An imaginary Number,when squared, provides a negative result.You are watching: What is the square root of negative 7 |

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### Try

Let"s shot squaring some numbers to watch if we can get a negative result:

No luck! always **positive**, or zero.

It seems prefer we cannot multiply a number by chin to get a an adverse answer ...

... But Would it it is in useful, and also what might we carry out with it? |

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

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

Which is actually really useful because ...

... By merely **accepting** that** ns **exists we can solve things**that require the square source of a an unfavorable number.**

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

In general:

So lengthy as we save that small "i" over there to remind us that us still**need to multiply by √−1 we room safe to continue with ours solution!**

**Using i**

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

Imaginary number can aid us fix some equations:

### Example: settle x2 + 1 = 0

Using genuine Numbers there is no solution, however now us **can** resolve it!

Subtract 1 native 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## Unit imaginary Number

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

In mathematics the price for√(−1) is **i** because that imaginary.

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

## Examples of imaginary Numbers

## Imaginary Numbers are not "Imaginary"

Imaginary numbers were when thought to be impossible, and so lock were dubbed "Imaginary" (to make funny of them).

But then civilization researched them more and uncovered they were in reality **useful** and **important** because they filled a void in mathematics ... But the "imaginary" name has actually stuck.

And that is also how the name "Real Numbers" came around (real is no imaginary).

## Imaginary Numbers space Useful

### Complex Numbers

Imaginary numbers become most beneficial when linked with real numbers come make complicated numbers like **3+5i** or **6−4i**

### Spectrum Analyzer

Those cool screens you see as soon as music is playing? Yep, complicated Numbers are provided to calculate them! using something dubbed "Fourier Transforms".

In fact plenty of clever things have the right to be done v sound using complex Numbers, choose filtering the end sounds, listening whispers in a crowd and so on.

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

### Electricity

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

**When we combine two AC currents they may not complement properly, and also it can be an extremely hard** to number out the brand-new current.

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

And the result may have actually "Imaginary" current, however it can still hurt you!

### Mandelbrot Set

The beautiful Mandelbrot collection (part of it is pictured here) is based on complex Numbers.

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### Quadratic Equation

The Quadratic Equation, i beg your pardon has countless uses,**can give results that incorporate imaginary numbers**

**Also Science, Quantum mechanics and Relativity use complicated numbers.**

**Interesting Property**

**The Unit imaginary Number, i, has an interesting property. It "cycles" v 4 different values each time us multiply:**

1 × i | = i | |

i × i | = −1 | |

−1 × i | = −i | |

−i × i | = 1 | |

Back to 1 again! |

So we have actually this:

i = √−1 | i2 = −1 | i3 = −√−1 | i4 = +1 |

i5 = √−1 | i6 = −1 | ...etc |

### Example What is i10 ?

i10= i4 × i4 × i2

= 1 × 1 × −1

= −1

And that leads united state into one more topic, the complex plane: