Number System

What is Number System?

A number system is a method of writing that expresses numbers using a consistent set of digits or symbols. This mathematical notation allows for the representation of numbers from a given set, providing a unique representation for each number. It also reflects the arithmetic and algebraic structure of these figures, enabling clear and accurate mathematical communication.

Types of Number Systems

1. Decimal Number System (Base 10)

• Digits Used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Description: The most widely used number system, particularly in everyday life. Each digit’s position represents a power of 10.
• Example: 345 = 3×10² + 4×10¹ + 5×10⁰

2. Binary Number System (Base 2)

• Digits Used: 0, 1
• Description: Used primarily in computing and digital systems. Each digit’s position represents a power of 2.
• Example: 1011 = 1×2³ + 0×2² + 1×2¹ + 1×2⁰

3. Octal Number System (Base 8)

• Digits Used: 0, 1, 2, 3, 4, 5, 6, 7
• Description: Used in some computing systems. Each digit’s position represents a power of 8.
• Example: 157 = 1×8² + 5×8¹ + 7×8⁰

4. Hexadecimal Number System (Base 16)

• Digits Used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
• Description: Commonly used in computing and digital electronics. Each digit’s position represents a power of 16.
• Example: 3A5 = 3×16² + 10×16¹ + 5×16⁰

5. Roman Number System

• Symbols Used: I, V, X, L, C, D, M
• Description: An ancient number system still used in certain contexts today. Numbers are represented by combinations of letters from the Latin alphabet.
• Example: XVI = 10 + 5 + 1 = 16

6. Natural Number System

• Description: Comprises all positive integers starting from 1 (1, 2, 3, …). Used for counting and ordering.
• Example: 1, 2, 3, …

7. Integer Number System

• Description: Includes all whole numbers, both positive and negative, including zero.
• Example: -3, -2, -1, 0, 1, 2, 3

8. Rational Number System

• Description: Consists of numbers that can be expressed as a fraction of two integers (where the denominator is not zero).
• Example: 1/2, -3/4, 5

9. Irrational Number System

• Description: Comprises numbers that cannot be expressed as a simple fraction. Their decimal expansions are non-repeating and non-terminating.
• Example: π, √2

10. Real Number System

• Description: Encompasses both rational and irrational numbers. It includes all numbers that can be found on the number line.
• Example: -2, 0, 1/2, π, √3

11. Complex Number System

• Description: Consists of numbers that have a real part and an imaginary part, expressed in the form a + bi, where “i” is the imaginary unit.
• Example: 3 + 4i, 2 – 5i

Decimal Number System (Base 10)

The decimal number system, also known as base 10, is the most widely used number system in the world. It forms the basis of our daily counting and arithmetic operations.

Key Features

• Digits Used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
• Base: 10
• Positional Notation: Each digit in a number has a value depending on its position, with each position representing a power of 10.

Positional Value

In the decimal system, the value of each digit depends on its place or position in the number. This positional value can be expressed using powers of 10. For example, in the number 345:

• The digit 5 is in the units place, so its value is 5×10⁰=5.
• The digit 4 is in the tens place, so its value is 4×10¹=40.
• The digit 3 is in the hundreds place, so its value is 3×10²=300.

Therefore, the number 345 can be expressed as:

345=3×10²+4×10¹+5×10⁰

Arithmetic Operations

The decimal system supports all basic arithmetic operations such as addition, subtraction, multiplication, and division. These operations are foundational in mathematics and everyday calculations.

Examples

• Example 1:
• 582=5×10²+8×10¹+2×10⁰
• 582=500+80+2
• Example 2:
• 1,237=1×10³+2×10²+3×10¹+7×10⁰
• 1,237=1,000+200+30+7

Advantages

• Universal Use: The decimal system is universally used in everyday life, commerce, science, and engineering.
• Simplicity: It is easy to understand and use for basic and advanced calculations.
• Compatibility: It is compatible with metric measurements, which are also based on powers of 10.

Applications

The decimal number system is applied in various fields including:

• Finance: For currency and financial calculations.
• Engineering: For measurements and design calculations.
• Science: For expressing quantities, measurements, and scientific notation.
• Daily Life: For counting, telling time, and managing budgets.

Binary Number System (Base 2)

The binary number system, also known as base 2, is fundamental in computing and digital electronics. It uses only two digits, 0 and 1, to represent all numbers.

