ZIMSEC O Level · Mathematics · Form 2
We normally count in base 10 (denary). Computers use base 2 (binary). Other bases follow the same idea: each place value is a power of the base.
Place values
In base 5, the place values are … 125, 25, 5, 1 (powers of 5). Only the digits 0 to 4 can be used.
Converting to base 10
Multiply each digit by its place value and add.
Worked example: Convert 1324₅ to base 10.
1 × 125 + 3 × 25 + 2 × 5 + 4 × 1 = 125 + 75 + 10 + 4 = 214
Converting from base 10
Divide repeatedly by the new base and write down the remainders. Read the remainders from the bottom up.
Worked example: Convert 45 to base 2.
45 ÷ 2 = 22 r 1; 22 ÷ 2 = 11 r 0; 11 ÷ 2 = 5 r 1; 5 ÷ 2 = 2 r 1; 2 ÷ 2 = 1 r 0; 1 ÷ 2 = 0 r 1
Reading upwards: 45 = 101101₂
Adding in other bases
Add as usual, but “carry” when you reach the base. In base 2: 1 + 1 = 10₂.
Worked example: Calculate 101₂ + 111₂.
Right column: 1 + 1 = 10, write 0 carry 1. Middle: 0 + 1 + 1 = 10, write 0 carry 1. Left: 1 + 1 + 1 = 11.
Answer: 1100₂
Check yourself
- Convert 213₄ to base 10.
- Convert 100 to base 8.
- Find the value of 34₅ + 23₅, giving your answer in base 5.
Show answers
- 2×16 + 1×4 + 3 = 39
- 144₈
- 112₅
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