Number System

Introduction to Number System

The number system is an important foundation in mathematics that helps us classify and work with numbers in a structured way. This allows us to easily perform basic operations such as addition, subtraction, multiplication, and division. Numbers are part of our daily lives, from counting objects to measuring distance and solving problems.

The number system is composed of different types of numbers, including natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Natural numbers (1, 2, 3, ...) are used for counting; the whole numbers include zero, integers contain both positive and negative numbers, rational numbers can be written as fractions, and irrational numbers cannot be expressed as simple fractions. Together, they create real numbers.

In this lesson, we will find out each type of number, its special features, and how they are connected. We will also see how the numbers can appear on the number line and why understanding them is important for solving mathematical problems in real life.

 

Table of Contents 

• What is a Number System?
• Importance of Number System
• Decimal Number System (Base 10)
• Binary Number System (Base 2)
• Octal Number System (Base 8)
• Hexadecimal Number System (Base 16)
• Conversions Between Number Systems
• Place Value and Base in Number Systems
• What is a Base?
• Importance of Number Systems in Daily Life
• Number System Examples and Practice Questions
• FAQs on Number Systems

 

What is a Number System? 

A number system is a way of writing and representing numbers using a set of symbols or digits. It is a method of expressing numbers by using a smooth structure based on value and base. In mathematics, many systems allow us to perform operations such as addition, subtraction, multiplication, and division. The total count of digits used in any number system indicates the base of the number system. For example, the base of a hexadecimal number system is 16 (numbers from 0-9 & letters from A-F). The most common number system we use every day is the decimal number system, which is a 10-digit system. It uses digits from 0 to 9. The other two number systems are binary (0,1) and octal (0-7) with bases 2 and 8, respectively. The value of each digit in a number depends on its position in the number. This is called the place value of a digit.

Key Features of Number Systems: 

• Base: Defines how many unique digits are used. For example, base-10 marks 0 to 9.

• Place value: The value of a digit depends on its place within the number from right to left. The numbers at the extreme right are at the unit place, followed by tens, hundreds, thousands, and so on.

• Symbol or numbers: Each system has a unique set of symbols/numbers allowed. In binary, we use the numbers 0 and 1. In the octal number system, we use 0–7 and so on. 

 

Importance of Number System

• The number system is the basis for all types of calculations in mathematics and computing.

• The number system forms the basis of data representation required for measurement, information storage, digital logic, etc. 

• Learning number systems is necessary for converting binary to decimal, octal to hexadecimal, etc.

• Whether it is calculating bills, measuring distance, making computer programs or decoding digital signals, the number system is used everywhere in our daily life.

• For example, computers use a binary number system to store and process data, while programmers often use hexadecimal values in coding.

 

Types of Number Systems

There are four main types of number systems. Each number system has its unique base depending on the set of symbols or digits used to represent numbers. Understanding these helps students perform conversions, solve real-world problems & process digital data. 

1. Decimal Number System (Base 10)

The decimal number system is most widely used in our daily lives. It is a basic-10 system, with base 1,0, which means it uses 10 digits from 0 to 9 to represent any number. 

Each digit has a place value depending on the powers of 10. For example, the number 472 is calculated as follows: (4 × 100) + (7 × 10) + (2 × 1) = 472 

Common Applications of the Decimal Number System: 

The decimal system is used in many ways in our daily lives. Here are some common uses:

• Counter: We use numbers to count things like books, pencils, or humans. The decimal system helps us to easily count by using the numbers from 0 to 9.

• Currency: All currencies, such as the rupee or the dollar, use the decimal system. It helps us read and write the amount as ₹ 25.50 or ₹ 100.75 correctly.

• Measurements: We measure length, weight, and time using numbers in the decimal system. For example, a ruler shows centimetres, and a clock shows hours and minutes.

2. Binary number system (Base 2)

The binary number system is a two-digit number system used in all digital devices, including computers, calculators, and microprocessors. It is a base-2 number system that uses only two digits, i.e., 0 and 1. 

• Each digit (called a 'bit') represents an on/off or correct/incorrect position ( 0 = ON and 1 = OFF).

• Example: Binary number 1011 is equal to 1 × 8 + 0 × 4 + 1 × 2 + 1 × 1 = 11 

Common Applications of the Binary Number System:

• Computer programming: A binary number is the language that computers understand. Each program or game you use is made of binary code inside the computer.

• Digital electronics: Devices such as calculators, digital watches, and mobile phones work using circuits that understand binary (0s and 1s), which turn things ON or OFF.

• Data Storage: Information such as photos, videos, songs, and documents is stored in the computer using binary numbers. Each file consists of a long series of 0s and 1s.

3. Octal Number System (Base 8)

The octal number system uses eight digits from 0 to 7. It is a base-8 system which is often used to represent binary values more efficiently because an octal digit represents three binary digits. 

