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Mathematics LibreTexts

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 C.
  • We call x the real part of z. This is denoted by x=Re(z).
  • We call y the imaginary part of z. This is denoted by y=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 i2=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)=47i
  • Multiplication:

(3+4i)(7+11i)=21+28i+33i+44i2=23+61i.

Here we have used the fact that 44i2=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

¯x+iy=xiy

If z=x+iy then its conjugate is ˉz=xiy and we read this as "z-bar = xiy".

Example 1.3.1

¯3+5i=35i.

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

zˉz=(x+iy)(xiy)=x2+y2

Note that zˉz is real. We will use this property in the next example to help with division.

Example 1.3.2 (Division).

Write 3+4i1+2i in the standard form x+iy.

Solution

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

3+4i1+2i=3+4i1+2i12i12i=112i5=11525i.

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|=x2+y2

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

Example 1.3.3

The norm of 3+5i=9+25=34.

Important. The norm is the sum of x2 and y2. It does not include the i and is therefore always positive.


This page titled 1.3: Terminology and Basic Arithmetic is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Jeremy Orloff (MIT OpenCourseWare) via source content that was edited to the style and standards of the LibreTexts platform.

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