Key Features

• Digits Used: 0, 1
• Base: 2
• Positional Notation: Each digit’s position represents a power of 2.

Positional Value

In the binary system, the value of each digit depends on its position in the number. Each position represents a power of 2. For example, in the binary number 1011:

• The rightmost digit (1) is in the 2⁰(units) place.
• The next digit (1) is in the 2¹(twos) place.
• The next digit (0) is in the 2² (fours) place.
• The leftmost digit (1) is in the 2³ (eights) place.

Therefore, the binary number 1011 can be expressed as:

1011=1×2³+0×2²+1×2¹+1×2⁰

1011=8+0+2+1=11

Arithmetic Operations

The binary system supports basic arithmetic operations similar to the decimal system:

• Addition: 1 + 1 = 10 (binary)
• Subtraction: 10 – 1 = 1 (binary)
• Multiplication: 10 × 10 = 100 (binary)
• Division: 100 ÷ 10 = 10 (binary)

Examples

• Example 1: 1101

1101=1×2³+1×2³+0×2¹+1×2⁰

1101=8+4+0+1=13

Advantages

• Simplicity in Digital Systems: Binary is the basis for digital circuits and computer systems, making it essential for computer programming and design.
• Ease of Implementation: Binary states (on/off) are easy to represent with electrical signals.

Applications

The binary number system is crucial in various fields, including:

• Computing: Used in all modern computers and digital systems.
• Digital Electronics: Fundamental for designing circuits and digital communication.
• Data Representation: Basis for binary code used in programming and data storage.

Octal Number System (Base 8)

The octal number system, also known as base 8, is a number system that uses eight digits to represent numbers. This system is commonly used in computing, especially in digital systems.

Key Features

• Digits Used: 0, 1, 2, 3, 4, 5, 6, 7
• Base: 8
• Positional Notation: Each digit’s position represents a power of 8.

Positional Value

In the octal system, the value of each digit depends on its position in the number. Each position represents a power of 8. For example, in the octal number 157:

• The rightmost digit (7) is in the 8⁰ (units) place.
• The next digit (5) is in the 8¹(eights) place.
• The leftmost digit (1) is in the 8² (sixty-fours) place.

Therefore, the octal number 157 can be expressed as:

157=1×8²+5×8¹+7×8⁰

157=64+40+7=111

Arithmetic Operations

The octal system supports basic arithmetic operations similar to the decimal system:

• Addition: 7 + 1 = 10 (octal)
• Subtraction: 10 – 1 = 7 (octal)
• Multiplication: 2 × 3 = 6 (octal)
• Division: 10 ÷ 2 = 4 (octal)

Examples

• Example 1:

345=3×8²+4×8¹+5×8⁰

345=192+32+5=229

• Example 2:

721=7×8²+2×8¹+1×8⁰

721=448+16+1=465

Advantages

• Simplifies Binary Representation: Octal is used as a shorthand for binary, as it is easier to convert between binary and octal.
• Reduces Length of Binary Numbers: Octal representation shortens long binary numbers, making them easier to read and understand.

Applications

The octal number system is used in various fields, including:

• Computing: Simplifies the representation of binary numbers, often used in programming and digital systems.
• Digital Electronics: Useful for representing and processing data in microprocessors and other digital devices.
• Data Entry: Employed in scenarios where binary data needs to be entered manually, reducing the likelihood of errors.

Hexadecimal Number System (Base 16 Number System)

The hexadecimal number system, also known as base 16, is widely used in computing and digital electronics. This system uses sixteen distinct symbols to represent numbers, making it an efficient way to express large binary numbers.

Key Features

• Digits Used: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
• Base: 16
• Positional Notation: Each digit’s position represents a power of 16.

Positional Value

In the hexadecimal system, the value of each digit depends on its position in the number. Each position represents a power of 16. For example, in the hexadecimal number 3A5:

• The rightmost digit (5) is in the 16⁰ (units) place.
• The middle digit (A, which represents 10) is in the 16¹ (sixteens) place.
• The leftmost digit (3) is in the 16¹ (256s) place.

