What is a base?
When there are many things, we count faster by making groups. The size of one group is called the base.
In our usual system the base is 10. Ten ones make one ten. Ten tens make one hundred. Each place is worth 10 times the place on its right.
347 means 3 × 100 + 4 × 10 + 7 × 1.
This only works because we have a sign for zero: 307 shows an empty tens place.
Grouping in other bases
Nothing forces us to use ten. In base 5, five ones make one "five", five fives make one "twenty-five", and so on. The places are 1, 5, 25, 125 …
Only the digits 0, 1, 2, 3, 4 are needed. (5 would already make a new group.)
Base b to base 10
Multiply each digit by its place value and add. 243₅ = 2 × 25 + 4 × 5 + 3 × 1 = 73.
Base 10 to base b
Divide by b again and again. Write the remainders from bottom to top. 73 ÷ 5 = 14 r 3; 14 ÷ 5 = 2 r 4; 2 ÷ 5 = 0 r 2. So 73 = 243₅.
Try it: take 23 beans. Make cups of 5. How many full cups, how many left over? You have written 23 in base 5.
Base 2: the language of computers
In base 2 (binary) there are only two digits: 0 and 1. The places are 1, 2, 4, 8, 16, 32 … each double the last.
13 = 8 + 4 + 1 = 1101₂. 10110₂ = 16 + 4 + 2 = 22.
Why do computers use it? A tiny switch inside a chip can be off (0) or on (1). Two states are easy to make and hard to confuse. Thousands of millions of switches together store numbers, letters, pictures and sound.
Base 60: an ancient system we still use
About 4000 years ago, people in ancient Mesopotamia (the Babylonians) counted in base 60. Their places were 1, 60, 3600 …
Why 60? It divides evenly by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20 and 30. That makes halves, thirds and quarters easy.
We still use it every day:
- 60 seconds = 1 minute, 60 minutes = 1 hour (3600 seconds).
- A full turn is 360°, and 1° = 60 minutes of arc.
2 h 5 min 30 s = 2 × 3600 + 5 × 60 + 30 = 7530 seconds. That is the number (2, 5, 30) in base 60.
Key formulas and definitions
- Place values in base b: 1, b, b², b³ …
- Digits allowed in base b: 0 to b − 1
- (d₂d₁d₀)_b = d₂ × b² + d₁ × b + d₀
- Base 10 → base b: divide by b, read remainders bottom to top
Worked examples
1. Write 341 as place values in base 10.
341 = 3 × 100 + 4 × 10 + 1 × 1.
2. Convert 1101₂ to base 10.
Places from the right: 1, 2, 4, 8. 1 × 8 + 1 × 4 + 0 × 2 + 1 × 1 = 13.
3. Convert 312₅ to base 10.
Places: 25, 5, 1. 3 × 25 + 1 × 5 + 2 × 1 = 75 + 5 + 2 = 82.
4. Convert 25 to base 2.
25 ÷ 2 = 12 r 1; 12 ÷ 2 = 6 r 0; 6 ÷ 2 = 3 r 0; 3 ÷ 2 = 1 r 1; 1 ÷ 2 = 0 r 1. Read upwards: 11001₂. Check: 16 + 8 + 1 = 25.
5. Convert 50 to base 5.
50 ÷ 5 = 10 r 0; 10 ÷ 5 = 2 r 0; 2 ÷ 5 = 0 r 2. So 50 = 200₅. Check: 2 × 25 = 50.
6. How many seconds are 1 h 20 min 15 s? Write the time in base 60 too.
1 × 3600 + 20 × 60 + 15 = 3600 + 1200 + 15 = 4815 s. In base 60 it is (1, 20, 15).
Common mistakes
- Using the digit 5 in base 5, or 2 in base 2. In base b the largest digit is b − 1.
- Reading the remainders from top to bottom. Read them from the last division up.
- Thinking 10₂ means ten. 10₂ means one 2 and no ones, which is 2.
- Forgetting zeros in empty places: 9 = 1001₂, not 11₂.