Number Bases

Number bases, also known as radix, represent the number of unique digits, including zero, used to represent numbers in a positional numeral system. For example, the most common system is the decimal system (Base 10), which uses digits 0-9.

1. Basic Concepts

• Binary (Base 2): Uses digits 0 and 1.
• Octal (Base 8): Uses digits 0 to 7.
• Denary/Decimal (Base 10): Uses digits 0 to 9.
• Base n: Uses digits 0 to (n-1). A number in base 5 cannot contain the digit 5.

2. Conversion Between Bases

A. Converting from Base n to Base 10

To convert from any base to base 10, use the expansion method. Multiply each digit by the base raised to the power of its position (starting from 0 on the right).

Example: Convert 1101₂ to base 10.
(1 × 2³) + (1 × 2²) + (0 × 2¹) + (1 × 2⁰) = 8 + 4 + 0 + 1 = 13₁₀.

B. Converting from Base 10 to Base n

Use the successive division method. Divide the decimal number by the target base and record the remainders from bottom to top.

C. Converting between Non-Decimal Bases

The standard procedure is to convert the given base to base 10 first, then convert the result from base 10 to the required base.

3. Fractional Parts

For fractions, the powers to the right of the decimal point are negative (e.g., n^-1, n^-2).

• Base n to 10: 0.101₂ = (1 × 2^-1) + (0 × 2^-2) + (1 × 2^-3) = 0.5 + 0 + 0.125 = 0.625₁₀.
• Base 10 to n: Multiply the fractional part by the target base repeatedly and record the integer parts.

4. Operations in Number Bases

Addition, subtraction, multiplication, and division follow the same logic as base 10, but you must 'carry over' or 'borrow' based on the value of the base.

• Addition: If the sum of digits equals or exceeds the base, divide by the base, write the remainder, and carry the quotient.
• Subtraction: When borrowing, the value borrowed is equal to the base.