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.

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

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

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

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 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.

$-2 - 6i$