Binary, denary and hexadecimal

A number base is just a set of column values. Denary uses powers of 10, binary powers of 2, hexadecimal powers of 16 (digits 0–9 then A=10, B=11, C=12, D=13, E=14, F=15).

Denary → binary: work left to right through the column headings, taking each one you can. 217 − 128 = 89; 89 − 64 = 25; 32 is too big (0); 25 − 16 = 9; 9 − 8 = 1; 4 and 2 too big; 1 − 1 = 0. Result 11011001.

Binary → hex: split into nibbles of 4 bits and convert each: 1101 = 13 = D, 1001 = 9. So 11011001 = D9. Check: D9 = 13 × 16 + 9 = 208 + 9 = 217 ✓

Why hex exists: it is human shorthand for binary. One hex digit = exactly 4 bits, so a byte is always 2 hex digits. It is shorter, far less error-prone to read, and maps back to binary with no arithmetic. Memory addresses, colour codes (#FF8800) and MAC addresses are all written in hex.

Units: AQA uses decimal prefixes — 1 kB = 10³ bytes, 1 MB = 10⁶, 1 GB = 10⁹, 1 TB = 10¹². The binary prefixes are separate: 1 KiB = 2¹⁰ = 1024 bytes, 1 MiB = 2²⁰ bytes.

Your turn — denary to binary

Your turn — binary to hex

Two's complement

To store negative numbers, the most significant bit is given a negative weight. In 8 bits:

To negate a number: write the positive value in binary, flip every bit, then add 1.

Range in n bits: from −2ⁿ⁻¹ to +2ⁿ⁻¹ − 1. For 8 bits that is −128 to +127 — note the asymmetry: there is one more negative value than positive, because 0 uses up a positive-looking pattern.

Why two's complement won: there is only one representation of zero (sign-and-magnitude has +0 and −0), and subtraction becomes addition: A − B is computed as A + (−B) using the ordinary adder circuit. One piece of hardware does both.

Your turn — negate in two's complement

Your turn — two's complement to denary

Fixed point binary fractions

Put a binary point in a fixed position and the columns to its right carry negative powers of 2:

With a fixed point, the number of bits before and after the point is decided in advance. That fixes both the range and the precision — and you cannot trade one for the other.

The fatal flaw: many everyday denary fractions have no exact binary representation. 0.1 in binary is 0.0001100110011… recurring forever. Truncating it introduces a tiny error — which is precisely why 0.1 + 0.2 does not equal 0.3 in most languages, and why money is stored in whole pence.

Floating point and normalisation

Floating point stores a number in two parts, in the form:

The mantissa carries the significant digits (and so sets the precision); the exponent says where the binary point goes (and so sets the range). For a fixed total word length, giving bits to one takes them from the other — that is the central trade-off.

Normalisation squeezes the maximum precision out of the mantissa by removing leading redundant bits:

• a positive normalised mantissa begins 0.1…
• a negative normalised mantissa begins 1.0…

In both cases the first two bits differ. The exponent is adjusted to compensate for each shift, so the value is unchanged.

Your turn — floating point to denary

Rounding errors, range and precision

Because a mantissa has finitely many bits, most reals can only be stored approximately. The gap between the true value and the stored one is a rounding error — and errors accumulate across millions of operations.

The relative error is usually the meaningful one: an absolute error of 1 km is catastrophic in surveying and irrelevant in astronomy.

The bug that costs lives: in 1991 a Patriot missile battery tracked time as a 24-bit fixed-point count of tenths of a second. 0.1 has no exact binary form, so after 100 hours of running, the accumulated rounding error was about a third of a second — enough to misplace an incoming missile by half a kilometre. 28 people died. Rounding error is not an academic curiosity.

Range and precision

Character sets and error checking

ASCII uses 7 bits for 128 characters (often stored in a byte with the top bit spare). It cannot represent Greek, Arabic, Chinese or an emoji. Unicode was created to fix exactly that: it assigns a unique code point to every character in every script, needing 16 bits or more per character.

Data in transit gets corrupted. Four standard defences:

• Parity bit — one extra bit set so the total number of 1s is even (even parity) or odd. It detects any single-bit error but cannot say where, and misses any two flipped bits.
• Majority voting — send each bit three times and take the most common value. It corrects as well as detects, at triple the bandwidth.
• Checksum — an arithmetic total sent with the data and recomputed on arrival. Detects, but does not locate or correct.
• Check digit — a digit computed from the others and appended (ISBNs, barcodes). Catches typing and transcription errors.

Detect vs correct: parity, checksums and check digits only detect — the data must be re-sent. Majority voting and other error-correcting codes can repair the data in place. That matters enormously for a spacecraft that cannot simply be asked to re-transmit.

Parity

Bitmapped images and sampled sound

A bitmapped image is a grid of pixels. Resolution = pixels across × down. Colour depth = bits per pixel, giving 2ⁿ possible colours (24-bit gives 2²⁴ ≈ 16.7 million).

A vector image instead stores objects (line, circle, fill) with properties. It scales to any size with no loss of quality and is usually far smaller — but it is hopeless for a photograph.

Sound is a continuous analogue wave, sampled by an ADC. Sampling rate (Hz) = samples per second; sample resolution = bits per sample.

Nyquist: to reproduce a frequency f you must sample at at least 2f. Human hearing tops out near 20 kHz — which is exactly why CD audio samples at 44.1 kHz. MIDI stores no sound at all: it stores instructions (note, pitch, velocity, duration), so it is tiny and endlessly editable, but it can never capture a real singer.

Your turn — image file size

Compression

Lossless compression discards nothing: the original is restored bit for bit. Essential for text, code and executables — a program with a few bits missing is not a program.

• Run-length encoding (RLE) — replace runs of identical values with (value, count). AAAAABBB → A5B3. Excellent on large flat areas; it can enlarge a file with no runs, since every single character becomes a pair.
• Dictionary-based (LZW) — build a dictionary of repeated sequences and send short index codes instead. The dictionary is rebuilt by the decoder, so it need not always be transmitted.

Lossy compression permanently throws data away, exploiting the limits of human perception — MP3 removes frequencies you cannot hear, JPEG discards fine colour detail your eye ignores. It achieves far higher compression ratios, and the discarded data is gone forever.

Exam trap: RLE is lossless, even though it is used on images. Being used on media does not make a method lossy — what matters is whether the original can be reconstructed exactly.

Encryption and hashing

Caesar cipher — shift each letter by a fixed amount. Trivially broken: there are only 25 keys, and frequency analysis cracks it instantly because E remains the commonest letter.

Vernam cipher (one-time pad) — XOR the plaintext with a truly random key that is at least as long as the message and is used exactly once. It is the only cipher that is mathematically unbreakable: every possible plaintext of that length is equally likely. Decryption is the same operation again, because XOR is its own inverse.

Hashing is not encryption. A hash is one-way — it is designed to be impossible to reverse. Passwords are stored as hashes so that a stolen database reveals nothing usable. Encryption is designed to be reversed by whoever holds the key; hashing is designed never to be reversed at all.

The Vernam cipher

Compression

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The big ideas to know

That is the whole of AQA 4.5. Press Finish to see your score.