# 1.3: Terminology and Basic Arithmetic

- Page ID
- 6468

Definition

**Complex numbers** are defined as the set of all numbers

\(z = x + yi\),

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

\(\overline{x + iy} = x - iy\).

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

\(|z| = \sqrt{x^2 + y^2}\).

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.**