**Introduction to Number System**

The world has witnessed a storm of change in the years of technology. Various subjects and mechanism concepts form up technology. And Mathematics plays an essential role in the growth of technology and science. The term Mathematics is outlined as the methodological implementation of matter. The various parts of mathematics assist us in making our lives systematic. Mathematics is not only vital in the field of science but is also crucial in computer science fields.

According to the Indian System of Numeration, the Number System, one of the vital parts of computer science, mainly deals with Mathematics. There are many types of number system that deals with different numerical values. The number system represents the order of a specific series of numbers. In simple words, Number System can be referred to as the Mathematical representation of a given number. Some of the essential types of number systems are – Binary, Decimal, Hexadecimal, Octal, etc.

The great Aryabhata in ancient age discovered the technique of numerals in the Hindu-Arabic Number system. Therefore, he is popularly known as the father of the Number System. This article will explore the possibilities of the Number System and try to develop a deep understanding of the topic.

**Definition of Number System in Mathematics **

The Number System can be highlighted as a writing system or a mathematical notation to express some numbers of a given set, in accordance with the Indian system of numeration. In different Number Systems, the identical system of numbers may define other numbers. For instance, 11, in the decimal system, represents a number, while in the binary system, it presents number three, and in the unary numeral system, it represents number two.

The properties of a Number System –

- Number System is used to depict a helpful set of numbers.
- Provides a standard representation to every number present in the Number System.
- Portrays the Arithmetic and Algebraic Structure of the numbers.

The Number System is currently based on the place-value systems, i.e., it depends upon positional characteristics. Therefore the value of the numbers is depicted by their position of the numbers in the respective representation. Let's make it clear with an example; the 4 in 40 and 4 in 400 represent 4 ten's and four hundred, respectively. This system appears in contrast to the ancient Number System in Egyptian civilization, Greek or Hebrew Number System, which were not position-based, making them more diplomatic to understand.

The field of computers and computer science implements two main number of systems. Specifically, the Binary Number System, which represents the numbers in the form of 0's and 1's, and the Hexadecimal Number System, which includes the use of 16 symbols – 0,1,2,3,4,5,6,7,8,9, A, B, C, D, E, F. The roman numbers one to hundred and so on are also a part of number systems. Therefore, we can use 1 to 100 roman numerals and more in the number system.

You can also grab the whole number system in Hindi number system Maths on the web for free.

**Types of Number System **

The Number System is a specific method to represent a few numbers of a given set by their positional values. The Number System is divided into mainly four types –

Decimal Number System (Base 10) – also known as the denary or decanary number system. It is one of the standard ways to represent integers and non-integers in a specific sequence. In the Decimal Number System, denoting the numbers is also referred to as decimal notation.

Binary Number System (Base 2) – also known as the base-2 number system. This number system typically uses two types of symbols to denote specific numbers i.e., 0's and 1's. The binary number system refers to the positional notion with a radix of 2. Every digit in this system is highlighted as a bit.

Octal Number System (Base 8) – also known as the base-8 number system. It implements numbers from 0 to 7.We can quickly form Octal numerals by grouping binary numerals into three (beginning from the right).

Hexadecimal Number System (Base 16) – also known as the base-16 number system. It is a positional number system arranging the numbers applying the base or radix of 16. The Hexadecimal usually uses 16 digits to represent the numbers, ranging from 0 to 9 and from A to F.

**Decimal Number System**

The Decimal Number System is also quoted as the denary number system. It is one of the standard mechanisms of representing the integers and non-integers. It's a Hindu-Arabic Number System, in which the way of expressing the numbers is popularly known as decimal notation. It is generally highlighted as a denotation of numbers in the decimal numeral system.

The decimal number system includes a base of 10 as it implements ten digits, from 0 to 9. In the decimal number system, the number specified on the left of the decimal point represents units, tens, hundreds, thousands, and so on.

Each position of the numbers presents a specific power to the base of 10. For instance, the decimal number 5687 includes seven at the unit's place, eight at the ten's place, six at the hundred's place, and five at the thousand's place. The value of the whole number is as follows –

(5 x 10³) + (6 x 10²) + (8 x 10¹) + (7 x 1)

= (5 x 1000) + (6 x 100) + (8 x 10) + (7 x 1)

= 5000 + 600 + 80 + 7

= 5687

The infinite decimal concept is also introduced in the number system, which defines a decimal value repeating infinite times.

**Binary Number System**

The modern Binary number system mechanism was developed in the 16th and 17th centuries by efficient scientists. However, the idea was inspired by the system of binary numbers highlighted earlier in the ancient era of Egypt, India, and China. In mathematics and electronics, a binary number can be portrayed as a number represented in the base-2 number system.

