complex numbers equation derivation

Question

How is the expansion of the square of |z1+z2| derived?

Answer 1

You can use dot product.

|z1+z2|²

= (z1+z2)·(z1+z2)

= z1² + z1·z2 + z2·z1 + z2²

= z1² + z2² + 2z1·z2

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Attn: z1·z2 = z2·z1

Answer 2

The absolute value, or modulus of the number z = a + bi is defined by |z| = √(a² + b²)

|z1 + z2|  = |(a1 + a2) + (b1 + b2)| = √[(a1 + a2)² + (b1 + b2)²]

|z1 + z2|² = (a1 + a2)² + (b1 + b2)²

Answer 3

z1 = x1 + i*y1

z2 = x2 + i*y2

z1 + z2 = (x1 + x2) + i*(y1 + y2) = X + iY

X = x1 + x2, Y = y1 + y2

i = sqrt(-1), i^2 = -1

|z1 + z2| = |X + iY| = sqrt(X^2 + (i^2)*Y^2) = sqrt(X^2 - Y^2)

|z1 + z2|^2 = (sqrt(X^2 - Y^2))^2 = X^2 - Y^2