Defining The Imaginary Number
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Lesson Priority: High
Objectives
- 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.Define: The Imaginary NumberThe imaginary unit, notated by $i$, is the number such that$$i^2 = -1$$The imaginary number $i$ is not part of the set of real numbers.
You Should Know
Right off the bat, defining something that doesn't actually exist seems suspect, especially for the logic-loving, by-the-books, true vs false, rules-based subject of mathematics. But if we seek to define an imaginary number, in spite of the fact that mathematics is rigid and logical, does that not suggest that there is a good reason to do so? In fact, including imaginary numbers in our analysis gives a complete answer to several facets of Algebra, including the fact that every nonzero real number has exactly two square roots, and the fact that every polynomial equation of degree $n$ has exactly $n$ solutions. We'll see each of these ($n$ $n$-th roots and the Fundamental Theorem of Algebra) in future advanced Algebra lessons.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 $4$.Remember!
Simplifying powers of $i$ is just a matter of remainder division by the number $4$. This is a skill we're often asked a few quick times and then never after. It can also show up as a question on the SAT / ACT exams.Put It To The Test
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$$\blacktriangleright \,\, i^{15} = i^{12} \cdot i^3 = 1 \cdot (-i)$$$$\therefore i^{15} = -i$$
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$\blacktriangleright$ Teachers sometimes throw a ridiculously unreasonable number on one of these problems for dramatic effect. The game is no different however.$$4697 \div 4 = 1174 \,\, \mathrm{R} \,\, 1$$$$\longrightarrow i^{4697} = i^{4696} \cdot i = 1 \cdot i$$$$\therefore i^{4697} = i$$
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$\blacktriangleright$ Once we learn about rationalizing expressions with $i$ in the denominator, we can take a direct algebra approach to this problem. For now, however, let's just assume the pattern we learned works backwards (it does).$$i^4 = 1$$$$i^3 = -i$$$$i^2 = -1$$$$i^1 = i$$$$i^0 = 1$$$$i^{-1} = -i$$$$i^{-2} = -1$$$$i^{-3} = i$$$$i^{-4} = 1$$$$i^{-5} = -i$$$$i^{-6} = -1$$$$\boxed{i^{-7} = i}$$
- 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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