The rule
each digit is its symbol × base^position
Why it works
A base is just how many symbols you count with before carrying. In base 10 the digit 3 in the hundreds column means 3 × 10², and in base 2 the digit 1 in the fourth column means 1 × 2³. The idea is identical; only the base changes.
How to do it by hand
- To go to decimal, multiply each digit by the base raised to its position and add
- To leave decimal, divide repeatedly by the target base
- Keep the remainders and read them backwards
- In hexadecimal, A to F stand for 10 to 15
What is worth knowing
Hexadecimal exists because it maps onto binary exactly: each hex digit is precisely four bits, so FF is 11111111 and no arithmetic is needed to see it. Octal did the same job for three-bit groupings on older hardware, which is why Unix file permissions are still octal. Binary itself is not a computing invention — Leibniz described it in 1703, inspired partly by the I Ching — but it is what makes digital circuits possible, since a switch has exactly two states.