Converting Bases — Definition, Formula & Examples

Converting bases means rewriting a number from one base (like base 10) into another base (like base 2 or base 16), so the value stays the same but the digits change.

Base conversion is the process of expressing a numeral represented in one positional number system (base $b_1$ ) as an equivalent numeral in a different positional number system (base $b_2$ ), preserving the quantity's value.

How It Works

To convert a base-10 number into another base, repeatedly divide by the new base and record each remainder. The remainders, read from bottom to top, form the digits of the new number. To convert back to base 10, multiply each digit by its place value (the base raised to that digit's position) and add the results together. Common bases include base 2 (binary), base 8 (octal), base 10 (decimal), and base 16 (hexadecimal).

Worked Example

Problem: Convert 25 from base 10 to base 2.

Step 1: Divide 25 by 2. Record the quotient and remainder.

25 \div 2 = 12 \text{ remainder } 1

Step 2: Keep dividing each quotient by 2.

12 \div 2 = 6 \text{ R } 0 \quad \rightarrow \quad 6 \div 2 = 3 \text{ R } 0 \quad \rightarrow \quad 3 \div 2 = 1 \text{ R } 1 \quad \rightarrow \quad 1 \div 2 = 0 \text{ R } 1

Step 3: Read the remainders from last to first to get the base-2 number.

25_{10} = 11001_2

Answer:

25_{10} = 11001_2

Why It Matters

Computers store all data in binary (base 2), so understanding base conversion is essential in computer science courses. Programmers also routinely use hexadecimal (base 16) to represent colors, memory addresses, and encoded values more compactly than binary.

Common Mistakes

Mistake: Reading the remainders in the wrong order (top to bottom instead of bottom to top).

Correction: Always read remainders from the last division to the first. The final remainder is the leading (leftmost) digit of the converted number.