Number System in Computer

A number system is a method used to mathematically represent the numbers of a particular set. This system expresses numbers logically through numbers or symbols. The number system presents numbers in a standard notation according to their arithmetic and algebraic structures and represents a large set of numbers.

All numbers can be created with digits 0 to 9 and an infinite number of numbers can be generated using these digits. For example, numbers such as 123, 4567, 891011, 1213, and 141599 can be expressed with this system.

Number systems play an important role in both daily life and scientific fields by facilitating mathematical operations and understanding of numbers.

There are four main number systems.

1. Binary Number System
2. Octal Number System
3. Decimal Number System
4. Hexadecimal Number System

1. Binary Number System: The Foundation of the Digital World

• Base: 2
• Symbols: 0, 1
• Use cases: Computers and digital electronic circuits.

Introduction

The binary number system is a number system that forms the basis of the digital world and is an indispensable part of computer science. This system is capable of representing all numbers using only two symbols "0" and "1". The binary number system is used effectively in the data processing processes of computers and digital devices and constitutes the basic building block of modern technology.

Basic concepts

Binary is a number system in which each digit (bit) can take one of two values "0" or "1". In this system, the position of each bit determines the weight of that bit, and these weights are expressed as powers of base two. For example, the number 1010 using the binary number system is equal to 10 in the decimal system (1x2³ + 0x2² + 1x2¹ + 0x2⁰ = 8 + 2 = 10).

Mathematical Foundations

In the binary number system, basic arithmetic operations (addition, subtraction, multiplication, and division) are similar to operations in the decimal system but are performed using only two digits. Binary addition is done by adding each bit and processing the results with overflow (carry) bits. For example, the binary addition of the numbers 101 and 110 works like this.

101 + 110 = 1011

Similarly, binary subtraction, multiplication and division operations are performed according to certain rules and are widely used in digital circuits and computer algorithms.

Scope of application

The binary number system has a wide range of applications in the fields of digital electronics and computer engineering. The binary number system is used in many areas such as digital circuit design, microprocessors, data transmission, cryptography and error correction codes. Central processing units (CPUs) of computers process and store data and instructions in binary form. Moreover, in modern communication systems, data transmission and error correction mechanisms are based on the binary number system.

Advantages and Disadvantages

The biggest advantage of the binary number system is that it is simple and reliable. Two-state "0" and "1" logic gates ensure stability and accuracy in digital circuits. Additionally, the use of the binary system simplifies the design and fabrication of electronic circuits. However, disadvantages of the binary system include that it requires more bits to represent large numbers and that some calculations become more complex than with the decimal system.

Sample Problems and Solutions

• Example 1: Converting binary number to decimal number.
• Binary number: 1101
• Decimal equivalent: 1x2³+1x2²+0x2¹+1x2⁰ = 8+4+0+1 = 13

• Example 2: Binary addition.
• Binary numbers: 1011 and 1101
• Addition: 1011+1101 = 11000
• Decimal equivalent: 11+13 = 24

• Example 3: Binary subtraction.
• Binary numbers: 1011 and 0110
• Subtraction: 1011-0110 = 0101
• Decimal equivalent: 11-6 = 5

Conclusion and Future Perspectives

The binary number system will continue to maintain its importance with the development of digital technology. Even as new technologies such as quantum computing emerge, the fundamental principles and applications of the binary system will remain valid. The binary system plays a critical role in data processing and storage in the digital world, and there will be further improved and optimized uses in the future.

2. Octal Number System: Alternative to the Digital World

• Base: 8
• Symbols: 0, 1, 2, 3, 4, 5, 6, 7
• Use cases: Old computer systems and some microcontroller applications.

Introduction

The octal number system is another important number system used in digital electronics and computer science. This system expresses numbers using eight digits "0-7", from zero to seven. It offers a more compact representation than the binary number system and can be more efficient in some special cases.

