Introduction to imaginary numbers

Pre-Calculus Tutorial

Introduction to imaginary numbers

Intro

-Imaginary numbers are represented as i
-You will see an imaginary number when there is a negative number within a square root. Imaginary numbers are typically accompanied by an exponent. It is important to understand the rules of exponents before continuing.

Before diving into the sample problem, it is important to understand the structure of imaginary numbers.

-i^0 = 1 *any number to the power of “0” will be “1”
-i^1= i *any number to the first power will equal itself
-i^2=-1 *this is simply a definition
-i^3= -i *take i^1 (which = i) and i^2 (which = -1) and when multiplied gives you -i
-i^4= 1 *i multiplied by i^3 =-i and -i (from what we saw on the previous example) = -1 multiplied by i. Therefore this is rewritten as (-1)(i)(i). We know that (i)(i)=-1 which leaves us with (-1)(-1)= 1
-i^5=i
-i^6=-1
-i^7=-i

*you will notice a pattern of (1,i,-1,-i) repeating itself. This pattern will continue as long as you are willing to count.

Sample Problem

Now that you have an understanding of how imaginary numbers work, what is the answer to the following equation?

i^20 = ___

i


-1


1


-i


20


Solution

There are a few ways to approach this problem. You can count out the pattern until you hit the power of 20 or you can break down the exponent which is the encouraged way to solve this equation. As numbers get into the hundreds, it will not be practical to count out the pattern.

-Step one: Break down the exponent into smaller numbers
i^20 = i^4×5

-Step two: use what we know and convert i^4 (which is an easy number =1)
i^4×5 = (1)^5

-Step three: simplify 1^5
1^5 = 1



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