If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Integrated math 2>Unit 5

Lesson 2: Complex numbers introduction

# Classifying complex numbers

Dive into expressions involving the imaginary unit 'i', simplifying them to reveal the real and imaginary components. Learn how real and imaginary numbers are subsets of complex numbers and study how to classify numbers as real, pure imaginary, or complex. Created by Sal Khan.

## Want to join the conversation?

• I understand that i = the square root of negative one, but does i equal an exact number, or is it just an imaginary one that no one seems to know. I thought about it for some time, but I never seemed to come up with a solution because the two numbers would have to be different, for example, i = -1*1, but that is obviously wrong because the two numbers are different which is incorrect for the principal root of negative one.
• Think of it this way. When you were very young, you answered addition questions such as 1 + 1. You then found the vast world of subtraction, which then led you to solving things such as 5 - 3. But when your teacher said, "Now find 3 - 5," then you were stumped. There was no number that you knew that would solve that equation. To solve that equation, somebody invented negative numbers. They were probably jeered at (because you cannot have a negative number of objects), but the inventor kept those numbers. The answer to the equation I gave you earlier is different from 2, but at the time, 2 was in the only set of numbers you knew. When your teacher told you about negative numbers, you now had a solution that was different than 2: -2. -2 cannot be expressed as a positive number, as a number that you already knew. So you now had two sets of numbers to work with.
Now we fast forward in history to... now. Square roots should be a breeze by now. You know that √4 = 2 and √225 = 15. But when your teacher said "What's √-1?" you were stumped... again. You thought it could be 1, or -1, or some other decimal number. But using basic facts about square roots, you decided that it was not positive, negative, or zero. But what if, just like the negative numbers, i were a number in its own space that cannot be expressed as a positive or negative number? As a side note, i is no more imaginary than the negative numbers... but it was labeled that way, and, well, the name stuck.
As we continue into higher systems of numbers, this is what we run in to all the time... Good question.
• Are there any numbers that are not complex?
• There are numbers that are not complex.
For example hypercomplex numbers which are complex numbers in multiple dimensions.
http://en.wikipedia.org/wiki/Hypercomplex_number
I'm not too clear about them, correct me if I'm wrong.
• What is the difference between complex and imaginary numbers?
• Imaginary numbers are simply numbers such as i, 2i, 3i, etc. whereas complex numbers are a combination of both real and imaginary numbers such as 2+3i, 6-7i, etc.
• I seem to have found a pattern, hopefully it's correct, but I noticed that if you start from i^1, that it is i, i^2 is -1, i^3 is -i, and lastly i^4 is 1.

This pattern seems to repeat itself. i, -1, -i, 1.

So for example since i^4 is 1, then i^5 would be i, i^6 would be -1, and so on?

Not sure if this is true or not so all help is appreciated.

And is there any simpler way to determine whether a group is complex or real?
• This concept has already been discussed in the earlier lesson on i
• I just noticed an interesting thing, we can write
i^3=i^2 *i
And also
i^3=i^4 /i
So that means- i^4 /i = i^2 *i
On simplifying, we get 1/i = -i, so that means that the reciprocal of i is also it's negative!
Can anyone confirm that what I have found is correct and expand on it a bit? It would be appreciated. Thanks
• Good work.
Yes, the -i is both the reciprocal (1/i = -i) and the negative (-i + i = 0) of i.
Can you show that i*(-i) = 1?
• So let me get this straight...Complex Numbers are EVERY number in existence, while "Real" and "Imaginary" numbers come under the category of "Complex"?
• The set of complex numbers is composed of all real and imaginary numbers, so all real and imaginary numbers are also complex. (There are numbers that are not part of the complex number set, but are typically only found in high-level math.)
• What does the Mandelbrot set have to do with imaginary numbers?
• Well, the Mandelbrot set is a set of complex numbers. So it has everything to do with them.

To determine if a complex number, c, is in the set, start with z₀ = 0 and generate a new number z₁ = z₀² + c. Keep repeating that process, generating z₂ from z₁, and so on. If the absolute value of the result keeps getting larger, then c is NOT part of the Mandelbrot set.

As you can imagine that's a LOT of work and it really needs a computer to reveal the complicated structure of the set.
• Why is 0 a real number, imaginary, and complex?
• 0 is a real number because that's part of how the reals are defined; the real numbers are a field, and a field needs a 0 element.

Imaginary numbers are numbers of the form bi, where b is real. 0 is real, and 0=0i, so 0 is also imaginary.

Complex numbers are numbers of the form a+bi, where a and b are real. 0 is real, and 0=0+0i, so 0 is a complex number.
• isn't a real number a complex number?
• Yes, the real numbers are a subset of the complex numbers. You can think of a real number as a complex number with the imaginary component as zero: a+0i.
• i see a pattern in this powering up of i with even and odd numbers but i couldnt comprehend it is there one?

## Video transcript

Now that we know a little bit about the imaginary unit i, let's see if we can simplify more involved expressions, like this one right over here. 2 plus 3i plus 7i squared plus 5i to the third power plus 9i to the fourth power. And I encourage you to pause the video right now and try to simplify this on your own. So as you can see here, we have various powers of i. You could view this as i to the first power. We have i squared here. And we already know that i squared is defined to be negative 1. Then we have i to the third power. I to the third power would just be i times this, or negative i. And we already reviewed this when we first introduced the imaginary unit, i, but I'll do it again. i to the fourth power is just going to be i times this, which is the same thing as negative 1 times i. That's i to the third power times i again. i times i is negative 1. So that's negative 1 times negative 1, which is equal to 1 again. So we can rewrite this whole thing as 2 plus 3i. 7i squared is going to be the same thing, so i squared is negative 1. So this is the same thing as 7 times negative 1. So that's just going to be minus 7. And then we have 5i to the third power. i to the third power is negative i. So this could be rewritten as negative i. So this term right over here we could write as minus 5i, or negative 5i, depending on how you want to think about it. And then finally, i to the fourth power is just 1. So this is just equal to 1. So this whole term just simplifies to 9. So how could we simplify this more? Well we have several terms that are not imaginary, that they are real numbers. For example, we have this 2 is a real number. Negative 7 is a real number. And 9 is a real number. So we could just add those up. So 2 plus negative 7 would be negative 5. Negative 5 plus 9 would be 4. So the real numbers add up to 4. And now we have these imaginary numbers. So 3 times i minus 5 times i. So if you have 3 of something and then I were to subtract 5 of that same something from it, now you're going to have negative 2 of that something. Or another way of thinking about it is the coefficients. 3 minus 5 is negative 2. So three i's minus five i's, that's going to give you negative 2i. Now you might say, well, can we simplify this any further? Well no, you really can't. This right over here is a real number. 4 is a number that we've been dealing with throughout our mathematical careers. And negative 2i, that's an imaginary number. And so what we really consider this is this 4 minus 2i, we can now consider this entire expression to really be a number. So this is a number that has a real part and an imaginary part. And numbers like this we call complex numbers. It is a complex number. Why is it complex? Well, it has a real part and an imaginary part. And you might say, well, gee, can't any real number be considered a complex number? For example, if I have the real number 3, can't I just write the real number 3 as 3 plus 0i? And you would be correct. Any real number is a complex number. You could view this right over here as a complex number. And actually, the real numbers are a subset of the complex numbers. Likewise, imaginary numbers are a subset of the complex numbers. For example, you could rewrite i as a real part-- 0 is a real number-- 0 plus i. So the imaginaries are a subset of complex numbers. Real numbers are a subset of complex numbers. And then complex numbers also have all of the sums and differences, or all of the numbers that have both real and imaginary parts.