Imaginary Numbers

 

An Imaginary Number, when squared, 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.

It seems like we cannot multiply a number by itself to get a negative answer ...

... 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 this:

equals the square root of -1
Which means that i is the answer to the square root of -1.

Which is actually very useful because ...

... by simply accepting that i exists we can solve things
that need the square root of a negative number.

Let us have a go:

Example: What is the square root of -9 ?

Answer: = √(9 × -1) = √(9) × √(-1) = 3 × √(-1) = 3i

(see how to simplify square roots)

Hey! that was interesting! The square root of -9 is simply the square root of +9, times i.

In general:

√(-x) = i√x

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!

Using i we can also come up with new solutions:

Example: Solve x2 = -1

Using Real Numbers there is no solution, but now we can solve it:

Answer: x = √-1 = i

 

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

i 12.38i -i 3i/4 0.01i -i/2

Imaginary Numbers are not "Imaginary"

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 because they filled a gap in mathematics ... but the "imaginary" name has stuck.

And that is also how the name "Real Numbers" came about (real is not imaginary).

Imaginary Numbers are Useful

Here are 2 cases where they are useful:

 

Electricity


 

AC (Alternating Current) Electricity changes between positive and negative in a sine wave.

When we 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

Quadratic Equation

The Quadratic Equation can give results that include imaginary numbers

Interesting Property

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

i cycle  
  • i × i = -1,
  • then -1 × i = -i,
  • then -i × i = 1,
  • then 1 × i = i (back to i again!)

So we have this:

i = √-1 i2 = -1 i3 = -√-1 i4 = 1 i5 = √-1 ...etc

 

Example What is i6 ?

i6 = i4 × i2
  = 1 × -1
  = -1

Conclusion

The unit imaginary number, i, equals the square root of minus 1

Imaginary Numbers are not "imaginary", they really exist, and you may need to use them one day.