Binary and number bases · Grade 11
In Grade 11 you learn: binary arithmetic.
We count in tens because we have ten fingers, but any whole number can be written in any base. Computers use base 2 (binary), and programmers use base 16 (hexadecimal) as a compact way to write binary. Converting between bases is all about place value.
Key facts
- In base b the columns are worth 1, b, b², b³, … from the right, and the digits run from 0 to b − 1.
- To convert to denary, multiply each digit by its column value and add.
- From denary, divide repeatedly by the base. The remainders, read from the bottom up, are the digits.
- Hexadecimal digits are 0–9 then A = 10, B = 11, C = 12, D = 13, E = 14, F = 15. Each hex digit is exactly 4 bits.
- n bits can store 2ⁿ different values, from 0 to 2ⁿ − 1.
Worked examples
Example 1
Convert 200 to binary and to hexadecimal.
Answer: 11001000₂ = C8₁₆