Imaginary Numbers Continued

Now you know how to convert negative square roots to real numbers multiplied by i, you must learn how to work with imaginary numbers.

The set of imaginary numbers is CLOSED to addition and subtraction, but not multiplication and division.  A set is closed to an operation if that operation can be performed on any numbers in that set and the answer will still be in the set.

The standard form of a complex number is written:
a + bi

a is called the real part, and b is called the imaginary part.  In real numbers, b is always 0.  In imaginary numbers, b must be unequal to 0, but a can be any real number.  When a = 0, the number is called a pure imaginary number.

To add complex numbers, you must add their real parts and their imaginary parts.  For example:
(3 + 2i) + (-5 + 7i) = (3 + -5) + (2 + 7)i = -2 + 9i

-8i + (9 + 3i) = (0 + 9) + (-8 + 3)i = 9 - 5i

To subtract imaginary numbers, use the definition of subtraction, that it is addition of the opposite.  Multiply the second number by -1 (every term) and add them.

(4 - 5i) - (3 - 12i) = (4 - 5i) + (-3 + 12i) = (4 - 3) + (-5 + 12)i = 1 + 7i

Multiplication and division are a bit different.  When multiplying, use the distributive property (or FOIL for binomials).  Then simplify the expression.  If i to any power appears, simplify it using the pattern.

4i(-2 + 10i) = 4i(-2) + 4i(10i) = -8i + 40i² = -8i + 40(-1) = -40 - 8i

(3 - 7i)(-6 + 4i) = 3(-6) + 3(4i) + -7i(-6) + -7i(4i) = -18 + 12i + 42i - 28i² =
-18 + 54i -28(-1) = -18 + 28 + 54i = 10 + 54i

When dividing complex numbers (there is an imaginary part in the denominator of a fraction), you must multiply the fraction by 1, written as the denominator's conjugate over itself.  This will clear the denominator of imaginary numbers.

Conjugates are two binomials in which the only difference is the middle sign (e.g. 5 +2i and 5 - 2i).

For complex numbers, (a + bi)(a - bi) = a² + b²

  6i     =     6i(8 + 3i)     =  48i +18i²   = -18 + 48i
8 - 3i     (8 - 3i)(8 + 3i)      64 + 9               73

Finally, you must put this in standard form.

-18 + 48i  = -18 + 48i
     73            73     73

Neither of these can be simplified any further (they could, for example, if 73 were 74), so you are done.

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