• For example, the binary digit 001 = 1 in octal; that is, '001' is represented as '1', and the binary digit '111' is '7' in the octal number system.

• Example: Octal 258 = 2 × 8 + 5 = 21 in decimal.

Common Applications of the Octal Number System

• Programming language: Octal numbers are sometimes used in coding to make binary numbers smaller and easier to read.

• Microcontroller System: Small computer chips use octal numbers to quickly store and process data.

• Low binary coding: Octal numbers help write long binary codes in a short and simple way.

4. Hexadecimal number system (base 16)

The hexadecimal number system uses 16 symbols: 0–9 and a–f, where a = 10, b = 11, ..., and f = 15. It is a base-16 system and is widely used in computer programming and memory addressing

• Example: Hex 2F 

• 2 × 16 + 15 = 47 in decimal.

Common Applications of the Hexadecimal Number System

• Web colour coding (e.g., #ffff): Hexadecimal numbers are used to display colour on websites.

• Memory addresses: Use hexadecimal numbers to display the location of computer-stored data.

• Digital systems: Hexadecimal Number helps in designing and reading digital devices such as calculators or microscopes.

 

Conversions Between Number Systems

Learning how to convert between number systems is a highly important skill that lays the foundation for advanced mathematical concepts and problem solving skills. It is applied in both mathematics and computer science. Conversion of number system is used in programming, data presentation & digital electronics. Here is a step-by-step explaination of methods to convert numbers from one system to another, with complete examples for better understanding.

1. Decimal to binary conversion

Follow these steps to convert a number from the decimal number system (base-10) to binary number system (base-2): 

Steps: 

1. Divide the decimal number by 2. 

2. Record the Remainder. 

3. Divide the quotient again by 2. 

4. Repeat the process until the quotient is 0. 

5. Read the remainders in reverse order (bottom to top). 

Example: 

Convert 25 to Binary:

• 25 ÷ 2 = 12 remainder 1 

• 12 ÷ 2 = 6 remainder 0 

• 6 ÷ 2 = 3 remainder 0 

• 3 ÷ 2 = 1 remainder 1 

• 1 ÷ 2 = 0 remainder 1 

Reading remainders from bottom to top: 11001 

Decimal 25 = Binary 11001

2. Binary to decimal conversion

To convert from binary (base-2) to decimal (base-10), multiply each piece 2 by 2 in position, starting from right to left. 

Steps: 

1. Write a binary number. 

2. Multiply each digit by 2ⁿ, where N is the correct position index (starts at 0). 3. Add all the results. 

Example: 

Convert 1011 to decimal: (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰) = 8 + 0 + 2 + 1 = 11

3. Octal to Decimal Conversion

The octal number system (base-8) uses digits from 0 to 7. To convert it to decimal: 

Steps: 

1. Write the octal number. 

2. Multiply each digit by 8ⁿ, where N is based on the position of digit from right to left. 

3. Add the sum of products. 

Example: 

Convert 157 (Octal) to Decimal: (1 × 8²) + (5 × 8¹) + (7 × 8⁰) = 64 + 40 + 7 = 111

4. Hexadecimal to Decimal Conversion

Hexadecimal number system (base-16) includes digits 0-9 & letters A-F (where A=10 to F=15). To convert to decimal: 

Steps: 

1. Replace each letter with its decimal equivalent. 

2. Multiply each digit by 16ⁿ 

3. Sum all the results. 

Examples: 

Convert 2F to decimal : (2 × 16¹) + (15 × 16⁰) = 32 + 15 = 47

These conversion techniques between number systems not only improve calculation skills but also lay the foundation of digital data processing, making them important for students to understand both basic and advanced math and technology concepts.

 

Place Value and Base in Number Systems

Understanding how numbers are represented and interpreted is a basic concept in mathematics. The terms place value and base play a central role in the number system, which helps us effectively read, write and understand both in daily life and mathematical operations. 

These two components place value and base-decimal (base-10), binary (base-2) & more as the basis for positive number systems. Let's break down what each is and how they work together to give value to the digits of a number.

Understanding Positional Notation

It is a method of representing or writing positional notation numbers where the location of each digit in a number determines its actual value. In this system, the same digit can represent different values, where it is shown in the number. 

Key concept: 

In number 753, the number 7 does not represent only 7 - it represents 700, as it is "hundreds" in place. 

Examples in Base-10 (decimal system):

Let's take number 4,382: 

As shown, the value of each digit = digit × place value. This system is called a positional number system. 

Positional Notation in Different Bases:

The nice thing about status signs is that it is not limited to the base-10. It works in any base system, where the base determines the value of each location. 

For example: 

• In binary (base-2), the digits are multiplied by the powers of 2. 

• In hexadecimal (base-16), the digits are multiplied by the powers of 16.

 

What is a Base?

