Imaginary Numbers

An Imaginary Number, when squared, gives a negative effect.
imaginary squared => negative

Try

Let's try squaring some numbers to encounter if we tin become a negative effect:

  • two × 2 = 4
  • (−2) × (−2) = iv (because a negative times a negative gives a positive)
  • 0 × 0 = 0
  • 0.1 × 0.ane = 0.01

No luck! Always positive, or zero.

It seems similar nosotros cannot multiply a number by itself to go a negative answer ...

thought

... but imagine that at that place is such a number (call it i for imaginary) that could practise this:

Would information technology be useful, and what could we practice with it?

Well, past taking the foursquare root of both sides we get this:

equals the square root of -1
Which means that i is the reply to the square root of −ane.

Which is actually very useful because ...

... past simply accepting that i exists we tin solve things
that need the foursquare root of a negative number.

Let united states of america have a go:

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

In full general:

√(−x) = i√x

And then long as nosotros keep that little "i" there to remind the states that we nevertheless
demand to multiply by √−1 we are condom to keep with our solution!

Using i

Example: What is (5i)2 ?

(5i)2 = fivei × fivei

= 5× 5× i × i

= 25 × i ii

= 25 × −ane

= −25

Interesting! We used an imaginary number (fivei) and ended upwardly with a real solution (−25).

Imaginary numbers tin can help united states of america solve some equations:

Instance: Solve x2 + 1 = 0

Using Existent Numbers there is no solution, but now we can solve information technology!

Subtract 1 from both sides:

x2 = −one

Take the foursquare root of both sides:

10 = ± √(−1)

x = ± i

Respond: x = −i or +i

Check:

  • (−i)2 + 1 = (−i)(−i) + 1 = +i2 + 1 = −1 + one = 0
  • (+i)2 +i = (+i)(+i) +1 = +i2 +1 = −ane + 1 = 0

i and j

Unit Imaginary Number

The square root of minus one √(−i) is the "unit" Imaginary Number, the equivalent of 1 for Real Numbers.

In mathematics the symbol for √(−1) is i for imaginary.

Can yous take the foursquare root of −one?
Well i tin can!

But in electronics they employ j (because "i" already means current, and the next alphabetic character after i is j).

Examples of Imaginary Numbers

i 12.38i −i 3i/four 0.01i πi

Imaginary Numbers are non "Imaginary"

Imaginary Numbers were once idea to be impossible, and so they were called "Imaginary" (to brand fun of them).

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

And that is also how the name "Real Numbers" came about (real is non imaginary).

Imaginary Numbers are Useful

complex plane vector add

Complex Numbers

Imaginary numbers go most useful when combined with existent numbers to make circuitous numbers like 3+5i or 6−4i

Spectrum Analyzer

spectrum analyzer

Those cool displays you lot come across when music is playing? Yep, Circuitous Numbers are used to summate them! Using something called "Fourier Transforms".

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

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

Electricity

plug
sine waves

Ac (Alternate Electric current) Electricity changes between positive and negative in a sine wave.

When we combine two Ac currents they may non match properly, and it can be very hard to figure out the new current.

Only using complex numbers makes information technology a lot easier to do the calculations.

And the event may take "Imaginary" current, but it tin can nevertheless hurt you lot!

Mandelbrot Set Zoomed In

Mandelbrot Set

The beautiful Mandelbrot Prepare (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

As well Science, Quantum mechanics and Relativity use complex numbers.

Interesting Property

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

ane × i = i
i × i = −1
−1 × i = −i
−i × i = ane
Back to 1 once again!
i cycle

So we accept this:

i = √−i itwo = −1 i3 = −√−1 i4 = +1
i5 = √−1 ihalf-dozen = −i ...etc

Instance What is i 10 ?

i 10 = i 4 × i four × i ii

= 1 × 1 × −1

= −1

Decision

i = square root of -1

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

Imaginary Numbers are not "imaginary", they really exist and take many uses.