Algebra II: Complex Numbers

On this page we hope to clear up problems that you might have with complex numbers and imaginary numbers. Unfortunately, imaginary numbers aren't all they're cracked up to be. For instance, a gazillion isn't suddenly a viable number. :-) Read on or follow any of the links below to better your understanding of complex and imaginary numbers.

Imaginary numbers
Complex numbers
Equations with complex numbers
Graphing complex numbers
Quiz on complex numbers

Imaginary Numbers

In the set of real numbers, negative numbers do not have square roots. A new kind of number, called imaginary was invented so that negative numbers would have a square root. These numbers start with the number i, which equals the square root of -1, or i^2 = -1.

All imaginary numbers consist of two parts, the real part, b, and the imaginary part, i.

Example:

1. Simplify: SQRT(-5)
   Solution: Write -5 as a product of prime factors.

             SQRT(-1 * 5)

             Write as separate square roots.

             (SQRT(-1))(SQRT(5))

             By definition, i = SQRT(-1), so the final answer is
             (SQRT(5))i.  (SQRT(5) is the real part, or
             b.)

Complex Numbers

A complete number system, one that includes both real and imaginary numbers, was devised. Numbers in this set are called complex numbers. Complex numbers consist of all sums a + bi where a and b are real numbers and i is imaginary.

Real numbers fit into the complex number system because a = a + 0i.

i behaves as any variable would.

Example:

1.  Problem: 7i + 9u
   Solution: Combine like terms.

             16i

Complex numbers, like real numbers, can be equal. For example, a + bi = c + di says that a and c must be equal and b and d must be equal.

Example:

1.  Problem: Find x and y in 3x + yi = 5x + 1 + 2i
   Solution: Using the above definition for equality of complex numbers,
             set the real parts of the equation equal and set the imaginary
             parts equal.

             3x = 5x + 1     yi = 2i
             -2x = 1         y = 2
             x = -(1/2)

Multiplication is done as if the imaginary parts of complex numbers were just another term. Always remember that i^2 = -1.

Example:

1.  Problem: 3i * 4i
   Solution: 12i^2

             Remember that i^2 equals -1.  Rewrite the answer.

             12(-1)
             -12

When dividing complex numbers, you multiply the problem by 1 (remember that anything divided by itself is 1). The conjugate of the divisor is usually used for 1.

Example:

1.  Problem: -5 + 9i
             -------
              1 - i 
   Solution: Multiply by 1.

             -5 + 9i   1 + i
             ------- * -----
              1 - i    1 + i

             Multiply out as you would normally multiply a binomial by a
             binomial.  FOIL might be useful.

             -5 - 5i + 9i + 9i^2
             -------------------
              1 +  i -  i -  i^2

             Perform the indicated operations, keeping in mind that i^2
             is equal to -1.  Combine like terms.

             -14 + 4i
             --------
              1 - i^2

             -14 + 4i
             --------
                 2   

             Perform the indicated division.

             -7 + 2i

Equations with Complex Numbers

Now that negative numbers have square roots, equations such as x^2 + 1 = 0 have solutions. They can also be factored!

Example:

1.  Problem: Solve for x: x^2 + 1 = 0.
   Solution: Subtract 1 from each side.

             x^2 = -1

             Take the square root of each side.  Remember that
             i = SQRT(-1).

             x = i, -i

2.  Problem: Show that (x + i)(x - i) is a factorization of x^2 + 1.
   Solution: Multiply.

             x^2 + ix - ix - i^2
             x^2 + 1

Graphing Complex Numbers

You graph real numbers on a number line, but you graph complex numbers the same way you would graph an ordered pair, such as (x, y), but the x-axis is replaced by the real axis, and the y-axis is replaced by the imaginary axis.

Example:

1.    Graph: A: 3 + 2i
             B:-4 + 5i
             C:-5 - 4i
             D:      i

   Solution: See the example figure.

Take the quiz on complex numbers. The quiz is very useful for either review or to see if you've really got the topic down.

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