The term “imaginary number” describes any number that gives a negative result when square. Keeping in mind that man invented all numbers, you can also work with imaginary numbers. It is acceptable to invent new numbers as long as it works within the framework of the rules that are already in place.

Put simply, an imaginary number is the square root of a negative number and has no tangible value. Although imaginary numbers are not real numbers and you cannot quantify them on a number line, those numbers are “real”. We use them all the time in advanced math classes.

## Solve problems with imaginary numbers

For a while it was believed that you couldn’t get the square root of a negative number. This resulted from the “nonexistence” of numbers that were negative after you squared them. It was impossible to work backwards by taking the square root as every number was positive after you squared it.

So you couldn’t square root a negative number and expect to find something practical. However, you don’t have to worry about finding the square roots of negative numbers. To solve this problem, you can use a new number.

This new number was invented during the Reformation period. During that time nobody believed that you could use this number for any “real world”. It was only used to simplify the calculations for solving certain equations. As such, the new number was widely viewed as “a fake number” that was invented just for convenience.

The new number we’re talking about is called “i” and means “imaginary”. People believed that this number – the square root of a negative number – wasn’t real. You write this imaginary number as:

`i = √-1`

- So:
`i² = (√-1)² = -1`

- And:
`i² = (√-1)² = √(-1)² = √1=1`

Use the following property to find the square root of a negative number in terms of the imaginary unit “i” *a *represents a non-negative real number:

`√-a = √-1.a =√-1.√a= I√a`

With this information we can write:

`√-9 = √-1.9 = √-1.√9= i.3 =3i`

We would expect 3i squared to be equal to -9. So `(3i)² =9i² =9(-1) = -9`

. You can write the square root of any negative number in terms of the imaginary unit. Such numbers are called imaginary numbers.

## Solving imaginary numbers with radicals

Since multiplication is commutative, the imaginary numbers are equivalent and are often misinterpreted as part of the radicand. To deal with this confusion, put the imaginary number in front of the radical and solve the problem. Let’s look at the complex number `21-20i`

.

### example 1

Solve the equation `21-20i`

.

#### solution

- After defining a square root, this number satisfies the following equation:
`21-20i=x2`

- Now press x as from <{e} {m}> + {b} {i} “/> where
*a*and*b*are real numbers:`21-20i=(a+bi)²`

- Then multiply the term on the right:
`21-20i=a²+(2ab)i+(b2)i²`

- How
`i²=-1`

Rearrange the equation by definition of “i” to give:`21-20i=(a²-b²)+(2ab)i`

.

Now compare the coefficients to get two equations in since both sides of the equation have the same shape *a* and *b*. You have `a²-b²=21`

(Name this equation 1). Next, compare the imaginary parts of the equation (the coefficients of i). You have `2ab=-20`

(Name this equation 2). You now have two equations with two unknowns. You can solve the simultaneous equations for *a* and *b*.

- First can do
*b*the subject of equation 2 by dividing both sides by 2a as follows:`b=-10/a`

. - You can then replace this expression with
*b*in equation 1 as follows:`a²-(-10/a)²=21`

. - Simplify and factor this equation to get:
`(a²+4)(a²-25)=0`

- The resulting equation is disguised as a square. Therefore:
`a²=-4 or a²=25`

Think of the assumption that *a* and *b* are real numbers. `a²=-4`

has no solutions that are of interest to you. This means that your solutions are `a=5`

and `a=-5`

. Now replace each one *a* Value in your previous expression for *b*. This means that if `a=5`

, `b=-2`

and when `a=-5`

, `b=2`

. Put last *a* and *b* in the context of the question and get the solution as a `5-2i`

and `-5+2i`

.

## Solve imaginary numbers with a single radical

If you have a single radical equation, do the following:

- Isolate the radical on the left side of the equation and leave everything else on the left side of the equation.
- Square both sides of the equation.
- Get the value of the unknown.
- Substitute the value of the unknown in the original equation to check this.

### example

Solve the equation `√7x -2 = 5`

#### solution

`√7x + 4 = 7`

`7x +4 = 7²`

`7x + 4 = 49`

`7x = 45`

`x = 45/7`

You can replace now `x = 45/7`

into the original equation `√7x -2 =5`

. So: `7-2 = 5`

## Solve equations of imaginary numbers with division

To divide imaginary numbers, multiply the numerator and denominator by the complex conjugate `a - bi`

. In this case, assuming `a - bi`

is a complex number then you have:

`(a + bi) (a - bi) = a²+ b²`

### example

share

`x = (5 - 3i) / 4 + 2i`

#### solution

Multiply by the top and bottom `4 - 2i`

`(5 - 3i)(4 - 2i) / (4 + 2i)(4 - 2i)`

`(20-10i-12i+6i²) / (16 - 8i + 8i - 4i2)`

`(20 - 22i + 6i²) / 16 - 4i2`

`(14 - 22i) / 20`

## Understand the practical application of the concept of imaginary numbers

Imaginary numbers, also called complex numbers, are applicable in real life. For example, on square planes, these numbers appear in equations that don’t touch the x-axis. Imaginary numbers are especially useful in advanced calculus.

Imaginary numbers are also very important in electricity, especially in AC electronic devices. Here the alternating current alternates between positive and negative in a sine wave. Therefore, combining alternating currents can be extremely difficult. The use of imaginary currents and real numbers helped solve this problem by making it possible to perform calculations to avoid electric shock.

After all, imaginary numbers are essential in signal processing. This is especially true if what you are measuring is cosine or sine wave. Signal processing is critical in cellular and wireless technologies, as well as in radar and brain waves.

**Imaginary doesn’t mean impossible**

Originally, imaginary numbers were considered impossible to solve. However, it emerges from this discussion that they are not as complex as they seem. you *can *Actually solve problems with these types of numbers. Knowing imaginary numbers has a deep meaning and a deep meaning in understanding physics and mathematics.

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