0)$ moved to $(0,1)$ — a 90° rotation counterclockwise. No scaling because $|i| = 1$.

Now take $z = 2 + 0i$ and $w = 2i$ (which has $r=2$, $\theta = 90^\circ$):

The point $(2,0)$ moved to $(0,4)$ — rotated 90° and scaled by factor 2.

Why This Matters

This transform is the foundation of: …