# 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!