Basic concepts

The octal number system (octal) represents numbers using eight different digits (0, 1, 2, 3, 4, 5, 6, 7). The position of each digit is defined by multiplying it by powers of base eight to determine its value. For example, the octal number 157 is expressed in decimal as:

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

Mathematical Foundations

In the octal number system, basic arithmetic operations (addition, subtraction, multiplication, and division) are similar to operations in the decimal system but are performed using only eight digits. Octal addition is done by adding each digit and processing the results with overflow (carry) digits. For example, adding the numbers 67 and 25 to octal is as follows.

67₈ + 25₈ = 114₈

Similarly, octal subtraction, multiplication and division operations are also performed according to certain rules, and these operations are widely used in digital circuits and computer algorithms.

Scope of application

The octal number system has been widely used, especially in older computer systems and digital circuit design. In UNIX-based systems, file permissions are specified in the octal system. Additionally, it is preferred in some low-level programming and hardware design applications because it offers a shorter and more compact representation than the binary number system.

Advantages and Disadvantages

The biggest advantage of the octal number system is that it takes up less space than the binary number system. This allows for a more concise representation of large binary numbers in the octal system. However, disadvantages of the octal system include difficulties in direct conversion to decimal and its lack of widespread use in modern computer systems.

Sample Problems and Solutions

• Example 1: Converting octal number to decimal number.
• Octal number: 345₈
• Decimal equivalent: 3x8²+4x8¹+5x8⁰ = 192+32+5 =229

• Example 2: Octal addition.
• Octal numbers: 157₈ and 246₈
• Addition: 157₈+246₈ = 425₈
• Decimal equivalent: 111 + 166 = 277

• Example 3: Octal subtraction.
• Octal numbers: 145₈ and 67₈
• Subtraction: 145₈-67₈ = 56₈
• Decimal equivalent: 101-55 = 46

Conclusion and Future Perspectives

The octal number system still has an important role in certain areas of the digital world. It will continue to be used, especially in situations requiring compatibility with legacy systems and in certain low-level programming applications. In the future, with the development of more modern systems and technologies, the use of the octal number system may decrease, but due to its historical and technical importance, it is not expected to disappear completely.

3. Decimal Number System: Fundamentals, Applications and Future

• Base: 10
• Symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Use cases: It is widely used in daily life, mathematical operations and financial transactions.

Introduction

The decimal number system is a number system that is widely used throughout the world and performs basic arithmetic operations. This base-ten system represents all numbers using the ten digits zero through nine. The decimal number system, which is frequently used in daily life, scientific calculations and engineering applications, is also one of the basic building blocks of digital technologies.

Basic concepts

The decimal number system represents numbers using ten different digits (0, 1, 2, 3, 4, 5, 6, 7, 8, 9). The position of each digit is multiplied by the powers of base ten to determine its value. For example, the number 1234 is expressed as:

1x10³+2x10²+3x10¹+4x10⁰ = 1000+200+30+4 = 1234

Decimal numbers can consist of integers and fractional parts. The decimal separator (usually a period or comma) indicates that the fractional part begins. For example, in the number 123.45, 123 represents the integer part and 45 represents the fractional part.

Mathematical Foundations

In the decimal number system, basic arithmetic operations (addition, subtraction, multiplication and division) are performed according to the rules in the decimal system. These operations are done mathematically as follows:

• Addition: Combining two or more numbers: 123+456 = 579
• Subtraction: Subtracting one number from another. 789-123 = 666
• Multiplication: Multiplying two numbers together. 12x34 = 408
• Division: Dividing one number by another. 144/12 = 12

Decimal fractions are fractions expressed as powers of 10. For example, 0.75 might be written as 75/100 or 3/4.

Scope of application

The decimal number system is used in almost all fields. It appears in many places in daily life, such as price tags, measurements and time calculations. In scientific research and engineering, precise measurements and calculations are made in the decimal system. It is used to express currencies in financial transactions, accounting and economics.

Advantages and Disadvantages

The advantages of the decimal number system include simplicity and universality. It is a system that people can easily understand and use. Additionally, it has a structure suitable for scientific and financial calculations.