Therefore, it can also be defined as positional notation using a radix of 2. Each digit of the numbers in the binary system is called a bit. The binary number system presents an easy and convenient mechanism for the users. Therefore it is implemented by almost all modern computer-based devices in today's date. The binary number system generally uses two binary digits to depict the numbers' positioning, i.e., 0's and 1's. For instance, 110101 is a binary number.

A number in Binary Number System can be converted into any other Number System according to the users' requirements. The binary number system is one of the most intensely used number systems across different computer-based devices today.

**Octal Number System**

The octal number system implements digits from 0 to 7, and therefore is a base-8 number system. We can create octal numbers from binary numerals by arranging consecutive binary digits in a group of three digits. For instance, we can represent a decimal 74 in binary format as 1001010.

Octal numbers possess immense importance in computer applications. The octal number is one of the most used number systems in today's modern world of technology and computers. We can easily convert octal number to decimal number.

**Hexadecimal Number System**

The Hexadecimal Number System or the Hex Number System is a positional number with a radix of 16. It is one of the most used number systems for long binary values. It presents a conveniently compact and easily understandable format compared to the long binary strings with 0's and 1's.

The hexadecimal number system uses sixteen different digits to position the numbers by various combinations. So, there are sixteen possible digits of the hexadecimal number system. In the hexadecimal number system, the numbers are depicted using digits from 0 to 9 and alphabets from A to F.

Although binary strings can be diplomatic and long, we can make it easier by dismantling these long binary numbers into smaller groups to make it convenient to understand and implement. For instance, the group of binary digits 1101, 0101, 1100, and 11112 are much suitable to comprehend compared to 11010101110011112 when they are all grouped.

Hexadecimal is one of the more fantastic sides of complexity while working with computers and memory address locations compared to binary or decimal number systems.

The numbers ranging from 0 to 10 are the inhabitants of the original decimal system. But we need the hexadecimal number system to represent the number from 10 to 15, using the capital alphabet from A to F.

**Number System Chart**

The Number System Chart can be highlighted as a chart that reflects the base value of different number systems –

Decimal Number System (Base 10) – also known as the denary or decanary number system. It represents integers and non-integers in a specific sequence. In the Decimal Number System, the numbers are denoted using a base value of 10.

Binary Number System (Base 2) – also quoted as the base-2 number system. This number system typically uses two types of symbols to denote specific numbers i.e., 0's and 1's. The binary number system highlights the positional notion with a radix or base of 2.

He uses numbers from 0 to 7. The octal number system defines the positional notation of the number implementing the base or radix of 8.

Hexadecimal Number System (Base 16) – also called the base-16 number system. It is a positional notation of numbers applying the base or radix of 16. The Hexadecimal generally implements 16 digits representing the numbers, ranging from 0 to 9 and from A to F.

Thus, the Number System Chart plays a significant role in portraying the base of all the actual number of systems.

**Number System Conversion**

The Number Systems includes different methods that allow the conversion of one number system into another.

**Binary to Decimal Number System**

The process for converting binary to decimal is by merely multiplying the binary number's bits with respective position weights. And then, we must add all those products. Therefore now we obtain the result after concerning binary to decimal.

**Binary to Octal **

The conversion of Binary into Octal involves two steps –

First, we must make a pair including three bits on both sides of the binary points. If we find one or two bits are left in a three-bit pair, the required number of zeros can be added to the extreme corners.

Next, we must collect and write all octal digits.

**Binary to Hexadecimal**

The conversion of binary to Hexadecimal involves two steps, which are as follows:

- First, we must create pairs of four bits on both sides of the binary point. If we find one, two, or three bits are left in a couple of four bits pairs, zeros can be added as required on the extreme ends.
- Now, we must write the hexadecimal digits respectively to each pair.

**Decimal to Binary Conversion**

We can convert decimal to binary, following two steps, which are as follows:

- Firstly, we need to perform the division on the integer and the respective quotient based on binary (2).
- We must now do the multiplication on the integer and the individual quotient having the base of binary (2).

**Decimal to Octal Conversion**

The procedure to convert octal into decimal is mostly the same as binary conversion into decimal. The only point of difference is that you must perform the division and multiplication operation with the quotient with base 8.

**Decimal to Hexadecimal Conversion **

The procedure to convert octal into decimal is mostly the same as binary conversion into decimal. The only point of difference is that you must perform the division and multiplication operation with the quotient with base 16.

You can use the vice-versa method to convert octal into other number systems or convert Hexadecimal into other number systems.

You can also convert Octal to Hexadecimal by the following methods –

- To convert octal to Hexadecimal, first, we must reverse the octal to binary equivalent.
- Then, to get the final octal to the decimal result, we must convert the binary number to a hexadecimal number.

