1.3: Terminology and Basic Arithmetic
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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)=−4−7i
- 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.
Complex conjugation is denoted with a bar and defined by
¯x+iy=x−iy
If z=x+iy then its conjugate is ˉz=x−iy and we read this as "z-bar = x−iy".
¯3+5i=3−5i.
The following is a very useful property of conjugation: If z=x+iy then
zˉz=(x+iy)(x−iy)=x2+y2
Note that zˉz is real. We will use this property in the next example to help with 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+2i⋅1−2i1−2i=11−2i5=115−25i.
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.
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.
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.