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## Algebra (all content)

### Course: Algebra (all content)>Unit 16

Lesson 2: What are the complex numbers?

# 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? • What is the difference between complex and imaginary numbers? • 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? • 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 • So let me get this straight...Complex Numbers are EVERY number in existence, while "Real" and "Imaginary" numbers come under the category of "Complex"? • 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.  