In this section, we shall look at a new concept called an **imaginary number**. Consider the square root of 2. We know that this yields the non-repeating decimal

Now, what is the square root of -2? You might think that there is no solution to the square root of a negative number. However, this is not true! In fact, this is where the imaginary number comes into play. The concept of an imaginary number stems from the **imaginary unit**, denoted by the letter* i, *and is represented by the following derivation:

Thus, the square root of -2 is simply

As a matter of fact, we can add real and imaginary numbers together. This structure of numbers leads us to the idea of a complex number.

A **complex number** is an algebraic expression that includes the factor i = √-1 and is written in the form z = a + bi.

## Standard Form of Complex Numbers

The standard form of complex numbers is

where

Re (z) = a is the

**real part**of the complex number zIm (z) = b is the

**imaginary part**of the complex number z

This is also denoted by

### Real and Imaginary Numbers

There are two important subclasses of complex numbers: for a complex number z = a + bi

If Im (z) = 0, then z = a is a

**real number**If Re (z) = 0, then z = bi is said to be

**purely imaginary**

### Why are Complex Numbers Important?

Complex numbers have a range of applications. For instance, they are widely used in the field of **electrical engineering** and **quantum mechanics**. Complex numbers also help us solve polynomial equations that do not have any real solutions: have a look at Graph and Solve Quadratic Equations which explains how to do this.

We can conduct basic arithmetic operations with complex numbers such as addition, subtraction, multiplication, and division.

## Operations with Complex Numbers; Addition and Subtraction

In this section we will explain the most important operations you should be able to perform with complex numbers:

- Addition and subtraction of complex numbers
- Scalar multiplication
- Multiplication and division of complex numbers

### Addition and Subtraction of Complex Numbers

To add complex numbers, simply **add the corresponding real and imaginary parts**. The same rule applies when subtracting complex numbers.

Let z_{1} and z_{2} be two complex numbers with z_{1} = a + bi and z_{2} = c + di, where a, b, c, and d are real numbers.

#### Addition of Complex Numbers Formula

Distributing the positive sign in the second term (to both the real and imaginary parts) and collecting like terms, we obtain

#### Subtraction of Complex Numbers Formula

Distributing the negative sign in the second term (to both the real and imaginary parts) and collecting like terms, we obtain

Let α = 3 - 2i and β = 5 + 7i be two complex numbers

**Calculate ****α + β**

**Determine**** ****α - β**** **

### Scalar Multiplication of Complex Numbers

The** Scalar Multiplication of Complex Numbers** is the multiplication of a real number and a complex number. In this case, the real number is also called the **scalar. **

To multiply a complex number by a scalar, simply **multiply both the real and imaginary parts by the scalar separately**.

Let z = a + bi be a complex number and *c* be a scalar, where *a*, *b* and* c* are real numbers.

#### Scalar Multiplication of Complex Numbers Formula

Let α = 3 - 2i and β = 5 + 7i be two complex numbers

**Find 7****α**** **

In this case, we are multiplying the complex number α by the real number 7 (also called *scalar). *

**Evaluate 2****β**** **

In this case, we are multiplying the complex number β by the real number 2 (also called scalar).

## Multiplication of Complex Numbers

Multiplying complex numbers is exactly the same as the binomial expansion technique: apply the FOIL method and combine like terms.

### Multiplication of Complex Numbers Formula

This is how the FOIL method works, step-by-step.

Let z_{1} and z_{2} be two complex numbers with z_{1} = a + bi and z_{2} = c + di, where a, b, c, and d are real numbers. To multiply them

- Write both in the standard form.
- Perform the binomial expansion.
- Combine like terms.

Noting that i^{2} = -1, we obtain

Simplifying this, we get

Let α = 3 - 2i and β = 5 + 7i be two complex numbers.

**Find α x β **

## Division of Complex Numbers

If you have a fraction of complex numbers, **multiply the numerator and denominator by the complex conjugate of the denominator.**

For a complex number z = a + bi, the **complex conjugate** of z is denoted by z* = a - bi.

After that, **expand and simplify the expression to the standard form** of complex numbers. The result is given by the following formula:

### Division of Complex Numbers Formula

When dividing complex numbers be sure to write the final answer in its standard form.

Let's see in practice and step-by-step how to perform complex numbers division. Let z_{1} and z_{2} be two complex numbers with z_{1} = a + bi and z_{2} = c + di, where a, b, c, and d are real numbers. Dividing z_{1} by z_{2}, we obtain

The complex conjugate of the denominator, z_{2} is z_{2}* = c - di.

Now multiplying both the numerator and denominator by z_{2}*, we get

Expanding this expression, we obtain

Finally, combining like terms, we have

Let α = 3 - 2i and β = 5 + 7i be two complex numbers. Here, β is the denominator. The complex conjugate of β is β* = 5 - 7i.

**Calculate α ****÷**** β **

Here, β is the denominator. The complex conjugate of β is β* = 5 - 7i. Thus, multiplying the numerator and denominator by β* yields:

## Operation with Complex Numbers - Key takeaways

Operation | Formula |

Addition | |

Subtraction | |

Scalar Multiplication | |

Multiplication | |

Division |

###### Learn with 10 Operation with Complex Numbers flashcards in the free StudySmarter app

We have **14,000 flashcards** about Dynamic Landscapes.

Already have an account? Log in

##### Frequently Asked Questions about Operation with Complex Numbers

How do you do operations with complex numbers?

How to do operations with complex numbers: To conduct operations with complex numbers, we must first identify the real and imaginary parts of the complex number.

How do you divide operations with complex numbers?

How to divide complex numbers:

- Multiply the numerator and denominator by the complex conjugate
- Expand and simplify the expression
- Write the final answer in standard form as a + b
*i*

What are the mathematical operations of complex numbers?

Operations with complex numbers include addition, subtraction, multiplication and division.

How do you solve operations with complex numbers?

How to solve operations with complex numbers: To solve operations with complex numbers, we must first identify the real part and imaginary part of the complex number and then perform the given arithmetic procedure

What are the rules of complex numbers?

The rules of a complex number refer to the relationship between a given complex number say z = a + bi and its complex conjugate z* = a - bi.

##### About StudySmarter

StudySmarter is a globally recognized educational technology company, offering a holistic learning platform designed for students of all ages and educational levels. Our platform provides learning support for a wide range of subjects, including STEM, Social Sciences, and Languages and also helps students to successfully master various tests and exams worldwide, such as GCSE, A Level, SAT, ACT, Abitur, and more. We offer an extensive library of learning materials, including interactive flashcards, comprehensive textbook solutions, and detailed explanations. The cutting-edge technology and tools we provide help students create their own learning materials. StudySmarter’s content is not only expert-verified but also regularly updated to ensure accuracy and relevance.

Learn more