Imaginary Numbers

Definition

A number that when squared (multiplied by itself) gives a negative result.

Try

Let's try squaring some numbers to see if we can get a negative result:

No luck! Always positive, or zero.

That is because we are squaring Real Numbers.

But imagine that there is such a number (call it i for imaginary) that could do this: i × i = -1 Would it be useful, and what could we do with it?

Well, by taking the square root of both sides we get an answer to the square root of -1:

And that is very useful ... by simply accepting that i exists we can solve things where we need to take the square root of a negative number.

So long as we keep that little "i" there to remind us that we still need to multiply by √-1 we are safe to continue with our solution.

Unit Imaginary Number The "unit" Imaginary Number (the equivalent of 1 for Real Numbers) is √(-1) (the square root of minus one). In mathematics we use i (for imaginary) but in electronics they use j (because "i" already means current, and the next letter after i is j).

Examples of Imaginary Numbers

Imaginary Numbers are not "Imaginary"

Actually Imaginary Numbers were once thought to be impossible, and so they were called "Imaginary" (to make fun of them).

But then people researched them more and discovered they were actually useful and important becaue they filled a gap in mathematics ... but the "imaginary" name has stuck.

Useful

Here are 2 cases where they are useful:

Electricity

		AC (Alternating Current) Electricity changes between positive and negative in a sine wave. If you combine two AC currents they may not match properly, and it can be very hard to figure out the new current. But using imaginary numbers and real numbers together makes it a lot easier to do the calculations.
		And the result may be "Imaginary" current, but it could still hurt you!

Quadratic Equation

	The Quadratic Equation can give results that include imaginary numbers ...
... but maybe after more calculations the "i" value gets cancelled out (or become real numbers because they get squared), leaving an answer that is all real.

Interesting Property

The Unit Imaginary Number, i, has an interesting property. It "cycles" through 4 different values each time you multiply:

So, i × i = -1, ... then -1 × i = -i, ... then -i × i = 1, ... then 1 × i = i (back to i again!)

Conclusion

The unit imaginary number, i, equals the square root of minus 1 Imaginary Numbers are not "imaginary", they are real and useful, and you may need to use them one day!