Disadvantages include reduced efficiency in some cases when working with large numbers or very small fractional numbers. In computer systems, some precision losses may occur when decimal numbers must be converted to binary.

Sample Problems and Solutions

• Example 1: Converting a decimal number to a fraction.
• Decimal number: 0.75
• Fractional equivalent: 75/100 = 3/4

• Example 2: Decimal addition.
• Decimal numbers: 12.34 and 56.78
• Addition: 12.34+56.78 = 69.12

• Example 3: Decimal subtraction.
• Decimal numbers: 100.50 and 25.25
• Subtraction: 100.50-25.25 = 75.25

Conclusion and Future Perspectives

The decimal number system will remain important in the future thanks to its universal acceptance and ease of use. With the development of digital technologies and scientific research, the use of the decimal system in areas requiring precision and efficiency will increase. New algorithms and calculation techniques will allow more effective use of the decimal number system.

4. Hexadecimal Number System: High-Efficiency Digital Representation

• Base: 16
• Symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
• Use cases: Used in computer science, memory addressing, low-level programming and colour coding.

Introduction

The hexadecimal number system is another number system widely used in digital electronics and computer science. This system represents numbers using sixteen digits (0-9 and A-F), from zero to fifteen. It offers a more compact and efficient representation than binary and decimal systems. The hexadecimal system has an important place, especially in computer science and digital circuit design.

Basic concepts

The hexadecimal number system (hexadecimal) uses sixteen different digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, and F. Where A is in the decimal system It corresponds to the value 10, B 11, C 12, D 13, E 14 and F 15. The position of each digit is multiplied by the power of base sixteen to determine its value. For example, the hexadecimal number 1A3 is expressed in decimal as:

1x16²+Ax16¹+3x16⁰ = 1x256+10x16+3 = 256+160+3 = 419

Mathematical Foundations

In the hexadecimal number system, basic arithmetic operations (addition, subtraction, multiplication, and division) are similar to operations in the decimal system but are performed using only sixteen digits. Hexadecimal addition is done by adding each digit and processing the results with overflow (carry) digits. For example, adding the numbers 1A and 2B in hexadecimal is as follows:

1A₁₆+2B₁₆ = 45₁₆

Similarly, hexadecimal subtraction, multiplication and division operations are also performed according to certain rules, and these operations are widely used in digital circuits and computer algorithms.

Scope of application

The hexadecimal number system has a wide range of applications, especially in computer science and digital electronics. Hexadecimal is used in many areas, such as memory addressing, colour coding (for example, specifying colours in HTML and CSS), machine language and assembly language programming. Additionally, it is widely preferred in digital circuit design and microprocessors because it allows direct conversions with the binary system.

Advantages and Disadvantages

The biggest advantage of the hexadecimal number system is that it allows direct conversions with the binary system and offers a more compact representation. This allows large binary numbers to be represented in hexadecimals more concisely and understandably. However, disadvantages of the hexadecimal system include that it can be difficult for some users to get used to and that it is not widely used in everyday calculations.

Sample Problems and Solutions

• Example 1: Converting hexadecimal numbers to decimal numbers.
• Hexadecimal number: 1C₁₆ and 2A₁₆
• Addition: 1C₁₆+2A₁₆ = 46₁₆
• Decimal equivalent: 28 + 42 = 70

• Example 2: Hexadecimal subtraction.
• Hexadecimal numbers: 3D₁₆ ve 1B₁₆
• Subtraction: 3D₁₆-1B₁₆ = 22₁₆
• Decimal equivalent: 61 - 27 = 34

Conclusion and Future Perspectives

The hexadecimal number system has a critical role in the digital world and will remain important in the future. It will continue to be used especially in the fields of computer science, digital electronics and programming. In the future, with the development of more modern systems and technologies, the use of the hexadecimal number system may further increase and find more applications.

Resources and Further Reading:

• Wikipedia: Binary numeral system.
• Wikipedia: Octal numeral system.
• Wikipedia: Decimal numeral system.
• Wikipedia: Hexadecimal numeral system.