How to Manually Convert Between Binary, Decimal, Octal, and Hexadecimal

Understanding Number Systems

Digital systems fundamentally operate using binary (base-2) numbers. However, humans are more accustomed to decimal (base-10). Octal (base-8) and hexadecimal (base-16) systems serve as convenient shorthand representations for binary, especially in computing contexts, as they align well with groups of binary digits (3 bits for octal, 4 bits for hexadecimal). This guide will equip you with the knowledge to perform these essential conversions manually.

Prerequisites

To effectively follow this guide, a basic understanding of:

• Positional Notation: The concept that the value of a digit depends on its position within a number.
• Exponents/Powers: Specifically powers of 2 (for binary), 8 (for octal), and 16 (for hexadecimal).
• Basic Arithmetic: Addition, subtraction, multiplication, and division.

Binary to Decimal Conversion

The decimal value of a binary number is found by summing the products of each binary digit with 2 raised to the power of its position. Positions are counted from right to left, starting at 0.

Formula

$Decimal = \sum_{i=0}^{n-1} (d_i \times 2^i)$ Where $d_i$ is the binary digit at position $i$ (from right, starting at 0).

Worked Example: Convert $1101_2$ to Decimal

1. Identify the binary digits and their positions:
   • $1$ (position 3)
   • $1$ (position 2)
   • $0$ (position 1)
   • $1$ (position 0)
2. Apply the formula: $ (1 \times 2^3) + (1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) $ $ = (1 \times 8) + (1 \times 4) + (0 \times 2) + (1 \times 1) $ $ = 8 + 4 + 0 + 1 $ $ = 13_{10} $

Common Pitfall

Forgetting to start the position count from 0 for the rightmost digit. Always remember that $2^0 = 1$.

Decimal to Binary Conversion

To convert a decimal number to binary, repeatedly divide the decimal number by 2 and record the remainder. The binary number is formed by reading the remainders from bottom to top.

Formula

Repeated division by 2, collecting remainders.

Worked Example: Convert $25_{10}$ to Binary

1. Divide 25 by 2:
   • $25 \div 2 = 12$ remainder $1$
2. Divide 12 by 2:
   • $12 \div 2 = 6$ remainder $0$
3. Divide 6 by 2:
   • $6 \div 2 = 3$ remainder $0$
4. Divide 3 by 2:
   • $3 \div 2 = 1$ remainder $1$
5. Divide 1 by 2:
   • $1 \div 2 = 0$ remainder $1$
6. Read the remainders from bottom to top: $11001_2$.

Common Pitfall

Reading the remainders from top to bottom instead of bottom to top, which would yield an incorrect result.

Binary to Octal Conversion

Octal numbers are a compact way to represent binary numbers. Since $2^3 = 8$, each group of three binary digits corresponds to exactly one octal digit.

Formula

Group binary digits in sets of three from right to left, then convert each group to its decimal (octal) equivalent. Pad with leading zeros if the leftmost group has fewer than three digits.

Worked Example: Convert $110101_2$ to Octal

1. Group the binary digits from right to left into sets of three:
   • 110 101
2. Convert each group to its decimal equivalent:
   • 110_2 = $(1 \times 2^2) + (1 \times 2^1) + (0 \times 2^0) = 4 + 2 + 0 = 6_{10}$
   • 101_2 = $(1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) = 4 + 0 + 1 = 5_{10}$
3. Combine the octal digits: $65_8$.

Worked Example with Padding: Convert $10110_2$ to Octal

1. Group binary digits: 10 110. The leftmost group has only two digits.
2. Pad the leftmost group with a leading zero: 010 110.
3. Convert each group:
   • 010_2 = $2_{10}$
   • 110_2 = $6_{10}$
4. Combine: $26_8$.

Common Pitfall

Incorrectly grouping bits from left to right, or forgetting to pad the leftmost group with zeros.

Octal to Binary Conversion

This is the reverse of binary to octal. Each octal digit is directly translated into its 3-bit binary equivalent.

Formula

Convert each octal digit to its corresponding 3-bit binary representation.

Worked Example: Convert $65_8$ to Binary

1. Convert each octal digit to 3-bit binary:
   • $6_8 = 110_2$
   • $5_8 = 101_2$
2. Concatenate the binary groups: $110101_2$.

Common Pitfall

Not using exactly three bits for each octal digit (e.g., writing $2_8$ as 10_2 instead of 010_2).

Binary to Hexadecimal Conversion

Hexadecimal numbers provide an even more compact representation than octal. Since $2^4 = 16$, each group of four binary digits corresponds to one hexadecimal digit. Hexadecimal uses digits 0-9 and letters A-F for values 10-15.

Formula

Group binary digits in sets of four from right to left, then convert each group to its decimal (hexadecimal) equivalent. Pad with leading zeros if the leftmost group has fewer than four digits.

Hexadecimal Equivalents

Worked Example: Convert $11010110_2$ to Hexadecimal

1. Group the binary digits from right to left into sets of four:
   • 1101 0110
2. Convert each group to its hexadecimal equivalent:
   • 1101_2 = $(1 \times 2^3) + (1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) = 8 + 4 + 0 + 1 = 13_{10} = D_{16}$
   • 0110_2 = $(0 \times 2^3) + (1 \times 2^2) + (1 \times 2^1) + (0 \times 2^0) = 0 + 4 + 2 + 0 = 6_{10} = 6_{16}$
3. Combine the hexadecimal digits: $D6_{16}$.

Common Pitfall

Misremembering the hexadecimal letter mappings (A-F) or incorrect grouping/padding.

Hexadecimal to Binary Conversion

Each hexadecimal digit is translated directly into its 4-bit binary equivalent.

Formula

Convert each hexadecimal digit to its corresponding 4-bit binary representation.

Worked Example: Convert $D6_{16}$ to Binary

1. Convert each hexadecimal digit to 4-bit binary:
   • $D_{16} = 13_{10} = 1101_2$
   • $6_{16} = 6_{10} = 0110_2$
2. Concatenate the binary groups: $11010110_2$.

Common Pitfall

Not using exactly four bits for each hexadecimal digit (e.g., writing $2_{16}$ as 10_2 instead of 0010_2).

When to Use an Online Converter

While understanding manual conversion is crucial, for large numbers, frequent conversions, or to quickly verify your manual calculations, an online binary converter tool offers significant advantages in speed and accuracy. It eliminates the potential for human error in repetitive arithmetic.