# COMPLEX ARITHMETIC

Complex arithmetic may sound like a real difficult exercise to master, but after getting use to imaginary numbers it can be as simple as 2+2.

The number known as i is equal to the square root of 1.  If you put that in your calculator you will find that the square root of 1 gives you an unreal answer.  i is the basis for the complex number system.

## First thing to learn is the number i in relation to its exponent.  These first four are something that is mostly used if memorized.

i^1=i

i^2=-1

i^3=-i

i^4=1

All the exponents to follow these four will fall along this pattern.

Example:

i^5=i

i^6=-1

i^7=-i

i^8=1

and so on

So what would i^56 equal?

First you would take 56/4, which equals 14. Since 56 is divisible perfectly by 4 it would equal 1 just like i^4.

From there you could go on to figure out what i^54 and i^57 are by just having the starting point from i^56.

COMPLEX NUMBERS

Complex numbers are written in the form of a+bi, so with numbers it would look something like 4+6i or 2-3i.

When adding and subtracting complex numbers you only add and subtract the whole numbers with the whole numbers and the imaginary numbers with the imaginary numbers.

Example:

(4+3i)+(6-2i)

First add 4+6, then add 3i + -2i.

So the answer would be 10+1i.

Lets try subtracting one.

(12-5i)-(3+8i)

Again subtract 3 from 12 first then subtract 8i from 5i.

Making the answer 9-13i.

Adding and subtracting complex numbers is just like normal arithmetic.  DONT LET THE is CONFUSE YOU!

MULTIPLYING COMPLEX NUMBERS

Multiplying complex numbers may be a little bit more difficult then adding and subtracting them but you still use a basic principle you learned back in middle school.

When multiplying complex numbers use the FOIL system as you used to make the general quadratic function.

First

Outer

Inner

Last

Example:

(2+3i) (5-i) = 2(5) + 2(-i) + 3i(5) + 3i(-i)

=10  2i + 15i  3i^2

=10 + 13i  3i^2

3i^2=-3 since i^2=-1

So 10 + 13i  3i^2 = 13 +13i.

## Lets do another!

(9-2i) (2+7i) = 18 + 63i  4i  14i^2

=18 + 59i  14i^2

=32 + 59i

Hopefully this web site has been a big help and will make complex arithmetic a little bit easier for you.  Try the quiz if you get the chance to see if you can master complex arithmetic!