Complex Numbers

The expression has no real answer. The symbol i is created to represent and is called an imaginary value. Since , i² = −1. Any expression that is a product of a real number with i is called a pure imaginary number.

Example 1: Simplify each of the following.

1. 
   

   
   

2. 
   

   
   

   This last expression is commonly written as so that the i is not mistakenly written under the radical.

3. (6 i)(4 i) = 24 i² = 24(−1) = −24

4. 
   

   
   

   For this last example, all imaginary values had to be put into their “ i-form” before any simplifying could be done. Note that
   

   
   

That is, the product rule for radicals does not hold (in general) with imaginary numbers.

When i is raised to powers, it has a repeating pattern.

When i is raised to any whole number power, the result is always 1, i, −1 or −i. If the exponent on i is divided by 4, the remainder will indicate which of the four values is the result.

Example 2: Simplify each of the following.

1. i³⁴

   Since 34 divided by 4 has a remainder of 2, i³⁴ = i² = −1.

2. i⁹⁵

3. Since 95 divided by 4 has a remainder of 3, i⁹⁵ = i³ = − i.

4. i¹⁰⁸

5. Since 108 divided by 4 has a zero remainder, i¹⁰⁸ = i⁰ = 1.

6. i⁵³

7. Since 53 divided by 4 has a remainder of 1, i⁵³ = i¹ = i.

Complex numbers and complex conjugates. A complex number is any expression that is a sum of a pure imaginary number and a real number. A complex number is usually expressed in a form called the a + bi form, or standard form, where a and b are real numbers. The expressions a + bi and a − bi are called complex conjugates. Complex conjugates are used to rationalize the denominator when dividing with complex numbers.

Arithmetic with complex numbers is done in a similar manner as arithmetic with polynomials. The following are definitions for arithmetic with two complex numbers call ( a + bi) and ( c + di).

• Combining like terms and factoring out the i,
  

  
  

• Using the distributive property,
  

  
  

• Rationalizing the denominator,
  

  
  

Example 3: Find the sum, difference, product, and quotient of (4 + 3 i) and (5 −4 i).

Quotient: Rationalize the denominator.

Example 4: Simplify .

Since 6 i is 0 + 6 i, its complex conjugate is 0 − 6 i. Therefore,

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