Imaginary Numbers

In algebra, imaginary numbers are part of the complex number system in which real numbers cannot measure.

The symbol for imaginary numbers is i. You can define what i is equivalent to, but you cannot measure it in real terms.

i is equal to the square root of -1.

$$i = \sqrt{-1}\,$$

This means that i2 is equal to -1.

$$i^2 = \sqrt{-1}\,\sqrt{-1}\, = -1$$

Therefore, i3 is equal to -i, or the negative square root of -1.

$$i^3 = \sqrt{-1}\,\sqrt{-1}\,\sqrt{-1}\, = -i = -\sqrt{-1}\,$$

Hence, i4 is equal to 1.

$$i^4 = \sqrt{-1}\,\sqrt{-1}\,\sqrt{-1}\,\sqrt{-1}\, = 1$$

This pattern of i, -1, -i, and 1 continues ad infinitum in the same exact manner shown.

Simplyfing In Terms of i
With i, you can simplify the square roots of negative numbers.

$$\sqrt{-7}\, = \sqrt{-1}\,\sqrt{7}\, = i\sqrt{7}\,$$

$$\sqrt{-16}\, = \sqrt{-1}\,\sqrt{16}\, = 4i$$

$$\sqrt{-50}\, = \sqrt{-1}\,\sqrt{25}\,\sqrt{2}\, = 5i\sqrt{2}\,$$

Defining Complex Numbers
All complex numbers have a real part and an imaginary part. This can be shown as follows:

$$a + bi$$, where a is real and b is imaginary.

Below are several examples of complex numbers.

$$16$$, $$7i$$,

$$\pi\,$$, $$2i\sqrt{7}\,$$,

$$4 + 8i$$, and even $$-8 + 3i\sqrt{19}\,$$ are all complex numbers.

Multiplying Complex Numbers
Multiplying complex numbers is not that much different than multiplying algebraic expressions. I'll show you how so below.

1. Copy the problem...

$$(8 + 7i)(-2 + i)$$

2. LOIF it (Last - Outside - Inside - First), which is like FOIL in reverse...

$$-16 + 8i - 14i - 7i^2$$

3. Combine like terms...

$$-16 - 6i - 7i^2$$

WATCH OUT! You're not done! Remember, i2 = -1.

$$-16 - 6i - 7i^2$$

4. Simplify the exponents...

$$-16 - 6i + 14$$

5. And combine the like terms again for the final answer.

$$-2 - 6i$$