1.3: Terminology and Basic Arithmetic

Definition

Complex numbers are defined as the set of all numbers

$$z = x + yi \nonumber$$

where $x$ and $y$ are real numbers.

• We denote the set of all complex numbers by $\mathbb{C}$.
• We call $x$ the real part of $z$. This is denoted by $x = \text{Re} (z)$.
• We call $y$ the imaginary part of $z$. This is denoted by $y = \text{Im} (z)$.

Important: The imaginary part of $z$ is a real number. It does not include the $i$.

The basic arithmetic operations follow the standard rules. All you have to remember is that $i^2 = -1$. We will go through these quickly using some simple examples. It almost goes without saying that it is essential that you become fluent with these manipulations.

• Addition: $(3 + 4i) + (7 + 11i) = 10 + 15i$
• Subtraction: $(3 + 4i) - (7 + 11i) = -4 - 7i$
• Multiplication:

$(3 + 4i)(7 + 11i) = 21 + 28i + 33i + 44i^2 = -23 + 61i.$

Here we have used the fact that $44i^2 = -44$.

Before talking about division and absolute value we introduce a new operation called conjugation. It will prove useful to have a name and symbol for this, since we will use it frequently.

Definition: Complex Conjugation

Complex conjugation is denoted with a bar and defined by

$$\overline{x + iy} = x - iy \nonumber$$

If $z = x + iy$ then its conjugate is $\bar{z} = x - iy$ and we read this as "z-bar = $x - iy$".

Example $\PageIndex{1}$

$\overline{3 + 5i} = 3 - 5i$.

The following is a very useful property of conjugation: If $z = x + iy$ then

$z\bar{z} = (x + iy)(x - iy) = x^2 + y^2$

Note that $z\bar{z}$ is real. We will use this property in the next example to help with division.

Example $\PageIndex{2}$ (Division).

Write $\dfrac{3 + 4i}{1 + 2i}$ in the standard form $x + iy$.

Solution

We use the useful property of conjugation to clear the denominator:

$\dfrac{3 + 4i}{1 + 2i} = \dfrac{3 + 4i}{1 + 2i} \cdot \dfrac{1 - 2i}{1 - 2i} = \dfrac{11 - 2i}{5} = \dfrac{11}{5} - \dfrac{2}{5} i$.

In the next section we will discuss the geometry of complex numbers, which gives some insight into the meaning of the magnitude of a complex number. For now we just give the definition.

Definition: Magnitude

The magnitude of the complex number $x + iy$ is defined as

$$|z| = \sqrt{x^2 + y^2} \nonumber$$

The magnitude is also called the absolute value, norm or modulus.

Example $\PageIndex{3}$

The norm of $3 + 5i = \sqrt{9 + 25} = \sqrt{34}$.

Important. The norm is the sum of $x^2$ and $y^2$. It does not include the $i$ and is therefore always positive.