# Complex Number|Definition & Meaning

**Definition**

In mathematics, a** complex number** is a part of a **number system** and is a **sum of** **real** and **imaginary** number parts of a number system. Every complex number may be written as **a + bi, **where **a** is the real part and** b** is the imaginary part.

**Graphing a Complex Number**

A complex number’s real and imaginary components, **Re(z)** and **Im(z)**, can be written as a pair of **ordered coordinates** on the **euclidean plane**. The c**omplex plane**, sometimes known as the **Argand Plane** after its discoverer Jean-Robert Argand, is the euclidean plane in the context of complex numbers. The **natural component a**, concerning the** x-axis**, and the **imaginary component ib**, concerning the** y-axis**, are used to depict the complex number **z = a + ib**. Let us make an effort to define the two key phrases associated with the argand plane’s representation of complex numbers: the **modulus** and **argument** of the complex number.

**Modulus of Complex Number**

In the Argand plane, a complex number is represented by the coordinates (a, ib), and its **distance from the origin** is known as its** modulus**. It is the **straight-line distance** between the starting point **(0, 0i)** and the given coordinates** (a, ib)**.

Furthermore, this can be interpreted as a derivation of the **Pythagorean theorem**, with the modulus standing in for the hypotenuse, the fundamental component for the base, and the imaginary portion for the apex of the right triangle. In other words, the** real part is squared** **along with the complex part**, and the entire bracket is **square-rooted**. The answer to the problem gives the **distance from the origin**, or the modulus.

**The Argument of a Complex Number**

Complex numbers have** arguments** that describe the **anticlockwise angle** formed by the line between the complex number’s **geometric representation** of the origin and the **positive x-axis**. We **calculate **it by **dividing the imaginary component** **by the real component** of the complex number.

**Cartesian Complex Plane**

The usage of **Cartesian coordinates** on the complex plane is **implicit in the definition of complex numbers**, which requires using two arbitrary absolute values. The **real part** is typically displayed with rising values to the right along the **horizontal (real) axis**, often just called the **x-axis**. In contrast, the **imaginary part** is typically displayed along the **vertical (imaginary)** **axis** or the **y-axis**, with increasing values to the top.

Coordinated point or position vector from the origin to the numbered location on the chart. A complex number **z** can have **Cartesian, rectangular, or algebraic representations** for its coordinate values.

In this context, **multiplying** **a complex number** by $i$ is the same as **turning the position vector** 90 degrees to the left of the x-axis.

**Properties of a Complex Number**

The following **complex number properties** will aid in your understanding of complex numbers and your ability to** do arithmetic** on them.

**Conjugate of a Complex Number**

The **conjugate of a complex number** is simply the original number with the imaginary component **replaced** by its** additive inverse**. The real component **remains the same** in the conjugate.

Two complex numbers are **conjugates of each other** if both their** sum** and **product** give a **real number**. Often, we use the conjugate by **multiplying and dividing** it with the provided equation to reduce the problem into a **simpler sum**.

**Reciprocal of a Complex Number**

If **z = a + bi** is a complex number, the **reciprocal is** (**1 / z) = 1 / (a + bi)**. Evidently, **multiplying** a complex number by its **reciprocal** gives the **complex unity 1 + 1i**. When dividing by another complex number, the reciprocal is a helpful tool.

**Ordering of a Complex Number**

**Ordering** is not possible with complex numbers, even though it is possible in real numbers and related number systems. It is **not possible** to rank a complex number since they l**ack the organization of a field**, and they cannot be sorted in any way that allows for addition or multiplication. A complex number’s** magnitude**, or how far it is from zero (the modulus), can be used to **express it** in a **two-dimensional Argand plane**.

**Operations on Complex Numbers**

Complex numbers are** amenable** to the same arithmetic operations as their natural-number counterparts, including **addition**, **subtraction**, **multiplication**, and **division**. The different arithmetic operations performed on complex numbers are described here. The addition and conjugate concept of the complex number is explained in the examples at the end.

**Addition of Complex Numbers**

Adding a natural number to a complex number requires adding the natural number to the real part of the complex number.

We add complex numbers the same way as we add natural numbers, though with a **small difference**. Since complex numbers contain real and imaginary parts, we need to ensure we add **compatible parts** separately (i.e., **real + real** and **imaginary + imaginary**). Thus, if **x = a + bi**, and **y = c + di**, then **z = x + y = (a + c) + (b + d)i.**

Clearly, if the second complex number is purely real or purely imaginary, only the real or imaginary part is summed in the result.

There are **5 laws** for the** additive property** of a complex number:

- Closure Law
- Commutative Law
- Associative Law
- Additive Identity
- Additive Inverse

**Subtraction of Complex Numbers**

The method of subtracting complex numbers from natural numbers is like the process of **subtracting natural numbers**, again keeping the compatibility of real and imaginary parts in mind. In this case, **subtracting any two complex numbers** involves performing the operation **across the real part of each number** and then **repeating** the procedure across the **imaginary part of each number**.

**Things To Remember**

These** two points** should be kept in mind while solving a problem with complex numbers:

**Real numbers**are essentially**complex numbers**with**purely real parts**. However, every complex number is**not necessarily real**, as it can be purely imaginary or a mixture of real and imaginary.- The
**distinction**between complex numbers and imaginary numbers is that**complex numbers can****exist without****becoming imaginary numbers**.

## Solved Examples of Complex Number Exercises

### Example 1

What is the final solution after multiplying (2 + 3i) (2 – 6i)?

### Solution

It should be noted that the real number of the problem is added or multiplied with the real part only; whereas, the imaginary number is added or multiplied with the complex region only. Therefore, the solution is as follow:

= 4 – 12i + 6i – 18i^2

= 4 – 6i + 18

**= 22 – 6i**

If plotted,** “22”** will be on the **real axis** or the x-axis, and the** “-6”** will be on the **imaginary axis** or the y-axis.

### Example 2

Simplify the following expression using the complex conjugate:

(7 – i)/(2 + 10i)

### Solution

Here the denominator’s conjugate is multiplied and divided by the problem given above.

= [(7 – i)/(2 + 10i)] x [(2 – 10i)/(2 – 10i)]

= (14 – 72i + 10i²)/(4 – 100i²)

= (4 – 72i)/104

= (4/104) – (72/104)i

Simplified Version:

**= (1/26) – (9/13)i**

*All images/mathematical drawings were created with GeoGebra.*