Number system

Number System: Definition

A number system is a mathematical scheme for representing and expressing numbers in a manner that is consistent and understandable. Every numeral or digit, we see as measuring, counting, coding, or computing is based on the number system. 

There are different number systems: Decimal, Binary, Octal and Hexadecimal, each with its own base and application. Number systems give order to numerical information from solving practical maths problems to running complex calculations on computers. 

In this chapter we are going to take a look at what a number system is, the types of number systems, the conversion of number systems, and the role of place value in representing numbers. Whether you are a student starting off with basic maths or advancing to computational logic, the number system is an essential skill to understand.

 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 indicates the number system’s base, & the value of each digit depends on its position, known as place value. 

The most common number system we use every day is the decimal number system, which is an original -10 system. It uses ten digits: 0 to 9. For example, in number 345, the points are '5' instead of equipment, '4' and '3' in hundreds of places - multiplied by the powers of 10. 

Key Features of Number Systems: 

● Base (or Radix): Defines how many unique digits are used. For example, base -10 marks 0 to 9. 

● Location value: The value of one digit depends from the right to the left position of it. 

● Symbol or numbers: Each system has a unique set of scars allowed. Uses binary {0, 1}, uses Octal {0–7}, and so on. 

Why is it important to understand the number system? 

● It is the basis for all types of calculations in mathematics and computers. 

● Data helps understand representation, measurement and digital logic. 

● It is necessary to learn conversion from binary to decimal or octal to hexadecimal. 

Whether it is calculating bills, measuring distance, making computer programming or decoding digital signals, the number systems are everywhere in 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 Explained 

Mathematics and computer science contain four main types of number systems commonly used. Each number system has its own base (radix) and a set of symbols or digits used to

represent numbers. Understanding these helps students perform conversions, solve real world problems & how they store your computer and process data. 

1. Decimal Number System (Base 10) 

The Decimal Number System is the most widely used system in everyday life. It is a basic -10 system, which means it uses 10 digits: 0 through 9. 

● 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 

Used in: 

● Counter 

● Currency 

● Measurements 

2. Binary number system (Base 2) 

The binary number system is used in all digital devices such as computers, calculators and microprocessors. This is a base -2 system, using only two digits: 0 and 1. 

● Each digit (called a bit ) represents an on/off or correct/incorrect position. 

● Example: Binary number 1011 is equal to 

1 × 8 + 0 × 4 + 1 × 2 + 1 × 1 = 11 

Used in: 

● Computer programming 

● Digital electronics 

● data storage 

3. Octal Number System (Base 8) 

The octal number system uses eight digits: 0 to 7. It is a base -8 system. The octal numbers are often used as a brevity for binary values because an octal digit represents exactly three binary digits. 

● Example: October 25 = 

2 × 8 + 5 = 21 in decimal.

Used in: 

● Programming language 

● Microcontroller system 

● Low binary coding 

4. Hexadecimal number system (base 16) 

Hexadecimal number system uses 16 symbols: 0–9 and a -f, where a = 10, b = 11, ..., f = 15. It is an base -16 system and is widely used in computer programming and Memory addressing ● Example : Hex 2F= 

2 × 16 + 15 =47 in decimal. 

Used in : 

● Web color coding (e.g., #FFFFFF) 

● Memory addresses 

● Digital systems 

This type of number makes the back bones into system mathematics and computer science. When we know how each system works, students can easily convert and solve more advanced problems associated with location value, base and digital argument. 

Conversions Between Number Systems 

Understand how to convert between number systems in both mathematics and computer science. These conversions help students understand how different grounds are related to each other and are widely used in programming, data presentation & digital electronics. There are step-by-step methods to convert numbers from one system to another, complete with examples for better understanding.

Determine 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 with 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: 

Copy code 

● 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 

Binary to decimal conversion 

To convert from binary (base -2) to decimal (base -10), multiply each piece from 2 to 2, in position force, 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

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 from right to left position. 

