Binary, denary & hexadecimal

Computers store all data as binary. AQA needs you to know three number bases:

• Binary — base 2. Only two digits: 0 and 1. Each digit is a bit.
• Denary (decimal) — base 10. The everyday numbers we count in, using digits 0–9.
• Hexadecimal — base 16. Uses digits 0–9 then letters A–F, where A=10, B=11, C=12, D=13, E=14, F=15.

Why hex? One hex digit represents exactly 4 bits (a nibble), so hex is a short, human-friendly way to write long binary numbers — 2 hex digits = 1 byte (8 bits).

Binary ↔ denary (8-bit place values)

Each bit in an 8-bit number has a place value. To convert binary to denary, add up the place values wherever there is a 1:

Denary → binary: work left to right, subtracting the biggest place value you can each time (put a 1), else put a 0. E.g. 45 = 32+8+4+1 → 00101101.

Your turn — binary to denary

Working with hexadecimal

Each hex digit is worth 4 bits (a nibble), so you can convert binary ↔ hex one nibble at a time, and hex ↔ denary using place values 16 and 1.

• Hex → denary: multiply the left digit by 16, add the right digit. E.g. 2F = (2×16) + 15 = 32 + 15 = 47. (F = 15.)
• Denary → hex: divide by 16. E.g. 156 = 9 remainder 12 → 9 and C → 9C.
• Binary ↔ hex: split the 8 bits into two nibbles. E.g. 1001 1100 → 9 and C → 9C.

Remember the letters: A=10, B=11, C=12, D=13, E=14, F=15. One byte (8 bits) always fits in exactly 2 hex digits.

Your turn — hex to denary

Binary addition

Add binary the same way you add denary, but you carry whenever a column reaches 2. The rules are:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 1 = 0 carry 1
• 1 + 1 + 1 = 1 carry 1

Overflow: if the answer needs more than 8 bits (a carry out of the leftmost column), an overflow error occurs — the byte can't hold the result.

Your turn — add in binary

Binary shifts

A logical shift moves every bit left or right by a number of places, filling the gap with 0s:

• Left shift by 1 place multiplies by 2. Shift by n places → × 2ⁿ.
• Right shift by 1 place divides by 2 (whole-number division). Shift by n places → ÷ 2ⁿ.

Careful with right shifts: bits shifted off the right-hand end are lost, so information (odd amounts / precision) can be lost when dividing.

Your turn — shift left

Bits, bytes and beyond

Data size is measured in units built up from the single bit:

• Bit — a single 0 or 1.
• Nibble — 4 bits.
• Byte — 8 bits.

Bigger units go up in thousands (AQA uses decimal prefixes, powers of 1000):

Watch out: AQA GCSE uses 1 kB = 1000 bytes (kilobyte, megabyte, gigabyte, terabyte), not 1024. Learn the order: bit → nibble → byte → kB → MB → GB → TB.

How big is a byte?

ASCII & Unicode

Text is stored by giving each character a number (a character code). Two character sets matter:

• ASCII — a 7-bit code, so it can represent 2⁷ = 128 characters (enough for English letters, digits and punctuation).
• Unicode — uses more bits, so it can represent a far larger set of characters, covering the world's languages and emoji.

Character codes: in ASCII, capital 'A' = 65 and lowercase 'a' = 97. Letters are in order, so 'B' = 66, 'C' = 67, and so on. The gap between 'A' and 'a' is always 32.

Which character set?

Bitmap images

A bitmap image is a grid of tiny dots called pixels. Two properties describe it:

• Resolution — the number of pixels (width × height). More pixels = more detail.
• Colour depth — the number of bits per pixel. Number of colours = 2^{colour depth} (so 1 bit = 2 colours, 8 bits = 256 colours).

Trade-off: more pixels or more colours make a nicer image but a bigger file.

Your turn — image file size

Representing sound

Sound is analogue (a continuous wave). To store it, the computer takes samples — measuring the wave's height at regular intervals:

• Sampling rate — how many samples per second are taken (measured in Hz). Higher rate = more accurate.
• Sample resolution — the number of bits per sample. More bits = finer detail.

Trade-off: a higher sampling rate or resolution gives better-quality sound but a larger file.

Lossy vs lossless

Compression makes a file smaller so it takes less storage and transfers faster. There are two types:

• Lossy — permanently removes some data to shrink the file (you can't get it back). Quality drops, but files are much smaller. e.g. JPEG images, MP3 audio.
• Lossless — no data is lost; the original can be perfectly restored. e.g. run-length encoding (RLE) and ZIP.

RLE (a lossless method) replaces runs of the same value with a count + value, e.g. "AAAA" → "4A". Great for images with large areas of one colour.

Which compression type?

Which number base?

Tap a description, then tap the number base it belongs to.

Match term to meaning

Tap a description on the left, then its matching term on the right.

The big ideas to know

You've covered all of AQA 3.3 — number bases, conversion, binary arithmetic, units, characters, images, sound and compression. Press Finish to see your score.