D.1 The definition of complex numbers

Let’s jump into the definition right away.

Definition 113. (Complex numbers)

The set of numbers written in the form

where i² = −1 is satisfied, is called complex numbers. We call a the real part and b the imaginary part of z, denoted by

If a + bi and c + di are two complex numbers, then we define addition and multiplication by

(a)

(b)

The set of complex numbers is denoted with ℂ. Thus, we write

According to the definition, addition is straightforward. However, multiplication seems a little bit convoluted. To see why it is defined this way, multiply them term by term, as you do with two polynomials.

An important property of complex numbers is their absolute value, or in other words, their distance from 0.

Definition 114. (Absolute value of complex numbers)

Let z = a + bi be a complex number. Its absolute value is defined by

In addition, each complex number has a conjugate, which, as we shall see later...