Therefore, the hexadecimal number 3A5 can be expressed as:

3A5=3×16²+A×16¹+5×16⁰

3A5=3×256+10×16+5

3A5=768+160+5=933

Arithmetic Operations

The hexadecimal system supports basic arithmetic operations similar to the decimal system:

• Addition: 1A + B = 25 (hexadecimal)
• Subtraction: 2E – 1C = 12 (hexadecimal)
• Multiplication: 7 × 9 = 3F (hexadecimal)
• Division: 30 ÷ 6 = 5 (hexadecimal)

Examples

• Example 1:

1F4=1×16²+F×16¹+4×16⁰

1F4=1×256+15×16+4

1F4=256+240+4=500

• Example 2:

7B²=7×16²+B×16¹+2×16⁰

7B²=7×256+11×16+2

7B²=1792+176+2=1970

Advantages

• Efficient Representation: Hexadecimal provides a compact and readable way to represent large binary numbers.
• Ease of Conversion: It is straightforward to convert between binary and hexadecimal, making it useful for programming and debugging.

Applications

The hexadecimal number system is crucial in various fields, including:

• Computing: Used in programming languages, memory addresses, and color codes in web design.
• Digital Electronics: Helps in the representation and processing of binary-coded data.
• Data Representation: Employed in defining machine-level instructions and error codes.

Number System Conversion

Number system conversion is the process of changing a number from one system to another. This is essential for various applications in computing, digital electronics, and mathematics. Below are the methods and examples for converting between different number systems.

1. Decimal to Binary

To convert a decimal number to binary, divide the number by 2 and record the remainder. Continue dividing the quotient by 2 until you reach 0. The binary number is the sequence of remainders read from 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

Binary: 11001

2. Binary to Decimal

To convert a binary number to decimal, multiply each bit by 2 raised to the power of its position from right to left, starting at 0, and sum the results.

Example: Convert 11001 to decimal

1×2⁴+1×2³+0×2²+0×2¹+1×2⁰

=16+8+0+0+1

=25

3. Decimal to Octal

To convert a decimal number to octal, divide the number by 8 and record the remainder. Continue dividing the quotient by 8 until you reach 0. The octal number is the sequence of remainders read from bottom to top.

Example: Convert 83 to octal

• 83 ÷ 8 = 10, remainder 3
• 10 ÷ 8 = 1, remainder 2
• 1 ÷ 8 = 0, remainder 1

Octal: 123

4. Octal to Decimal

To convert an octal number to decimal, multiply each digit by 8 raised to the power of its position from right to left, starting at 0, and sum the results.

Example: Convert 123 to decimal

1×8²+2×8¹+3×8⁰

=64+16+3

=83

5. Decimal to Hexadecimal

To convert a decimal number to hexadecimal, divide the number by 16 and record the remainder. Continue dividing the quotient by 16 until you reach 0. The hexadecimal number is the sequence of remainders read from bottom to top, using digits 0-9 and letters A-F.

Example: Convert 255 to hexadecimal

• 255 ÷ 16 = 15, remainder 15 (F)
• 15 ÷ 16 = 0, remainder 15 (F)

Hexadecimal: FF

6. Hexadecimal to Decimal

To convert a hexadecimal number to decimal, multiply each digit by 16 raised to the power of its position from right to left, starting at 0, and sum the results.

Example: Convert 1A3 to decimal

1×16²+A×16¹+3×16⁰

=1×256+10×16+3

=256+160+3

=419

7. Binary to Octal

To convert a binary number to octal, group the binary digits into sets of three, starting from the right. Convert each group to its octal equivalent.

Example: Convert 110101 to octal

• Group: 110 101
• Convert: 110 = 6, 101 = 5

Octal: 65

8. Octal to Binary

To convert an octal number to binary, convert each octal digit to its 3-bit binary equivalent.

Example: Convert 65 to binary

• Convert: 6 = 110, 5 = 101

Binary: 110101

9. Binary to Hexadecimal

To convert a binary number to hexadecimal, group the binary digits into sets of four, starting from the right. Convert each group to its hexadecimal equivalent.

Example: Convert 10111011 to hexadecimal

• Group: 1011 1011
• Convert: 1011 = B, 1011 = B

Hexadecimal: BB

10. Hexadecimal to Binary

To convert a hexadecimal number to binary, convert each hexadecimal digit to its 4-bit binary equivalent.

Example: Convert 1F to binary

• Convert: 1 = 0001, F = 1111

Binary: 00011111

FAQs