Basics

A complex number is an expression of the form z=a+bi where a and b are real numbers and i=Sqrt[-1]. The number a is called the real part, denoted Re(z) and b is called the imaginary part, denoted Im(z). Note that i^2=-1. You add, subtract, multiply and divide complex numbers in the manner that you would think from elementary algebra. In Mathematica, the CAPITAL letter I denotes Sqrt[-1].

For addition and subtraction, add and subtract the real and imaginary parts.

In[2]:=

  (2+3 I)+(1+2 I)

Out[2]=

  3 + 5 I

In[3]:=

  (4-6 I)-(6-4 I)

Out[3]=

  -2 - 2 I

FOIL is used, along with i^2=-1, to do multiplication.

In[4]:=

  (2-3 I) (1+4 I)

Out[4]=

  14 + 5 I

In[5]:=

  (2-I)/(5+2 I)

Out[5]=

  8    9 I
  -- - ---
  29   29

Rationalizing the denominator was used in this calculation. The number 29 in the above calculation is the square of the modulus (absolute value) of (5+2i). In general, if z=a+bi, |z|=Sqrt[a^2+b^2]. Alternately,
|z|=Sqrt[ [Re(z)]^2+[Im(z)]^2 ].

In[6]:=

  z=5+2 I

Out[6]=

  5 + 2 I

In[7]:=

  Re[z]
  Im[z]
  Abs[z]

Out[7]=

Out[8]=

Out[9]=

  Sqrt[29]

The complex conjugate of z=a+bi, denoted z bar (z with a bar over it), is
a-bi.

In[10]:=

  Conjugate[z]

Out[10]=

  5 - 2 I

Up to Complex Numbers