Number bases and binary numbers

Our system is base 10

We count in base 10 (decimal): ten digits, 0–9, and each place is worth 10 times the place to its right.

Why 10? Probably because we have 10 fingers. But any whole number greater than 1 can be a number base.

Binary: base 2

Binary uses only two digits: 0 and 1. Each place is worth double the place to its right.

Computers use binary because electronic switches have two states: off (0) and on (1). Each binary digit is a bit, and 8 bits make a byte.

Binary to decimal

Add the place values where there’s a 1.

Example: 101₂ → 4 + 0 + 1 = 5

Example: 110110₂ → 32 + 16 + 0 + 4 + 2 + 0 = 54

The small 2 shows the base.

Decimal to binary

Method 1: Subtract powers of 2.

Example: 45

• Largest power ≤ 45 is 32 → 45 − 32 = 13 → 1 in the 32s place
• 16 > 13 → 0 in the 16s
• 8 ≤ 13 → 13 − 8 = 5 → 1 in the 8s
• 4 ≤ 5 → 5 − 4 = 1 → 1 in the 4s
• 2 > 1 → 0 in the 2s
• 1 → 1 in the 1s

45 = 101101₂

Method 2: Divide by 2 repeatedly and read the remainders from bottom to top.

Reading up: 101101₂ ✓

Counting in binary

0, 1, 10, 11, 100, 101, 110, 111, 1000, …

Other bases

Hexadecimal (base 16) uses 0–9 and A–F (A = 10, …, F = 15). It’s a compact way to write binary: each hex digit = 4 bits. Web colours like #FF0000 (red) are hexadecimal.

Worked examples

Example 1: 101₂ in decimal? 5.

Example 2: 13 in binary? 8 + 4 + 1 → 1101₂.

Example 3: How many values can one byte store? 2⁸ = 256 (0 to 255).

Common mistakes

• Reading binary 101 as "one hundred one". Say "one zero one".
• Putting place values in the wrong order (the ones place is on the right).
• Reading remainders top to bottom in the division method.

Try it yourself

1. 1001₂ in decimal
2. 11111₂ in decimal
3. 20 in binary
4. 100 in binary
5. Hexadecimal B in decimal

Answers: 1) 9 2) 31 3) 10100₂ 4) 1100100₂ 5) 11

Quick recap

• A number base sets how many digits are used and the place values.
• Binary (base 2) uses 0 and 1; places double: 1, 2, 4, 8, …
• Bits and bytes store all computer data.