Number Systems Binary, decimal, octal, hexadecimal and conversions

Subject: DIGITAL TECHNOLOGY

Class: SS 1

Topic: Number Systems: Binary, Decimal, Octal, Hexadecimal and Conversions

Core Concepts

In digital technology, information is represented using different number systems. While humans commonly use the decimal system, computers primarily use the binary system. Understanding these systems and how to convert between them is fundamental to digital literacy.

1. Decimal Number System (Base 10)
   • This is the number system we use every day.
   • It uses ten unique digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
   • Each digit's position (place value) is a power of 10. For example, in 245, the 5 is in the units (10⁰) place, 4 in the tens (10¹) place, and 2 in the hundreds (10²) place.
   • Example: 245₁₀ = (2 × 10²) + (4 × 10¹) + (5 × 10⁰) = 200 + 40 + 5.
2. Binary Number System (Base 2)
   • This is the language of computers.
   • It uses only two unique digits: 0 and 1. Each digit is called a "bit" (binary digit).
   • Each digit's position is a power of 2.
   • Example: 1011₂ = (1 × 2³) + (0 × 2²) + (1 × 2¹) + (1 × 2⁰) = 8 + 0 + 2 + 1 = 11₁₀.
3. Octal Number System (Base 8)
   • It uses eight unique digits: 0, 1, 2, 3, 4, 5, 6, 7.
   • Each digit's position is a power of 8.
   • Octal is often used as a shorthand for binary because three binary digits (bits) can be represented by one octal digit (2³ = 8).
   • Example: 17₈ = (1 × 8¹) + (7 × 8⁰) = 8 + 7 = 15₁₀.
4. Hexadecimal Number System (Base 16)
   • It uses sixteen unique symbols: 0-9 and A, B, C, D, E, F.
   • A represents 10, B represents 11, C represents 12, D represents 13, E represents 14, and F represents 15.
   • Each digit's position is a power of 16.
   • Hexadecimal is also a shorthand for binary because four binary digits (bits) can be represented by one hexadecimal digit (2⁴ = 16). It is commonly used in computer memory addresses and colour codes.
   • Example: 1A₁₆ = (1 × 16¹) + (A × 16⁰) = (1 × 16) + (10 × 1) = 16 + 10 = 26₁₀.

Conversions Between Number Systems

The ability to convert numbers between these systems is essential.

1. To Decimal from Any Base (Binary, Octal, Hexadecimal)
   • Multiply each digit by its place value (base raised to the power of its position, starting from 0 from the right).
   • Sum the results.
   • Example (Binary to Decimal): 1101₂ = (1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰) = 8 + 4 + 0 + 1 = 13₁₀.
2. From Decimal to Any Other Base (Binary, Octal, Hexadecimal)
   • Repeatedly divide the decimal number by the target base.
   • Record the remainders from each division.
   • The new number is formed by reading the remainders from bottom to top.
   • Example (Decimal to Binary): Convert 13₁₀ to binary.
     • 13 ÷ 2 = 6 remainder 1
     • 6 ÷ 2 = 3 remainder 0
     • 3 ÷ 2 = 1 remainder 1
     • 1 ÷ 2 = 0 remainder 1

     Reading remainders from bottom up: 1101₂.

3. Binary to Octal/Hexadecimal & Vice Versa (Shortcut Method)
   • Binary to Octal: Group binary digits in threes from the right. Convert each group to its octal equivalent. (e.g., 110101₂ → 110 101 → 6 5 → 65₈).
   • Octal to Binary: Convert each octal digit to its 3-bit binary equivalent. (e.g., 65₈ → 110 101 → 110101₂).
   • Binary to Hexadecimal: Group binary digits in fours from the right. Convert each group to its hexadecimal equivalent. (e.g., 11010110₂ → 1101 0110 → D 6 → D6₁₆).
   • Hexadecimal to Binary: Convert each hexadecimal digit to its 4-bit binary equivalent. (e.g., D6₁₆ → 1101 0110 → 11010110₂).

Key Points

• Every number system has a "base" which determines the number of unique digits it uses.
• The position of a digit in a number determines its "place value", which is a power of the base.
• Binary (base 2) is fundamental for computers, using only 0s and 1s (bits).
• Decimal (base 10) is our everyday number system.
• Octal (base 8) and Hexadecimal (base 16) are convenient for humans to represent long binary numbers more compactly.
• Mastering conversion techniques is crucial for understanding how digital systems process and represent data.

Summary

Number systems like Binary, Decimal, Octal, and Hexadecimal are different ways to represent quantities. Each system is defined by its base and the set of digits it uses. While decimal is for human convenience, binary is the core language of computers. Octal and hexadecimal serve as efficient shortcuts for representing binary data. Understanding their structure and being able to convert between them is a foundational skill in digital technology.