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# 1.3: Terminology and Basic Arithmetic

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## 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.