Complex Arithmetic

Samples - Algebraic

1. The sum of a complex number and its conjugate is twice the real part of the complex number:

(a + bi) + (a - bi) = (a + a) + (b - b)i = 2a

2. A complex number minus its conjugate is (i times) twice the imaginary part of the complex number:

(a + bi) - (a - bi) = (a - a) + (b - (-b))i = 2bi

3. The reciprocal of a complex number is its conjugate divided by the square of its modulus:

(a + bi)*((a - bi)/(a² + b²)) = ((a + bi)*(a - bi))/(a² + b²) = (a² + b²)/(a² + b²) = 1

4. The modulus of the square of a complex number is the square of the modulus of the complex number:

(a + bi)2 = (a2 - b2) + 2abi

So the modulus of (a + bi)2 is

Sqrt[(a2 - b2)2 + (2ab)2]

= Sqrt[a4 - 2a2b2 + b4 + 4a2b2]

= Sqrt[a4 + 2a2b2 + b4]

= Sqrt[(a2 + b2)2]

= a2 + b2

the square of the modulus of a + bi

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