# What are imaginary numbers?

If you are wondering what are imaginary numbers and thinking that there must be a meaning for imaginary number, Then lets get into the article to learn about what exactly are imaginary numbers.

**Imaginary numbers :**

The imaginary numbers are the numbers which, when squared, give a negative number. Imaginary numbers are the square roots of negative numbers where they do not have any definite value.

The imaginary numbers are represented as the product of a real number and the imaginary value** i**.

**Example –**

3i, 5i, 25i are some examples of imaginary numbers.

The value of i^{2} is given as -1.

So the value of (5i)^{2} is -25, and that implies is **i.**

**Complex number :**

Complex numbers are a combination of real numbers and imaginary numbers.

The complex number is of the standard form – a + bi.

Where a and b are real numbers. i is an imaginary unit.

Now let’s get a quick understanding by using some examples:

**Real numbers –**

-1,2, 10, 10000.**Imaginary numbers –**

4i, -5i, 2400i.**Complex numbers –**

2+3i, -5-4i.

**Conjugate pair of an imaginary number :**

- a+bi is a complex number and the conjugate pair of
**a +bi**is**a-bi**. - When an imaginary number is multiplied by its conjugate pair, then the result will be a real number.

**Rules of imaginary numbers :**

**Arithmetic Operations on Imaginary Numbers :****1. Addition –**

- When two imaginary numbers are added, then the real part is added into one , and then the imaginary part is added into one.

**Examples –**

**1.** (2 + 2i) + (3 + 4i)

= (2 + 3) + (2 + 4)i

= (5 + 6i)

**2.** (3 + 4i) + (5 + 3i)

= (8) + (7)i

= 8 + 7i

**3.** (5 + 3i) + (4 + 2i)

= (5 + 4) + (3 + 2)i

= 9 + 5i.

**2. Subtraction –**

- When two imaginary numbers are subtracted, then the real part is subtracted, and then the imaginary part is subtracted.

**Examples –**

**1.** (2 + 2i) – (3 + 4i)

= (2 – 3) + (2 – 4)i

= (-1 – 2i)

**2.** (3 + 4i) – (5 + 3i)

= (-2) + (1)i

= -2 + i

**3.** (5 + 3i) – (4 + 2i)

= (5 – 4) + (3 – 2)i

= 1 + i.

**3. Multiplication**

- When two imaginary numbers are multiplied, then the result will be as follows –

**Examples –**

**1.** (a + bi) (c + di)

= (a + bi)c + (a + bi)di

= ac + bci + adi+

= (ac – bd)+i(bc + ad)

** 2. **(3 + 4i)*(3 – 4i)

= (9 + 12i -12i – 16)

= 25

** 3. **(2 + 2i) * (3 – 4i)

= (6 + 6i -8i – 8)

= (14 – 2i)

**4) Division**

The numerator and denominator will be multiplied by its conjugate pair of denominators.

**Examples –**

**1. **(2 + 2i) / (3 + 4i)

Multiplying the numerator and denominator with the conjugate pair of denominators.

=(2 + 2i) * (3 – 4i) / (3 + 4i)*(3 – 4i)

=(6 +6i -8i – 8) / (9 + 12i -12i – 16)

=(14 – 2i) / 25

**2.** (3 + 4i) / (2 + 2i)

Multiplying the numerator and denominator with the conjugate pair of denominators.

= ((3 + 4i) * (2 – 2i)) / ((2 + 2i) * (2 – 2i))

=(6 + 2i – 8) / (4-4)

=(14 + 2i) / (8)

=(7 + i) / 4

**3.** (2 + 2i) / (2 + 2i)

Multiplying the numerator and denominator with the conjugate pair of denominators.

= ((2 + 2i) * (2 – 2i)) / ((2 + 2i) * (2 – 2i))

= (4 – 4) / (4 – 4)

= 8 / 8

= 1

**Where do we use imaginary numbers :**

- Imaginary numbers are very useful in various mathematical proofs.
- Imaginary numbers are used to represent waves.
- Imaginary numbers show up in equations that donâ€™t touch the x axis.
- Imaginary numbers are very useful in advanced calculus.
- Combining AC currents is very difficult as they may not match properly on the waves.
- Using imaginary currents helps in making the calculations easy.