A base is the number of unique digits (including zero) that a positional number system uses.

• Base-10 (Decimal): Digits used are 0–9
• Base-2 (Binary): Digits used are 0 and 1
• Base-8 (Octal): Digits used are 0 to 7
• Base-16 (Hexadecimal): Digits used are 0 to 9 and A to F (where A=10, B=11, … F=15)  

How Base Affects Place Value: 

In any positional number, the value of each digit is calculated as: Digit × (Base ^ Position) 

Where: 

• The position is counted from right to left, starting from 0.
• ^ denotes exponentiation (power).

1. Example: Decimal Number 327 in Base 10 

Using powers of 10: 327 = (3 × 10²) + (2 × 10¹) + (7 × 10⁰) 

                                          = (3 × 100) + (2 × 10) + (7 × 1) 

                                          = 300 + 20 + 7 

                                          = 327 

2. Example: Binary Number 1011 in Base 2 

Using powers of 2: 1011 = (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰) 

                                          = (1 × 8) + (0 × 4) + (1 × 2) + (1 × 1) 

                                          = 8 + 0 + 2 + 1 

                                          = 11

Change the entire interpretation of the same digit as you can see, change the base. This is why it is important to understand the base system in areas such as computer science, electronics and mathematics.

Summary Table: Base vs Place Value

Why it matters in learning?

Understanding the place value and base system strengthens students: 

• Break down and construct a large number correctly.

• Understand the binary logic used in computers.

• Change the numbers from one base to another.

• Strengthen number sense and mental math skills.

Real-Life Applications 

• Digital electronics base-2 (binary) depends on the logic.

• IP addresses use base-8 (octal) or base-16 (hex).

• Mathematical coding systems use several places for encryption, calculation, and data structuring.

 

Importance of Number Systems in Daily Life

Understanding the number system is more than just an educational practice; it plays an important role in our daily lives. From simple arithmetic to complex data processing, the number of systems forms the basis for how we interpret, use, and communicate quantitative data. Let's see how these systems affect different aspects of modern life.

Use in Computers and Technology

The binary number system (base-2) is the main language of all modern computers. Each digital device, from smartphones to advanced servers, processes data using binary code. These systems use combinations of 0s and 1s to represent instructions and data, making the number system important in the computer for its operation. 

For example: 

• Text, photos, and videos are stored in the form of binary digits.

• Programming uses Boolean algebra, which is based on binary numbers.

• Network addresses (e.g., IP addresses) depend on binary and hexadecimal formats.

Understanding the number systems allows students to understand basic data processing principles and lays the foundation for subjects such as computer science, coding, and digital electronics.

Role in Mathematics and Education

In mathematics, the number system provides structure and clarity. Students use the decimal number system (base-10) from the beginning and gradually detect binary, octal, and hexadecimal systems as they advance.

Applications in education include: 

• Performing arithmetic operations (addition, subtraction, multiplication, and division).

• Understanding place value and numerical representation.

• Solving conversion problems between various number systems. 

Mastering these systems helps increase logical reasoning, problem-solving capacity, and computational thinking skills essential for academic success and real-world application. 

 

Number System Examples and Practice Questions

To develop a strong understanding of number systems, it's important to solve a variety of examples & practice problems.

Examples: 

1. Convert 45 Decimal to Binary: 45 ÷ 2 = 22 R1 → 101101 (Binary)

2. Convert 1101 Binary to Decimal: (1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰) = 8 + 4 + 0 + 1 = 13

3. Convert 72 Decimal to Octal: 72 ÷ 8 = 9 R0 → 110 (Octal)

4. Convert 1F Hexadecimal to Decimal: (1 × 16¹) + (15 × 16⁰) = 16 + 15 = 31

Practice Questions: 

1. Convert 58 (decimal to binary).

2. Convert 100101 (binary to decimal).

3. Convert 63 (decimal to octal).

4. Convert 3E (hexadecimal to decimal).

5. Write the decimal equivalent of 1110 (binary).

 

FAQs on Number Systems

1. What are the types of number systems?

Ans: There are four main types of number systems:

• The binary number system uses only 0 and 1.

• The octal number system uses numbers from 0 to 7.

• The decimal number system uses numbers from 0 to 9.

• The hexadecimal number system uses numbers from 0 to 9 and the letters A to F.

2. What is the number 10 system?

Ans: The number 10 system is called the decimal number system. It uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. This is the most common system we use in daily life for counting and calculation.

3. What is the Base 1 number system called?

Ans: The basis 1 number system is called the unary number system. It only uses one symbol, usually "1", to count. For example, number 3 in unary will be written as "111".

4. What is the equivalent binary number for the decimal number 43?

Ans: The decimal number 43 is written as 101011 in the binary number system.

5. What numbering system uses 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9?

Ans: The numbering system that uses digits is the 0 to 9 is the decimal number system.