Defining The Imaginary Number
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- Define the imaginary number
- Learn how to simplify powers of i
The commonly used imaginary number completes the picture in many advanced math topics, even though it doesn't exist. Here we seek to understand what the imaginary number represents, and the basics of working with it.
Practice problems and worksheet coming soon!
Math's Imaginary Friend
Taking the square root of negative numbers is one of many things in math that we are told we cannot do. The problem is that, when you square any nonzero number, the result is positive, regardless of whether your starting number was positive or negative. Therefore, when you reverse this process (aka square roots), you cannot find a number that, when squared, gives you a negative result.Enter the imaginary number, $i$. We define the imaginary number to be the number that, when squared, gives you a negative result. Specifically,$$i^2 = -1$$Another way to say this is$$i = \sqrt{-1}$$Although many stickler teachers and profs will assert that the first definition is the proper one, and for that reason as well as simplicity, we will stick to that here.What Can We Do With i?
Like a kid with a new toy on Christmas, we want to know - what can we do with it? Equipped with this new tool, what is now possible to achieve that was not possible before? There are three major tasks that we'll learn right away - the first two are in the next lessons, and the third we will tackle now. Additionally, there are many subtle uses for this number throughout advanced algebra topics, so don't be surprised when it occasionally shows up down the road.Task 1 - Square RootsFor now, one of the most common and useful purposes of the imaginary number is to allow us to take square roots of negative numbers. The next lesson » will focus on the how-to of simplifying square roots of negative numbers.Task 2 - Algebra ManipulationIn some situations, we will be presented with expressions that contain $i$. It is our job to work with $i$ algebraically, which, for the most part, means treating it like a variable.Using imaginary numbers with arithmetic » and rationalizing with imaginary numbers » are two common processes, which each have a dedicated lesson. Again, most of the time we're just treating $i$ like a variable.Task 3 - Simplifying Powers of iThis is a quick trick to learn, and beside the conceptual knowledge about what $i$ is and what it represents, it is the only real "skill" that we'll pick up in this lesson. Let's see what it's all about.The Cyclical Powers of i
When you raise $i$ to a power, there are only a handful of possibilities for the result you ultimately end up with after simplifying. To see the pattern, let's start with $i$ itself.$$i^1 = i$$Nothing special to look at here - in fact all we really said was $i = i$. Next look at $i^2$:$$i^2 = -1$$This still doesn't tell us anything new or profound. We actually defined $i$ with this exact fact - that $i$ is the number such that $i^2 = -1$. Next, let's find $i^3$ by multiplying $i^2$ by $i$:$$i^3 = i^2 \cdot i = -1 \cdot i = -i$$Interestingly enough, $i^3$ simplifies to $-i$. Now let's find $i^4$ by multiplying $i^3$ by $i$:$$i^4 = i^3 \cdot i = -i \cdot i = -i^2 = -(-1) = 1$$Therefore $i^4$ really simplifies to $1$. If we keep going, we'll see that $i^5 = i$, since it is really$$i^5 = i^4 \cdot i = 1 \cdot i = i$$There is a repeating pattern to be observed here:$$i = i$$$$i^2 = -1$$$$i^3 = -i$$$$i^4 = 1$$$$i^5 = i$$$$i^6 = -1$$$$i^7 = -i$$$$i^8 = 1$$$$i^9 = i$$$$....$$The most efficient way to describe and use this pattern is to remember that when $i$ is raised to a power that is a multiple of $4$, the result is $1$. E.g.$$i^4 = 1$$$$i^8 = 1$$$$i^{12} = 1$$$$i^{16} = 1$$$$....$$Therefore, if we are asked how to simplify something like $i^{63}$, we just need to break it into pieces, where one of the pieces is a multiple of $4$.Put It To The Test
- Understand what the imaginary number is and what is represents, even though it isn't real and can't actually exist
- Though it will take experience, begin to understand why allowing this number to exist is useful for math
- Familiarize yourself with the three main things we will want to accomplish using $i$
- Be able to take $i$ raised to any power and simplify it into one of four results: $i$, $-i$, $1$, or $-1$.
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