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Introduction to Imaginary Numbers - Concept

Teacher/Instructor Carl Horowitz
Carl Horowitz

University of Michigan
Runs his own tutoring company

Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!

An imaginary number bi has two parts: a real number, b, and an imaginary part, i, defined as i^2 = -1. Imaginary numbers are applied to square roots of negative numbers, allowing them to be simplified in terms of i. When a real number, a, is added to an imaginary number, a + bi is said to be a complex number. Beware that in some cases the letter j is used instead of i for the imaginary number.

Intro to imaginary numbers. Now imaginary numbers contrary to their name actually do exist, sort of a weird concept that we're going to be getting into.
So to start that out, let's do a little bit of a review on radicals. So say I say something like square root of 8, and I say simplify that. Your logic is like okay. What goes into 8. We have square root of 4 times 2. Square root of 4 is a perfect square so we can take that out giving us 2 root 2. Where imaginary numbers come in is when we are taking the square root of a negative number. Okay? And up until now what we've said is you can't take the square root of a negative number which is sort of a lie, I apologise. But in order to do that we have to use these imaginary numbers, okay?
So take a look at the square root of -9. Okay. We've been saying this isn't real which is true, okay? But just like we did up here, we can split this up into a real component which is the 9 part we know and the -1 part which is the part that sort of throws us off. Okay?
So what we're going to say is this is the same thing as the square root of 9 times -1. We can split up square roots and then write it in up here like you know how to do it. So this is the square root of 9 times the square root of -1. Square root of 9 is 3. So what we're really left with here is 3 times the square root of -1. This is where imaginary numbers come into play. And square root of -1 is really the basis of all imaginary numbers we deal with. And going to give you a little bit of definition. Okay?
What we do is we call the square root of -1 the letter i. Lower case case of i. So going back to the problem we have over here we can replace that in, and this ends up being 3i, i standing for the imaginary number, square root of -1. Okay. In addition there is one other part of the definition of i, okay? And that is i squared.

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