Math & Education

How Positional Number Systems Work Across Bases

Every positional number system — decimal (base 10), binary (base 2), octal (base 8), hexadecimal (base 16) — represents a number as a sum of digit values multiplied by increasing powers of the base, so the only thing that changes between bases is how many unique digit symbols are available and what power each position represents; converting between bases is simply re-expressing the same underlying quantity using a different base's place values.

Once you see decimal as just one specific case of this general pattern, converting between bases stops feeling like a separate skill for each one.

The pattern decimal already follows

In decimal, the number present already works this way without anyone thinking about it: present in present means (3 × 100) + (4 × 10) + (7 × 1) — each digit's contribution depends on both its own value and the power of ten tied to its position. Binary, octal, and hex simply swap the base from 10 to 2, 8, or 16, and use powers of that base instead of powers of ten.

Converting by hand between any two bases

The most reliable manual method is to convert through decimal as a middle step: to go from any base to decimal, multiply each digit by its place value (a power of the source base) and sum the results; to go from decimal to any other base, repeatedly divide by the target base and read the remainders in reverse order.

Frequently asked questions

Why does hexadecimal use letters A through F?

Hexadecimal is base 16, which needs 16 unique digit symbols — after using 0 through 9, there aren't enough decimal digits left, so the convention extends the digit set with A through F to represent the values 10 through 15 in a single character.

Is there a fastest way to convert between binary and hexadecimal specifically?

Yes — since 16 is a power of 2 (2^4), each hex digit corresponds exactly to a group of 4 binary digits, so you can convert directly by splitting a binary number into groups of 4 (from the right) and converting each group to its single hex digit, without needing to go through decimal at all.