Number System in Computer Organization: Its Types with Examples

A number system is a way of expressing numbers in a consistent manner using digits or other symbols. In the context of computers, a number system represents data and performs operations at the machine level. Computers typically work with binary, octal, decimal, and hexadecimal systems. These systems serve as the foundation for data encoding, arithmetic operations, and other computations in computers.

There are 4 types of number systems in computer organization:

1. Binary Number System

The binary number system is the most fundamental system used in computers. It uses only two digits, 0 and 1, to represent all data. This is because the underlying electronic circuits in a computer can only understand two states on (1) and off (0). Every bit in binary can represent a single binary digit, making it the building block for all computer operations.

Example: 

For 11010,

= 1 × 2⁴ + 1 × 2³ + 0 × 2² + 1 × 2¹ + 0 × 2⁰ 

= 16 + 8 + 0 + 2 + 0

= (26)₁₀

2. Octal Number System

The octal number system is a base-8 system that uses digits from 0 to 7. It is used as a shorthand representation of binary numbers because each octal digit corresponds to a group of three binary digits (bits).

Example: 

For 726,

=7 × 8² + 2 × 8¹ + 6 × 8⁰ 

= 7 × 64 + 2 × 8 + 6

= 448 + 16 + 6 = (470)₁₀

Decimal Number System

The decimal number system is used in everyday life, based on base 10. It uses digits from 0 to 9. Although computers don't natively use the decimal system, it is important for human-computer interaction, as users generally interact with computers using decimal numbers.

Example: 

In 720, the value of 7 is 700 (7 × 10²), the value of 2 is 20 (2 × 10¹), and the value of 0 is 0 (0 × 10⁰).

Hexadecimal Number System

The hexadecimal number system is a base-16 system that uses digits from 0 to 9 and letters A to F (representing values 10 to 15). It is commonly used in computer programming to represent binary data, as each hexadecimal digit corresponds to four binary digits.

Example: 

For number 27FB:

2 × 16³ + 7 × 16² + 15 × 16¹ + 11 × 16⁰ 

= 2 x 4096 + 7 x256 + 15 x 16 + 11

= 8192 + 1792 + 240 + 11

= (10235)₁₀

One important aspect of number systems in computers is their ability to convert between different bases. Conversion between number systems is crucial for understanding how data is processed and stored in various formats. Here are some common conversions:

• Binary to Decimal Conversion: Multiply each binary digit by 2 raised to the power of its position (starting from 0), and then sum the results.
• Decimal to Binary Conversion: Repeatedly divide the decimal number by 2, recording the remainder for each step.
• Binary to Hexadecimal Conversion: Group binary digits in sets of four (starting from the right) and convert each group to its equivalent hexadecimal digit.
• Hexadecimal to Binary Conversion: Replace each hexadecimal digit with its 4-bit binary equivalent.

The importance of number systems in computers cannot be overstated. Number systems form the core of all computer operations and have a direct impact on several key areas:

• Number systems are used to represent various types of data in computers, such as integers, floating-point numbers, and characters.
• The ability to perform operations with binary or hexadecimal numbers allows computers to perform fast arithmetic calculations.
• Number systems influence how data is stored in memory. For example, binary representation is used in RAM and hard drive storage, while hexadecimal representation is commonly used for memory addresses.
• Programming languages often require the use of different number systems to manage and manipulate data effectively, especially for low-level system programming or hardware interaction.

The numbers are represented in binary, octal, decimal, and hexadecimal in the below table:

Here are the advantages of the number system in computer organization:

• Number systems can help convert one number system to another.
• Using binary or hexadecimal numbers allows computers to process data and perform calculations more quickly and efficiently.
• The binary number system is simple to use, with only two digits, 0 and 1.

Here are the disadvantages of the number system in computer organization:

• Hexadecimal numbers are difficult to read and write, and complex mathematical operations like multiplication and division can be difficult to perform.
• The decimal number system is less suitable for specific applications
• Binary and octal numbers are not easily understandable for humans, requiring conversion to and from more familiar decimal or hexadecimal systems.

In conclusion, number systems are the backbone of computers to process and represent data. They provide a way for machines to understand and manipulate numbers efficiently. Whether it's binary, hexadecimal, octal, or decimal, understanding these systems is vital for anyone studying computer organization or involved in computer programming.

1. What are the four types of number systems? 

The four main types of number systems used in computer organization are:

• Binary (base 2)
• Octal (base 8)
• Decimal (base 10)
• Hexadecimal (base 16)

2. What is the application of the number system? 

The application of number systems in computers is crucial for data representation, computation, memory management, and interaction with hardware. Different number systems help optimize storage, increase the speed of operations, and make data more manageable in programming.

3. What is the use of a number system in computers? 

The use of a number system in computers is to represent, store, and process data in a way that electronic circuits and processors can understand. Binary, hexadecimal, and other systems are essential for carrying out calculations, managing memory, and interacting with software applications. They are fundamental for everything from basic arithmetic to complex algorithms in modern computing.