3. Add products. 

Example: 

Convert 157 (Octal) to Decimal: 

(1 × 8²) + (5 × 8¹) + (7 × 8⁰) = 64 + 40 + 7 = 111 

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 make the origin of digital data processing, making them important for students to understand both basic and advanced math and technology concepts. 

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Place Value and Base in Number Systems 

Understanding how numbers are represented and interpreted is a basic concept in mathematics. The term place value and base plays 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 what each 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 position 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 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 position 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. 

Role of Base in Value Representation 

A number defines the base of a number system (also known as Radix): 

● How many digits are used in the system. 

● The multipliers or powers that determine the value of each position. 

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).

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 

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 ?

Understand the place value and base system strengthens students: 

• Break down and construct a  large number correctly.

• Understand the binary logic used in the computers.

● Change the numbers from one base to another. 

● Strengthen number spirit 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, calcultion, 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 their operation. 

For example: 

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

● Programming uses Booler 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 help increase logical reasoning, problem-solving capacity, and computational thinking—skills essential for academic success and real-world application. 

Number System Examples 

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 

Here are some frequently asked questions related to number systems that can help clear common doubts: 

Q1. What are the 4 types of number system? 

Answer: The four primary types of number systems are binary (base-2), octal (base-8), decimal (base-10) & hexadecimal (base-16). These systems are fundamental in computing and other fields for representing & manipulating numbers.

Binary (Base-2):

• Uses only two digits: 0 and 1.

• The foundation of computer systems and digital electronics, where data is stored & processed as binary code. 

Octal (Base-8):

• Uses eight digits: 0, 1, 2, 3, 4, 5, 6, and 7. 

• Can be used to represent binary numbers in a more compact format, with each octal digit representing a group of three binary digits. 

Decimal (Base-10): 

• The most common number system used daily, with ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. 

• The system we typically use for counting & everyday calculations. 

Hexadecimal (Base-16): 

• Uses sixteen symbols: 0-9 and A-F, where A-F represents 10-15. 

• Often used in computer programming and networking to represent memory addresses & colors.

Q2. What is the 5 system of numeration? 

Answer: Quinary (base 5 or pental) is a numeral system with five as the base. A possible origination of a quinary system is that there are five digits on either hand. 

Q3. What is the 7 number system? 

Answer: The 7 number system, also known as the septenary or base-7 system, is a positional notation system where each digit's place value is a power of 7. It uses the digits 0, 1, 2, 3, 4, 5, & 6. 

Here's a breakdown:

• Place Values: In base-7, the place values are powers of 7, starting from the rightmost digit as 7^0 (1), then 7^1 (7), 7^2 (49), and so on. 

• Digits: The allowable digits in a base-7 number are 0, 1, 2, 3, 4, 5 & 6. 

• Example: The base-7 number 2347 would be interpreted as (2 * 7^2) + (3 * 7^1) + (4 * 7^0) = (2 * 49) + (3 * 7) + (4 * 1) = 98 + 21 + 4 = 123 in base-10.

 Q4. Is 16 an odd or even number? 

Answer: Yes, 16 is an even number because it is completely divisible by 2. This can be checked by observing the last digit of 16 which is 6. If the last digit of a number is 0, 2, 4, 6, or 8 it is an even number. 

Q5. Where do we use hexadecimal numbers in real life? 

Answer: Hexadecimal numbers are commonly used in computer programming, memory addressing, color coding in web design (e.g., #FF5733) & digital circuits.

 Conclusion: 

The study of number systems is essential for both mathematical understanding and practical applications in fields like computer science, electronics, and data processing. By mastering the four primary number systems—Decimal (Base-10), Binary (Base-2), Octal (Base-8), and Hexadecimal (Base-16)—you can build a strong foundation in converting between these systems, understanding digital logic, and solving real-world problems efficiently.

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