Binary and denary

Denary (base 10) uses digits 0–9. Binary (base 2) uses only 0 and 1. Each binary digit is a bit. In an 8-bit number the place values double from right to left:

Method: to convert binary → denary, add the place values under each 1. To convert denary → binary, subtract the biggest place value you can, put a 1 there, repeat.

Your turn — binary to denary

Denary to binary

Hexadecimal

Hexadecimal (base 16) uses digits 0–9 then A–F for 10–15. One hex digit = exactly 4 bits (a nibble), so hex is a short, human-friendly way to write binary.

Fast method: convert each 4-bit nibble to one hex digit. 1010 = 10 = A, 1101 = 13 = D. To go the other way, expand each hex digit back into 4 bits.

Your turn — hex to denary

Binary addition

Binary addition uses four rules: 0+0=0, 0+1=1, 1+1=10 (0 carry 1), 1+1+1=11 (1 carry 1).

Overflow: if a result needs more bits than are available (e.g. a 9th bit for an 8-bit number), an overflow error occurs — the extra carried bit is lost, so the answer is wrong.

Your turn — binary addition

Binary shifts

Shifting the bits sideways multiplies or divides by powers of 2:

• Left shift by n places = multiply by 2ⁿ (zeros fill in on the right).
• Right shift by n places = integer divide by 2ⁿ (bits fall off the right).

Careful with right shifts: because bits drop off the end, dividing an odd number rounds down (integer division) — e.g. 7 right-shifted by 1 gives 3, not 3.5.

Your turn — a left shift

Units of storage

The smallest unit is the bit (a single 0 or 1). Bigger units build up:

• 1 nibble = 4 bits · 1 byte = 8 bits
• 1 kibibyte (KiB) = 1024 bytes
• 1 mebibyte (MiB) = 1024 KiB · 1 gibibyte (GiB) = 1024 MiB · 1 tebibyte (TiB) = 1024 GiB

Note: each step up multiplies by 1024 (2¹⁰). In everyday use people often say "kilobyte = 1000 bytes", but the exact binary units go up in 1024s.

Match each amount to its unit

Tap an amount on the left, then its matching unit on the right.

Characters & ASCII

Text is stored as numbers using a character set. In ASCII, each character has a 7-bit code (stored in a byte), so it can represent 128 different characters.

• The codes are in order, so 'A' = 65, 'B' = 66, … and 'a' = 97.
• Unicode uses more bits, so it can represent far more characters — including other alphabets and emoji.

Handy fact: because codes run in sequence, if 'A' is 65 then 'C' is 65 + 2 = 67. Digit characters like '0'–'9' also have their own codes (they are not the same as the numbers 0–9).

Your turn — ASCII codes

Storing images

A bitmap image is a grid of pixels. Each pixel's colour is stored as a binary number. The number of bits per pixel is the colour depth.

• 1 bit per pixel → 2 colours (black/white). 8 bits → 256 colours. 24 bits → about 16.7 million colours.
• Resolution = the number of pixels (width × height). More pixels and higher colour depth = better quality but bigger file.

File size (bits) ≈ width × height × colour depth. Doubling the colour depth roughly doubles the file size.

Colour depth

Storing sound

Sound is analogue, so it is sampled: the height of the wave is measured many times a second and each measurement stored as a binary number.

• Sample rate — how many samples per second (measured in hertz). Higher rate = more accurate sound, bigger file.
• Bit depth (sample resolution) — how many bits store each sample. More bits = finer detail.

Trade-off: higher sample rate and bit depth give better sound quality but larger files. This is why streaming services offer different quality settings.

Compression

Compression makes files smaller so they use less storage and transfer faster. Two types:

• Lossless — no data is lost; the original can be perfectly restored. Used for text and program files. (e.g. run-length / ZIP-style methods.)
• Lossy — some data is permanently removed to shrink the file more; quality drops slightly and it can't be fully restored. Used for photos, music and video (e.g. JPEG, MP3).

Choosing: use lossless when every bit matters (a spreadsheet or program). Use lossy when a small quality loss is acceptable in exchange for a much smaller file (a holiday photo).

Lossy or lossless?

Tap a statement, then tap where it belongs. (Some are true of both.)

Which compression?

Bits per hex digit

What is overflow?

The big ideas to know

You've covered the whole of Edexcel Data representation. Press Finish to see your score.