An imaginary Number,when squared, gives a negative result.

You are watching: Is the square root of a negative number a real number

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Let"s shot squaring part numbers to check out if we can obtain a an unfavorable result:


No luck! always positive, or zero.

It seems favor we can not multiply a number by chin to gain a an adverse answer ...

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... Yet imagine the there is such a number (call it ns for imaginary) that might do this:


Would it it is in useful, and what can we perform with it?

Well, by taking the square source of both sides we get this:

Which method that i is the answer to the square root of −1.

Which is actually really useful due to the fact that ...


... By merely accepting that ns exists we deserve to solve thingsthat need the square source of a an adverse number.


Hey! the was interesting! The square source of −9 is merely the square source of +9, times i.

In general:


So long as we save that small "i" there to remind united state that we stillneed to multiply by √−1 we space safe to proceed with our solution!

Using i


Interesting! We provided an imagine number (5i) and also ended up with a actual solution (−25).

Imaginary number can assist us fix some equations:


Example: fix x2 + 1 = 0

Using actual Numbers over there is no solution, however now we can deal with it!

Subtract 1 from both sides:


Answer: x = −i or +i

Check:

(−i)2 + 1 = (−i)(−i) + 1 = +i2 + 1 = −1 + 1 = 0(+i)2 +1 = (+i)(+i) +1 = +i2 +1 = −1 + 1 = 0

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Unit imaginary Number

The square root of minus one √(−1) is the "unit" imagine Number, the identical of 1 for actual Numbers.

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

Can you take the square source of −1?Well i can!

But in electronic devices they use j (because "i" already means current, and also the following letter after ns is j).

Examples of imaginary Numbers


Imaginary Numbers space not "Imaginary"

Imaginary number were when thought to be impossible, and also so lock were dubbed "Imaginary" (to make fun of them).

But then civilization researched them more and found they were in reality useful and also important due to the fact that they fill a gap in math ... However the "imaginary" name has stuck.

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

Imaginary Numbers room Useful

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Complex Numbers

Imaginary numbers come to be most useful when merged with genuine numbers come make facility numbers prefer 3+5i or 6−4i


Spectrum Analyzer

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Those cool displays you see when music is playing? Yep, complicated Numbers are supplied to calculation them! using something dubbed "Fourier Transforms".

In fact plenty of clever things deserve to be done with sound using complicated Numbers, choose filtering the end sounds, hearing whispers in a crowd and also so on.

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

Electricity

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AC (Alternating Current) electricity changes between positive and an adverse in a sine wave.

When we combine two AC currents they may not enhance properly, and also it deserve to be an extremely hard to figure out the new current.

But using complex numbers provides it a lot simpler to perform the calculations.

And the an outcome may have "Imaginary" current, however it have the right to still ache you!

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Mandelbrot Set

The beautiful Mandelbrot set (part of that is pictured here) is based on complex Numbers.

See more: Which Is An Acid-Conjugate Base Pair? ? + Example What Is A Conjugate Acid And Base Pair


Quadratic Equation

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The Quadratic Equation, which has plenty of uses,can offer results that include imaginary numbers

Also Science, Quantum mechanics and also Relativity use facility numbers.

Interesting Property

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

1 × i= i
i × i= −1
−1 × i= −i
i × i= 1
Back to 1 again!
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So we have actually this:


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

Example What is i10 ?


i10= i4 × i4 × i2
= 1 × 1 × −1
= −1

And the leads us into an additional topic, the complex plane: