Imaginary Numbers

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

Try

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

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

No luck! Always positive, or zero.

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

thought

... merely imagine that there is such a number (telephone call it i for imaginary) that could do this:

Would it exist useful, and what could we do with it?

Well, by taking the foursquare root of both sides nosotros get this:

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

Which is really very useful because ...

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

Permit us have a go:

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

In general:

√(−x) = i√x

So long as we keep that petty "i" at that place to remind us that we still
need to multiply by √−1 we are rubber to go along with our solution!

Using i

Example: What is (5i)ii ?

(5i)2 = vi × 5i

= 5× five× i × i

= 25 × i 2

= 25 × −one

= −25

Interesting! We used an imaginary number (5i) and ended upwards with a real solution (−25).

Imaginary numbers tin can assist us solve some equations:

Example: Solve tenii + ane = 0

Using Existent Numbers at that place is no solution, merely now we can solve information technology!

Subtract 1 from both sides:

102 = −i

Accept the square root of both sides:

10 = ± √(−1)

10 = ± i

Respond: ten = −i or +i

Cheque:

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

i and j

Unit Imaginary Number

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

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

Can you lot take the square root of −ane?
Well i can!

Just in electronics they use j (considering "i" already means current, and the next letter after i is j).

Examples of Imaginary Numbers

i 12.38i −i 3i/four 0.01i πi

Imaginary Numbers are not "Imaginary"

Imaginary Numbers were once thought to be incommunicable, and then they were chosen "Imaginary" (to brand fun of them).

Merely then people researched them more and discovered they were actually useful and important because they filled a gap in mathematics ... but the "imaginary" proper noun has stuck.

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

Imaginary Numbers are Useful

complex plane vector add

Complex Numbers

Imaginary numbers get most useful when combined with real numbers to make complex numbers like 3+5i or six−4i

Spectrum Analyzer

spectrum analyzer

Those cool displays yous see when music is playing? Aye, Complex Numbers are used to summate them! Using something chosen "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 "Indicate Processing".

Electricity

plug
sine waves

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

When nosotros combine 2 Ac currents they may not match properly, and it tin be very hard to figure out the new electric current.

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

And the effect may accept "Imaginary" current, but it can nonetheless hurt you!

Mandelbrot Set Zoomed In

Mandelbrot Set

The cute Mandelbrot Set (office of it is pictured hither) is based on Complex Numbers.

Quadratic Equation

Quadratic Equation

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

Besides Science, Quantum mechanics and Relativity utilise circuitous numbers.

Interesting Property

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

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

Then we take this:

i = √−ane i2 = −ane i3 = −√−1 ifour = +1
i5 = √−1 i6 = −1 ...etc

Instance What is i 10 ?

i 10 = i 4 × i 4 × i 2

= 1 × 1 × −1

= −one

Determination

i = square root of -1

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

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