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!

Take the square root of both sides:

x = ± √(−1)
x = ± i

Answer: x = −i or +i

Check:

  • (−i)2 = (−i)(−i) = +i2 = −1
  • (+i)2 = (+i)(+i) = +i2 = −1

 

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

 

Complex Numbers

Imaginary numbers become most useful when combined with real numbers to make complex numbers.

 

Spectrum Analyzer

Those cool displays you see when music is playing? Yep, Complex Numbers are used to calculate them! Using something called "Fourier Transforms".

In fact many clever things can be done with sound using Complex Numbers, like filtering out sounds, hearing whispers in a crowd and so on.

It is part of a subject called "Signal Processing".

 

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 complex numbers makes it a lot easier to do the calculations.

And the result may be "Imaginary" current, but it could still hurt you!

Mandelbrot Set Zoomed In

 

Mandelbrot Set

The beautiful Mandelbrot Set (part of it is pictured here) is based on Complex Numbers.

 

 

Quadratic Equation

Quadratic Equation

The Quadratic Equation, which has many uses,
can give results that include imaginary numbers

Also, in Science, Quantum mechanics and Relativity also use complex 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.