Imaginary Numbers

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

Try

Let's endeavor squaring some numbers to meet if nosotros tin become a negative consequence:

  • 2 × 2 = 4
  • (−ii) × (−2) = 4 (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 like we cannot multiply a number by itself to get a negative answer ...

thought

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

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

Well, by taking the square 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 −i.

Which is really very useful considering ...

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

Let u.s.a. take a get:

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

In general:

√(−x) = i√10

So long as nosotros proceed that fiddling "i" in that location to remind us that we however
demand to multiply by √−i nosotros are prophylactic to continue with our solution!

Using i

Example: What is (5i)2 ?

(5i)two = 5i × 5i

= five× v× i × i

= 25 × i 2

= 25 × −i

= −25

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

Imaginary numbers can help united states solve some equations:

Instance: Solve x2 + 1 = 0

Using Existent Numbers in that location is no solution, but now we can solve it!

Subtract 1 from both sides:

x2 = −1

Take the square root of both sides:

10 = ± √(−one)

x = ± i

Reply: x = −i or +i

Cheque:

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

i and j

Unit Imaginary Number

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

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

Can yous take the foursquare root of −1?
Well i can!

Merely in electronics they use j (because "i" already ways electric current, and the adjacent letter after i is j).

Examples of Imaginary Numbers

i 12.38i −i 3i/iv 0.01i πi

Imaginary Numbers are not "Imaginary"

Imaginary Numbers were in one case thought to be impossible, and and so they were called "Imaginary" (to brand fun of them).

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

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

Imaginary Numbers are Useful

complex plane vector add

Complex Numbers

Imaginary numbers become nigh useful when combined with existent numbers to make complex numbers like 3+5i or 6−4i

Spectrum Analyzer

spectrum analyzer

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

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

Information technology is part of a subject field called "Betoken Processing".

Electricity

plug
sine waves

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

When we combine ii AC currents they may not match properly, and it can be very hard to figure out the new electric current.

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

And the consequence may take "Imaginary" electric current, but it can nevertheless hurt you!

Mandelbrot Set Zoomed In

Mandelbrot Set

The cute Mandelbrot Gear up (part of information technology is pictured here) is based on Circuitous Numbers.

Quadratic Equation

Quadratic Equation

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

Also Science, Quantum mechanics and Relativity employ complex numbers.

Interesting Property

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

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

So nosotros take this:

i = √−1 i2 = −one i3 = −√−i i4 = +one
i5 = √−i i6 = −ane ...etc

Example What is i 10 ?

i x = i four × i 4 × i two

= i × 1 × −1

= −1

Conclusion

i = square root of -1

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

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