Data storage — OCR GCSE Computer Science

Denary is base 10, using digits 0–9. Binary is base 2, using only 0 and 1. An 8-bit binary number uses place values 128, 64, 32, 16, 8, 4, 2 and 1. To convert denary to binary, subtract the largest place value that fits, put 1 in that column, and repeat with the remainder until zero, filling unused columns with 0. For example, 157 = 128 + 16 + 8 + 4 + 1, so 157 in 8-bit binary is 10011101. To convert binary to denary, add the place values where the bit is 1. For example, 00101101 = 32 + 8 + 4 + 1 = 45. Always write exactly 8 bits when the question asks for 8-bit binary.

Binary addition operates column by column from the least significant bit (rightmost), similar to denary addition but using only 0 and 1. The single-bit rules are: 0+0=0, 0+1=1, 1+0=1, 1+1=10₂ (write 0, carry 1), and 1+1+1=11₂ (write 1, carry 1). For example, adding 0110 1101₂ and 0011 0111₂: starting at the right, 1+1=10₂ (write 0, carry 1); next 0+1+1=10₂ (write 0, carry 1); then 1+1+1=11₂ (write 1, carry 1); continuing leftwards yields 1010 0100₂. In an 8-bit system, the maximum representable value is 255. If an addition exceeds this (e.g., 200+100=300), a carry is generated out of the most significant bit (the 128s column). Because there is no ninth bit to store this carry, it is discarded, causing an overflow error. The stored result is incorrect, specifically 256 less than the true sum (e.g., 300 becomes 44).

Hexadecimal is base 16, using the digits 0–9 and the letters A–F, where A = 10, B = 11, C = 12, D = 13, E = 14 and F = 15. A 2-digit hexadecimal number has place values 16¹ = 16 and 16⁰ = 1, so it represents values from 0 to 255. To convert denary to hexadecimal, divide the number by 16: the quotient is the first digit and the remainder is the second. For example, 202 ÷ 16 = 12 remainder 10, so 202 = CA₁₆. To convert hexadecimal to denary, multiply the first digit by 16, multiply the second digit by 1, then add: CA₁₆ = 12 × 16 + 10 × 1 = 192 + 10 = 202.

Hexadecimal is base 16, so each hex digit stands for a group of four binary bits (a nibble). To convert binary to hex, split the binary integer into nibbles from the right, pad the leftmost group with leading zeros if needed, then replace each nibble with its hex digit: 1010 becomes A, 1111 becomes F. For example 11011011 splits as 1101 1011, giving DB. To convert hex to binary, do the reverse: write each hex digit as its four-bit pattern and join them, so 2F becomes 0010 1111, or 101111 after dropping leading zeros. This works because 16 = 2⁴, so four binary place values map exactly onto one hex place value.

A binary shift moves every bit in a binary pattern a stated number of places left or right. A left shift by n places multiplies the value by 2ⁿ, because each bit moves to a place value n positions higher. A right shift by n places divides the value by 2ⁿ, discarding any bits shifted past the right-hand end and losing fractional parts. For example, 00010110 (22) shifted left by 2 gives 01011000 (88), which is 22 × 2². The same pattern shifted right by 1 gives 00001011 (11), which is 22 ÷ 2. Bits shifted out of the fixed-width register are lost, and zeros fill the vacated positions. This means a left shift can cause an overflow error if significant bits are lost, changing the value unexpectedly, while a right shift causes a loss of precision.

Computers store characters as binary codes because a byte can hold only 0s and 1s. A character set defines which code maps to which character. For example, in ASCII the letter A is decimal 65, which is 01000001 in 8-bit binary. To encode a word, each character is converted to its code and the codes are stored in order. To decode, the binary is split into equal-length codes and each code is looked up in the character set. The same binary pattern can represent different characters in different character sets, so the correct character set must be known. Unicode uses more bits, allowing many more characters from different languages and symbols.

