College Algebra Lecture Notes Section 1.4Page 1 of 4

Section 1.4: Complex Numbers

Big Idea:.Calling the square root of negative one “i” allows us to state the solution of many algebra equations that would otherwise be unsolvable with real numbers.

Big Skill:You should be able to perform arithmetic (add, subtract, multiply, and divide) with complex numbers.

A. Identifying and Simplifying Imaginary and Complex Numbers

Imaginary Numbers and the Imaginary Unit

  • Imaginary Numbers are those of the form , where k is a positive real number.
  • The imaginary unit i represents the number whose square root is -1:

i2 = -1 and

Rewriting imaginary numbers

  • For any positive real number k, .

Practice:

  1. Evaluate .
  1. Evaluate .
  1. Evaluate .
  1. Evaluate .

Complex Numbers

  • Complex numbers are numbers that can be written in the form a + bi, where a and b are real numbers and is the imaginary unit.
  • The real number a is called the real part of the complex number.
  • The real number b is called the imaginary part of the complex number.
  • The standard form of a complex number is a + bi.

Practice:

  1. Write in standard form: .

The Subset Hierarchy of Numbers:

B. Add and Subtract Complex Numbers

  • Adding, subtracting, multiplying, or dividing complex numbers results in an answer that is also a complex number.
  • To add complex numbers, add “like terms”:
  • To subtract complex numbers, subtract “like terms”:

Practice:

C. Multiply Complex Numbers and Find powers of i

  • To multiply complex numbers, use the distributive property, then combine “like terms” (like using FOIL):
  • One thing to keep in mind: the product property of radicals does not apply when the radicand is a complex number, so evaluate the square root of the negative number first, then multiply the radicals.

Correct:

 / Incorrect:


Product of Complex Conjuagtes

  • For a complex number a + bi and its conjugate a –bi, theor product (a + bi)(a –bi), is the real number a2 +b2.
  • To compute powers of i, use an exponent that is the remainder of dividing the original exponent by 4. This works because the powers of i repeat themselves every four factors of i, as can be seen from making a list of powers of i.

Practice:

D. Divide Complex Numbers

  • To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator, then multiply and simplify. Note: the denominator will always multiply out to a difference of squares, which will be a real number.

Example:

Practice: