2.4 Complex Numbers

2.4 Complex Numbers

The real number system has been evolving for many years. The real number system includes rational and irrational numbers. The real number system is shown in Venn Diagram form below. 

However, some quadratic equations have no real solutions. This prompted mathematicians to expand the number system to include the imaginary unit i.

The Imaginary Unit i

• The imaginary unit i is defined as  

• Due to this definition: 










• Real numbers and multiples of this imaginary unit make up the set of complex numbers
• Each complex number can be written in the standard form a + bi. In this standard form, the real number a is called the real part of the complex number and the number bi (where b is a real number) is called the imaginary part of the complex number. 
• If b = 0, the number a + bi = a is a real number.
• A number of the form bi, where , is called a pure imaginary number.

The chart illustrating complex numbers and the subsets of complex numbers, as well as the subsets of real numbers, is shown below.

Equality of Complex Numbers

Two complex numbers a + bi and c + di, written in standard form, are equal to each other if and only if a = c and b = d.

Operations with Complex Numbers

• To add or subtract two complex numbers, you add or subtract the real and imaginary parts of the numbers separately. 
• If a + bi and c + di are two complex numbers, written in standard form, their sum and differences are defined as follows. 
• Sum: 

• Difference: 
  
• The additive identity in the complex number system is zero. 
• The additive inverse of the complex number a + bi is: 

Example

Make sure that you write your answer in standard form. 

To multiply complex numbers, simply use the FOIL method. 

Example

Complex Conjugates and Division

To find the quotient of two complex numbers, multiply the numerator and denominator by the conjugate of the denominator.

Example

  

Applications

• Complex numbers can be used in fractal geometry
• On a coordinate system called the complex plane, the vertical axis is the imaginary axis and the horizontal axis is the real axis. 
• The point that corresponds to the complex number a + bi is (a, b).