**Number System in Computer**

The Number System is not only the king of mathematics, but it also plays a crucial role in the field of computers. The computer, although being a brilliant machine, can't understand the human language. Therefore, whenever we enter something in the computer, it first translates it into a number for the sake of understanding. The computer is only familiar with the positional number system, including only a few symbols quoted as digits. These symbols depict different values according to the positions it occupies in the computer memory. Therefore, we can determine the value or position of each digit in a number using the following ways –

- Collecting Information about the digit
- Searching for the position of the digit in a given number
- Knowing the number system (here, the bottom of the number system resembles the number of digits that the respective system works with)

There are four different systems widely used: Decimal Number System, Binary Number System, Octal Number System, and Hexadecimal Number System. Therefore the primary mechanism to represent and work with numbers can be referred to as a number system. The binary number system is one of the most widely used number systems, following the decimal number system in technology and computers.

**Frequently Asked Questions **

**Q1: Define Number Systems. And mention it types?**

Ans - The Number System can be outlined as a writing mechanism or a mathematical notation to portray some numbers of a given set in an efficient matter. In various Number Systems, an identical system of numbers may define different numbers. For instance, 11, in the decimal system, represents a number, while in the binary system, it presents number three, and in the unary numeral system, it represents number two.

The four types of Number systems are – Decimal Number System (with base value 10, Binary Number System (with base value 2), Octal Number System (with base value 8), Hexadecimal Number System(with base value 16).

**Q2: Why is the Number System essential?**

Ans – The number System is one the vital method to denote numbers. It is a collection of values used to indicate specific quantities. Let's take an example, and the number system can be used to represent the number of students in a school or the number of students standing in the assembly line. The radix or the base number depicts the number of digits used in the Number System.

The base number or radix can be different depending upon the four types of Number systems are – Decimal Number System(with base value 10, Binary Number System(with base value 2), Octal Number System(with base value 8), Hexadecimal Number System(with base value 16).

**Q3: What is the Base number, and what is base number 1, Number System Called?**

Ans – The base number or radix is the number that portrays how many digits can be worked upon by a Number System. For example, binary has a radix or base of 2, decimal has a base of 10, octal decimal has a base of 8, and Hexadecimal has a base of 16.

Number System with base 1, can be quoted as Unary System, the simplest and the most convenient number system to denote natural numbers. In a unary number system, we indicate zero by an empty string, with a symbol's absence. For example, the numbers of 2, 4, 5 will be represented in the unary number system as 11,1111,11111.

**Q4: What is a Binary Number System? Why is it important?**

Ans – A binary number can be defined as a number represented in the base-2 number system. Therefore, it can also be outlined as positional notation using a radix of 2. Each digit of the numbers in the binary system is known a bit. The binary number system presents an easy and convenient mechanism for the users. Therefore it is implemented by almost all modern computer-based devices in today's date. The binary number system generally utilizes two binary digits to depict the numbers' positioning, i.e., 0's and 1's. For instance, 101110 is a binary number. The importance of the binary number has touched new heights in the modern world of technology.

**Q5: How to convert the Octal Number System to Binary Number System?**

Ans – Octal are the numbers with radix or base 8. In contrast, binary is the number with radix 2.

The conversion of octal to decimal is a two-step procedure. First, we need to convert the octal to the decimal number system. Then, we need to convert decimal to the binary number system.

**Convert octal to decimal – **

- Step 1 - Identify the number of digits in the given number. Suppose the number of digits to be 'n'.
- Step 2 – Then, we must multiply each digit of the number with 8n-1 when the digit comes to the nth position from the number's right corner. And if the number contains a decimal part, we need to multiply each digit in the decimal part by `8-m` as the digit takes the mth position from the decimal point.
- Now, we must add all the terms obtained after multiplication.

Now, convert decimal to binary to obtain the results.

**Q6: What is the International Number System?**

Ans – In International Number System, we provide an efficient mechanism to understand large numbers conveniently. For instance, let's know International Number System with numbers – ones (1), ten's (10), hundred's (100), and so on.

**Q7: How to convert Hexadecimal to Decimal?**

Ans – To convert the Hexadecimal to decimal number system, we need to begin by multiplying the hexadecimal number by 16. Next, you must raise the power of zero to it, and next, you must increase the capacity by one each time.

**Q8: How to solve the Number System problems efficiently?**

Ans – 98To solve the problems of the number system efficiently; you must develop a deep understanding of the topic. You must spend quality time learning the concepts and various methods. You can also refer to NCERT solutions of class 9 Maths paper to grab the latest questions to practice them. NCERT solutions of Class 9 Maths paper and number system in Hindi number system in Maths are freely accessible on the Internet.