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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.












  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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