Understand the decimal, binary and hexadecimal number systems, why computers use binary and hexadecimal, and how to convert between the three bases.

What this dot point is asking

AQA wants you to describe the decimal, binary and hexadecimal number systems, explain why computers use binary and why hexadecimal is convenient, and convert numbers between the three bases reliably under exam time.

The three number systems

A positional number system gives each column a weight equal to a power of the base. In binary the column weights from the right are $1, 2, 4, 8, 16, 32, 64, 128$ and so on, each double the one before. In hexadecimal the weights are $1, 16, 256, 4096$. The same idea (digit times place value, summed) recovers the denary value in any base, which is the single rule that underpins every conversion you will be asked to do.

Why binary and hexadecimal

Hexadecimal does not change how the machine stores data; the hardware is still binary. Hex is purely a notation that programmers and the exam use because it compresses binary by a factor of four without any awkward arithmetic, since the base is a power of two.

Converting between bases

Binary to decimal: write the place values above the bits and add the values where there is a 1. So $1011_2 = 8 + 0 + 2 + 1 = 11_{10}$.

Decimal to binary: repeatedly divide by 2, recording remainders, then read the remainders from bottom to top. So $13 \div 2$ gives quotient 6 remainder 1, then 3 r 0, then 1 r 1, then 0 r 1, read upward as $1101_2$. An equivalent method is place-value subtraction: take out the largest power of two that fits, set that bit, and repeat with the remainder.

Binary and hexadecimal: split the binary into groups of four bits from the right (pad the left with zeros if needed) and convert each group to one hex digit. So $1011\,1110_2 = \text{BE}_{16}$, and each hex digit expands back to four bits. Hexadecimal to decimal multiplies each digit by its power of 16: $\text{2F}_{16} = 2 \times 16 + 15 = 47_{10}$.