The history of Number systems is quite intriguing. The most popular type of the Number System was also known as the Hindu-Arabic Number System which was developed by *Aryabhat *in the 5th century BC. This set has been used by many ancient civilisations for practical purposes.

Talking about the modern age, we just cannot* imagine* our lives without numbers, considering the fact that computers only understand the language of numbers. This means that the modern technology surrounding us is entirely based on the “Number System.”

*Are you ready to dive into the world of the Number system?*

*Let’s get started!*

## What is a Number System?

The technique to represent numbers is called the **Number system**. This is a set of values used to represent different quantities. For example, a number system is used to represent the number of audience in a movie hall or the number of people standing in a queue to collect tickets. The base or radix of a number system is the total number of digits used in it.

Computers represent all kinds of information and data from audio, images, videos, etc. in binary numbers.

Our general mode of communication with each other is made of letters or words. We send our message or information through the keyboard by typing letters or words; But computers don’t have brains like us. So how do they understand us?

Thanks to the compiler that makes computers understand our language. Compilers change the human language to numbers. Thus computers understand only numbers.

In Computer System Architecture there is a unique technique of representing numbers called Number System. The number system supported by Computer architecture is shown in the image below. We need to study them and understand the conversion technique between them.

The important number systems used by computers are:

We use the decimal number system in general. However, the computers understand the binary number system. The octal and hexadecimal number systems are also used now-a-days by computers.

**Decimal number system:**

- The decimal number system uses the following ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

- Hence, the numbers in this system have a base of 10.

- If we come across a number without a base, it means that its base is 10.

- This is the system that use in order to represent numbers in real life.

Examples: 18_{10}, 222_{10}, 7877_{10}

Applications: We use decimals everyday while dealing with money, weight, length etc. Whole numbers provide less precision, therefore decimal numbers are used where more precision is required.

**Binary number system**

- Binary number system uses just two digits: 0 and 1.

- Hence, the numbers in this system have a base 2.

- 0 and 1 are called bits.

- 8 bits together make a byte.

- The data in computers is stored in the form of bits and bytes.

Examples: 1011_{2}, 111_{2, }etc.

Application: In modern technology, the binary number system is an indispensable part of computer science. Every computer language (like C, C++, java, python) and programs are based on the binary **number system**. It is also used in digital encoding ( the process of representing data as discrete bits of information).

**Octal number system: **

- The octal number system uses the following eight digits: 0, 1, 2, 3, 4, 5, 6, and 7.

- Hence, the numbers in this system have a base 8.

- The advantage of this system is that it has less number of digits and therefore, there would not be many computational errors.

Examples: 245_{8}, 51_{8} etc.

Application**:** The Octal number system was used widely within IT similar to how hexadecimal is used today. However, there are some places where octal is still widely utilised such as file permissions within UNIX (the source code for Linux, mac OS and android along with other operating systems). It is also used within digital displays, which does not support symbols.

**Hexadecimal number system: **

- The hexadecimal number system uses the following sixteen digits/alphabets: The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 and following are the alphabets A, B, C, D, E, and F.

- Therefore, the numbers in this system have a base of 16.

- Here, A−F of the hexadecimal system indicate the numbers 10−15 of the decimal number system respectively.

- This system is widely used in computers in order to reduce the large-sized strings of the binary system.

Examples: A7B_{16}, BC_{16} etc.

**Applications:** A computer can understand number systems that use only a few symbols called digits. These symbols denote different values corresponding to the position they occupy in the number. Computers are usually designed to process hexadecimal numbers.

## Steps for the conversion of number system:

**How to convert a number from binary/octal/hexadecimal system to decimal system?**

Multiply each digit of the given number with the exponents of the base, starting from the rightmost digit. Make sure that the exponents start from 0 and increase by 1 every time. At last add the above products.

**Example 1:**

Convert **1000010**_{2}_{ }to the decimal system.

**Solution:**

Starting from the rightmost digit, we will multiply each digit with the corresponding powers of 2. Let’s start.

020=0, 121=2, 022=0, 023=0, 024=0, 025=0, 126=64

Now, upon adding all the numbers, we get 0+2+0+0+0+0+64 = 66

Pictorially,

Thus,

**1000010**_{2}** = 66**_{10}** **

**Example 2:**

Convert ABC_{16 }into the decimal system system.

**Solution:**

In the Hexadecimal System, remainders greater than 9 are represented by alphabets.

E.g. A = 10, B = 11, C = 12, D = 13, E = 14, F = 15

To convert a hexadecimal number into binary, first convert it into decimal and then from decimal into binary.

Conversion of ABC_{16} to decimal: Multiply each digit with corresponding powers of 16, starting from the rightmost digit.

C160=12, B161=1116=176, A162=10256=2560

Adding all the numbers, we get 12+176+2560 = 2748

Pictorially,

So, ABC_{16} = 2748_{10}

### How to convert a number from **decimal system to binary/octal/hexadecimal system**?

First divide the given number by the base of the required number. Then note down the quotient and the remainder in the “quotient-remainder” format repetitively. It is shown in an example given below. Then stop the process until we get the quotient to be less than the base. The new number in the required number system is obtained by reading all the remainders and the last quotient from the bottom to top.

**Example 1:**

Let us convert **700**** _{10}** into the hexadecimal system using the above-mentioned process.

**Solution:**

Thus,

**700**_{10 }**= 2BC**_{16 }

**Example 2:**

Convert **300**** _{10}** into the binary system (base-2).

**Solution:**

300_{10} is in the decimal system.

We will divide 300 by 2 and note down the quotient and the remainder.

We will repeat this process for every quotient until we get a quotient which is less than 2.

The equivalent number in the binary system can simply be obtained by reading all the remainders and just the last quotient from bottom to top as shown in the above figure.

Thus,

**300**_{10 }**= 100101100**_{2}

## I hope you had a great time exploring the Number System.The number system plays an important role in the functioning of a computer. The base conversion helps both the computer and the user to understand the data and information.

You can learn all about the Number System on Cuemath by just clicking here. Cuemath is one of the leading Math platforms which makes Math super easy and fun! Because at Cuemath, students can learn at their own pace through various learning tools like Numeric puzzles, worksheets, tab exercises, puzzles, math boxes, and much more. And in case you want to build some* math muscles, ***Cuemath **also has a *math gym! *

If you have any queries related to this topic, don’t forget to comment down below, we will shortly get back to you. Thank you for spending your valuable time here. I hope it was a good read.