# Complex number

A number that contains a real number and an imaginary number in either addition or subtraction form is called a complex number.

## Introduction

A complex number is a quantity and it is formed by the combination of a real number and an imaginary number. They are interconnected by either a plus sign $(+)$ or a minus $(-)$ sign to form a complex numbers mathematically.

For example, $a$ and $b$ are literals but represents real numbers. Take the imaginary number as the product of imaginary unit and $b$. A complex number is usually represented by either $a+bi$ or $a-bi$ in mathematics.

### Examples

Complex numbers are formed in three different ways possibly.

01

#### Complex numbers with Real and Imaginary Parts

A complex number is usually have both real and imaginary parts. A real number represents the real part of the complex number and the product of a real number and an imaginary unit represents the imaginary part of the complex numbers.

##### Examples

$(1) \,\,\,\,\,\,$ $3+5i$

$(2) \,\,\,\,\,\,$ $-7+4i$

$(3) \,\,\,\,\,\,$ $-9-\dfrac{4}{3}i$

$(4) \,\,\,\,\,\,$ $0.75+3i$

$(5) \,\,\,\,\,\,$ $8-\sqrt{5}i$

02

### Complex numbers without real part

The real part of a complex number can be zero because zero is also a real number. Therefore, a complex number can only have imaginary quantity in some cases. In these types of cases, you have to assume that the real part of the complex number is zero. For example, $0+7i = 7i$

##### Examples

$(1) \,\,\,\,\,\,$ $2i$

$(2) \,\,\,\,\,\,$ $-5i$

$(3) \,\,\,\,\,\,$ $\dfrac{1}{7}i$

$(4) \,\,\,\,\,\,$ $-\sqrt[4]{3}i$

$(5) \,\,\,\,\,\,$ $80.9i$

03

### Complex numbers without imaginary part

Similarly, the imaginary part of a complex number can also be zero. In fact, an imaginary quantity is a product of imaginary unit and a real number. It is possible, an imaginary number is formed by the product of imaginary unit and zero. For example, $4-0i = 4$. Hence, every real number can be imagined as a complex number.

##### Examples

$(1) \,\,\,\,\,\,$ $\sqrt{6}$

$(2) \,\,\,\,\,\,$ $\dfrac{8}{9}$

$(3) \,\,\,\,\,\,$ $-0.17$

$(4) \,\,\,\,\,\,$ $3$

$(5) \,\,\,\,\,\,$ $-9$