What are complex numbers? Well, as the name implies, it’s… complicated. In fact, most complex numbers can’t even be found on the number line!
Complex numbers include everyday real numbers like 3, -8, and 7/13, but in addition, we have to include all of the imaginary numbers, like i, 3i, and -πi, as well as combinations of real and imaginary. You see, complex numbers are what you get when you mix real and imaginary numbers together — a very complicated relationship indeed!
By definition a complex number is any number of the form a + bi, where both a and b are real numbers, and i is the imaginary unit, defined by its main property: i2 = -1.
We’ll have a lot more to say about i in this article. And we’ll explore the definition and properties of complex numbers. There’s a lot to say about these amazing and mysterious gadgets, so buckle up and hold on tight!
Imagining the Impossible
Mathematicians have a tendency to invent new tools as the need arises. Need to count losses as well as profits? Invent the negative numbers. Need to keep track of parts of a whole? Introduce fractions.
Need to take a square root of a negative number? Dream up imaginary numbers!
For example, the equation x2 = -1 cannot be solved by any real number. Why? Because if you square either a positive or a negative real number, the result is always positive. In particular, x = -1 is not a solution to the equation because (-1)2 = 1.
But if we could imagine a number like √-1, then we can solve the equation! Let’s give it a name: i for imaginary.
i = √-1
The trouble is: i is still not a real number! Oh well, just because something doesn’t exist, that’s no reason to discriminate against it. In fact, if we’re willing to think like a mathematician, then we can use i just like any other number if we’re careful and follow the rules.
For example, if you follow the basic rule: i2 = -1, then you get two solutions to the equation x2 = -1. Both x = i and x = –i work!
(±i)2 = -1
A Brief History of the Imaginary
Up until the mid-sixteenth century, certain kinds of equations were impossible to solve. For example, some kinds of cubic equations (degree 3 polynomial equations) resisted all known methods at the time.
Then along comes Niccolò Fontana (Tartaglia), who managed to crack the code of the cubic equation! He, along with his rival Gerolamo Cardano and Cardano’s student Lodovico Ferrari, were using strange “imaginary” numbers to find real solutions to the cubic.
The rest, as they say, is history!
Working with Complex Numbers
So remember, a complex number has the form a + bi.
For example, -7 + 3i and 8.23 + (43/11)i are complex numbers.
There are a few basic rules and properties that we need to establish in order to work with these kinds of numbers.
- The Most Important Rule: i2 = -1
- The Second Most Important Rule: Purely real terms and purely imaginary terms are not like terms.
We’ve seen that first rule already, so let me say something about the second rule.
Essentially, you have to keep the real and imaginary parts separate at all times. The complex number -7 + 3i is NOT the same as -4i. Just think of i almost as a variable (like x or y in a polynomial expression).
Simplifying Complex Radicals
If there is a negative under a radical, then chances are that you’ll need to use complex numbers to simplify it.
Just remember that the square root of -1 is equal to i.
And don’t forget to pull out any perfect squares while you’re at it!
Example: √-200 = √(-1)(100)(2) = i10√2 (or 10√2 i in standard form)
Complex numbers have to follow most of the same rules as real numbers, such as the commutative, associative, and distributive laws.
The rules for adding, subtracting, and multiplying follow pretty easily.
- Adding: (a + bi) + (c + di) = (a + c) + (b + d)i.
This formula just means we have to group like terms.
Example: (4 – 3i) + (-1 + 8i) = (4+(-1)) + (-3+8)i = 3 + 5i
- Subtracting: (a + bi) – (c + di) = (a – c) + (b – d)i
Again, just group like terms. Don’t forget to distribute the minus!
Example: (4 – 3i) – (-1 + 8i) = 4 – 3i + 1 – 8i = 5 – 11i
- Multiplying: (a + bi) × (c + di) = (ac – bd) + (ad + bc)i
Instead of memorizing that formula, just think “FOIL” multiplication, and remember that i2 = -1
Example: (4 – 3i)(-1 + 8i) = (4)(-1) + (4)(8i) + (-3i)(-1) + (-3i)(8i)
= -4 + 32i + 3i + (-24)i2
= -4 + 35i + (-24)(-1)
= -4 + 35i + 24
= 20 + 35i
- Dividing: On the other hand, the rule for division is a bit complicated. You have to use the complex conjugate and then simplify your result completely.
The complex conjugate of a complex number a + bi is: a – bi.
When you divide (a + bi) / (c + di), you have to multiply both the numerator and denominator by the complex conjugate of the denominator and then simplify.
It helps to know the difference of squares formula by heart, as it will play a role in cleaning up the denominator.
(x + y)(x – y) = x2 – y2
Let’s see how this works by example. Notice in the first line where the complex conjugate of the denominator has to be multiplied to the top and bottom.
As you might realize, there’s a lot more to be said about complex numbers! This article represents just the tip of a very large iceberg.
Complex numbers are used to describe the electromagnetic fields and waves that allow your cell phone to operate. They help to define the fundamental particles of our universe, such as the electron and proton. And they can even generate beautiful fractal images.
Image by realworkhard
I hope that you have gained a better understanding of imaginary and complex numbers! Keep the basic rules and definitions in mind, and perhaps you will learn to love these fascinating and complicated mathematical tools as much as I do!