Lesson 5-5: Imaginary and Complex Numbers
We’ve been doing a great job learning how to solve quadratic equations! One of the things we learned in the last lesson is the quadratic formula. We also learned that the discriminant (the part inside the radical of the quadratic formula) tells us the number of solutions the equation has.
Do you recall the one situation in which the discriminant tells us there is no solution? That’s right! When the discriminant is negative, we have no solution. Why is that? Do you recall what the discriminant is inside of? Well, can we take the square-root of a negative number? Nope! Not at least using numbers that we’re used to!
Today we’re going to explore a way of actually talking about a solution for quadratic equations that have a negative discriminant. It is going to require you to have a bit of an open mind. To get there, let’s talk about the history of numbers a little…
Mommy, where do numbers come from???
We take numbers for granted. We even take weird numbers like π for granted. It is a pretty weird number…it never ends, never repeats and represents the ratio of the circumference to the diameter of any circle. If you stop and think about it, that is weird.
What about the number zero? Zero is a very weird number. Think about it…zero is something that represents…nothing. How can something represent nothing? Beats me, but the number zero does a great job of it!
Numbers were first thought of as a way of saying how much of something you had. From this we got the counting (also referred to as the whole) numbers: 0, 1, 2, 3…
Mainly numbers were used for counting money…or cows…or whatever was considered wealth. And where there is wealth and the making of money, there is… the loss of money. As soon as people started trying to explain how much they’d lost, we ran into another very bizarre set of numbers: the negative numbers. They are strange…they are numbers that are less than zero right? Think about that. Zero represents nothing. How in the world can you have something less than nothing???
Well, as soon as you accept negative numbers, you have what we call the integers.
So far, numbers were only seen as something that described a quantity (or a lack of a quantity). But people like to compare don’t they? So soon we had to come up with a way to say that it takes three of your piles to make one of mine. And now…we have…fractions!
As soon as you accept fractions, you have what we call the rational numbers.
The ancient Greeks really loved this idea and ran with it. In fact they liked it so much they believed that all numbers were rational numbers. Until…some poor guy used the Pythagorean Theorem on an isosceles triangle and discovered that the length of the hypotenuse isn’t…a…rational number. It is something different. Well, he got all excited and started telling all the other math guys about this. They…didn’t like this idea. So they tied a weight around his feet and threw him over the side of a ship. Problem solved…we don’t need those weird new pesky numbers do we?
Well, we do. And we now call them the irrational numbers.
Put all of the above together and we get the number system we spend most of our time in…the real number system.
So what’s my point? My point is that all along, we kept bumping into weird new circumstances that required us to think about numbers in a new way. Today is such a day. Today we need to stretch our thinking to find a way to talk about the numbers that actually pop out of these quadratic equations that supposedly have no solution.
Sorry, I don’t see any numbers that are the square root of a negative!
We now know that a quadratic equation can easily have a discriminant that is negative. So what??? Who cares? Where do you find one of these?
Well, it turns out, all over the place. Tonight, when you go home and play your Wii or Playstation, you are bumping around with these things. When you pull out your iPod to listen to some music, you are bumping around with these things.
Two very common places we find this type of quadratic is in electrical circuits (the Wii and Playstation) and in signal-processing (the iPod).
So how do we work with these things?
First we need to know what to call them. The name given to is the imaginary unit and we use the lowercase letter i to represent it. So in math we’d say .
Let’s try this out: how would you represent using the imaginary unit i?
We “separated” the -1 and the 2 and took them one by one.
Write the following using the imaginary unit i: and and :
Complex numbers
You can add an imaginary number and a real number. We call the result a complex number. They have a real part and an imaginary part. In general they look like this:
where a and b are real numbers; ais the real part, bi is the imaginary part.
Does it make any sense for b to be zero? Nope…if b = 0 then the imaginary part disappears and we just have a real number.
If a is zero then the real part disappears and we’re left with an imaginary number.
How do you think you can decide if two complex numbers are equal? Well, it seems kind of obvious…the real parts need to be equal and the imaginary parts need to be equal.
So does? Yup because
Adding and subtracting complex numbers
When you add or subtract complex numbers, just combine the real parts and then combine the imaginary parts. All the normal rules of arithmetic that you’re used to work with complex numbers. For practice, combine the following:
Multiplying complex numbers
To multiply two complex numbers, multiply the real parts and multiply the imaginary parts. Remember that . For practice, multiply the following:
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