A character set is a defined list of characters and the unique binary code assigned to each one. It allows a computer to convert between binary and text. For example, in ASCII the character A is code 65, which is 01000001 in 8-bit binary. The character set must be known by both the sender and receiver; otherwise the same binary code may be displayed as a different character. ASCII is a small character set, while Unicode is a much larger character set that includes characters from many languages and many symbols. The size of a character set affects how many bits are needed for each code.

A character set maps each character to a unique binary code. If a set uses n bits per character, each character is one n-bit pattern, so the number of distinct patterns is 2ⁿ. With 7 bits, 2⁷ = 128 characters can be represented, which covers standard ASCII. With 8 bits, 2⁸ = 256 characters, covering extended ASCII. Unicode uses more bits per character, so it can represent far more characters, including symbols and scripts from many languages. To work out capacity, identify the bits per character, then evaluate 2ⁿ. To work out bits needed for a required number of characters, find the smallest n where 2ⁿ is at least that number. For example, 300 characters need 9 bits because 2⁸ = 256 is too few and 2⁹ = 512 is enough.

A bitmap image is a grid of pixels. Each pixel stores one colour value as a binary code. The number of bits per pixel is the colour depth, so n bits per pixel allow 2ⁿ different colours. For example, 1 bit per pixel gives 2 colours, 3 bits gives 8 colours, and 8 bits gives 256 colours. The image file stores the binary codes for the pixels, usually row by row. Image resolution is the number of pixels in the width and height. File size in bits can be estimated by multiplying width by height by colour depth. For example, a 10 by 10 pixel image at 3 bits per pixel needs 10 × 10 × 3 = 300 bits. More pixels or more bits per pixel increase file size and can improve quality, but use more storage.

Metadata is data about data: it describes a file's content, format, origin, and properties so that software and users can interpret and manage it. In a bitmap image, metadata may include width and height in pixels, colour depth, resolution, file format, creation date, camera settings, and geolocation. For a sound file, it may include sample rate, bit depth, duration, and artist. Metadata does not usually form the main content, but without it the raw data may be meaningless. For example, a bitmap stores pixel values, but the image viewer needs metadata to know how many pixels wide the image is and how many bits represent each colour. Metadata also supports searching, sorting, copyright tracking, and automatic processing. In exams, you should explain what metadata is, give relevant examples, and describe how it is used, rather than simply listing file properties.

In a bitmap image, resolution is the number of pixels, often expressed as width × height. Colour depth is the number of bits used to represent each pixel's colour. Increasing resolution adds more pixels, allowing finer detail, but increases file size. Increasing colour depth allows more distinct colours, improving colour accuracy, but also increases file size as each pixel needs more bits. Uncompressed file size in bits is calculated as width × height × colour depth. For example, a 1000 × 800 image at 24-bit colour depth has 19,200,000 bits. Dividing by 8 gives 2,400,000 bytes, which is 2.4 MB (using base-10 conversions: 1000 bytes = 1 kB, 1000 kB = 1 MB). Higher resolution or colour depth improves quality but requires more storage.

Sound is a continuous pressure wave, so a computer cannot record every instant of it. Instead, a microphone converts the wave into a varying analogue voltage, and an analogue-to-digital converter measures that voltage at regular intervals. Each measurement is a sample, and the number of samples taken each second is the sample rate, measured in hertz (Hz). Each sample is rounded to the nearest available level, and the number of bits used per sample is the bit depth. The resulting stream of binary values is stored as a file, for example a WAV file. Playback reverses the process: a digital-to-analogue converter turns each stored value back into a voltage, and a speaker produces the sound. A higher sample rate and bit depth capture the wave more accurately but create a larger file.

Three factors control the size of a sound file, while two of these also determine the quality of played-back sound. Sample rate is how many samples are taken each second, measured in hertz (Hz); a higher rate captures rapid changes in the wave more accurately. Bit depth is the number of bits per sample; a greater depth gives more amplitude levels, so each sample is rounded less coarsely and quiet detail is preserved. Duration is how long the sound lasts. File size in bits is calculated as sample rate × bit depth × duration in seconds × number of channels. Doubling any of these three factors roughly doubles the file size. Higher sample rates and bit depths improve playback quality, but all three factors increase storage and processing demands.