Complex Arithmetic

Samples - Algebraic

1. (5 + 2i) + (5 - 2i) = (5 + 5) + (2 - 2)i = 10. In general, what is the sum of a complex number and its conjugate?

2. (5 + 2i) - (5 - 2i) = (5 - 5) + (2 - (-2))i = 4i. In general, what is a complex number minus its conjugate?

3. Write 1/(5 + 2i) in the form a + bi.

(1/(5 + 2i)) = (1/(5 + 2i))*((5 - 2i)/(5 - 2i)) = (5 - 2i)/(5² + 2²)

The number sqrt(a² + b²) is called the modulus of the complex number a + bi. The modulus is denoted |a + bi|. We shall see a simple geometric interpretation of the modulus. Find a general expression for the reciprocal of a complex number.

4. (5 + 2i)² = (5 + 2i)*(5 + 2i) = (5*5 - 2*2) + (5*2 + 2*5)i = 21 + 20i. The general formula is

(a + bi)² = (a² - b²) + 2abi

straightforward, but lacking a clean interpretation. We shall see a simple expression using the polar representation of complex numbers. Find the general expression for the relation between the modulus of a complex number and the modulus of its square.

Here are the answers to these general questions.

Return to Samples.