Using the Standard Form for Complex Numbers

Instructor: David Karsner
Complex numbers are numbers that have two components. One component is the real part and the other one is the imaginary part. The standard form of a complex number is ''a''+''b''i where ''a'' is real and ''b''i is imaginary. This lesson will cover the standard form and provide examples.

What about i?

If you have spent any time around algebra, you are probably used to seeing and using different variables, X, of course, being the most common. You have probably used most of our twenty six letters as variables. You might even be comfortable using Greek letters as variables. There is probably one letter that you haven't used as a variable. Think about it. Have you use the letter i as a variable? The letter i is rarely used as a variable. It has been set apart to represent something else. The letter i is the name for the imaginary square root of -1. This lesson will explain the use of i in the standard form of complex numbers. It will also give you instructions on how to add, subtract, multiply, and divide complex numbers.

Complex Numbers

Complex numbers are numbers in which the real component and the imaginary part of the number are both represented. The numbers in standard form will be a + bi, where a is the real part and bi is the imaginary part. An example of a complex number would be 3 +5i. 3 is the real part, and 5i is the imaginary part. Real numbers with no imaginary part can be represented using complex numbers, such as 12 + 0i. Imaginary numbers with no real component can also be represented as a complex number, such as 0 + 7i.

Adding and Subtracting Complex Numbers

To add or subtract complex numbers, add/subtract the real parts with real parts and the imaginary parts with the imaginary parts.

For example, (4+6i) + (7-9i) = (4+7)+(6i-9i) = (11)+(-3i) = 11-3i

Notice, that if you had mistakenly taken i as a variable and performed the sum of two binomials you would have gotten an answer that looks the same.

Multiplying Complex Numbers

Do you remembers how to multiply binomials? The handy trick called F.O.I.L. stands for first, outer, inner, and last. Even though it's not the same thing, this trick will still work for multiplying complex numbers. The only difference in multiplying complex numbers from multiplying binomials is you have to remember the role of i. i is defined as the square root of negative one. i2=-1. Any time you have i2, you can replace it with -1.

For example (3+2i)(5-4i):

  1. First: 3x5=15
  2. Outer: 3 x -4i = -12i
  3. Inner: 2i x 5 = 10i
  4. Last: 2i x -4i = -8i2
  5. Put them together: 15-12i+10i-8i2
  6. Combine like terms: 15 -2i-8i2
  7. Replace i2 with -1: 15-2i-8(-1)
  8. Simplify: 23-2i

Multiplying Complex Numbers

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