Data Representation

Inside every computer are millions of tiny electronic switches called transistors. A transistor can only be in one of two states: it is either letting current through (ON) or it is not (OFF). We represent these two states with the digits 1 (ON) and 0 (OFF).

Because hardware naturally has two states, the simplest and most reliable way to store and move data is to use a number system with just two digits. That system is binary. Any form of data, whether it is a number, a letter, an image or a sound, must first be converted into binary so the computer can process it.

Two more terms you must know: data is processed using logic gates (covered fully in Chapter 10) and is held inside the CPU in tiny stores called registers (Chapter 3).

The syllabus expects you to know three number systems and to convert between them. The key idea behind any system is its base: the number of different digits it uses.

Key vocabulary

An 8-bit binary number has a fixed set of place values, each one double the value to its right, starting from 1 on the far right:

Binary to denary

Wherever there is a 1, add that column's place value. Ignore the columns with a 0.

Denary to binary (repeated division by 2)

Divide the number by 2 over and over, writing down the remainder each time. Then read the remainders from bottom to top.

Long strings of binary are hard for humans to read and easy to copy wrongly. Hexadecimal solves this: one hex digit represents exactly 4 binary bits, so hex is four times shorter than binary while still mapping cleanly onto it.

Why hexadecimal is useful

• It is much shorter than binary, so it is easier to read and write
• It is less error-prone when humans copy values by hand
• It converts to and from binary very easily (4 bits per digit)

You will meet hex again in MAC addresses, IP addresses (IPv6), colour codes in HTML, and memory addresses.

Why do programmers use hexadecimal?

Programmers in particular reach for hexadecimal because a single hex digit is equivalent to exactly 4 bits of binary. Compressing every 4 bits into one symbol makes long binary values:

• Easier to read, because the value is a quarter of the length
• Easier to understand, since each digit maps cleanly to one nibble
• Easier to debug, as mistakes are far quicker to spot in a short hex string than in a long row of 1s and 0s

Seeing it at scale: 32 bits

The benefit becomes obvious with longer values. Here is a 32-bit binary number split into its four bytes (eight nibbles), with the matching hexadecimal underneath. Notice how 32 digits of binary collapse into just 8 digits of hex.

11011001101001111111000011010110  =  D9A7F0D6

Binary to hexadecimal

Split the binary into groups of 4 bits, working from the right. Convert each nibble to its hex digit.

The 16 hex values to memorise

It pays to know these by heart. Each row shows one hex digit, the nibble it stands for, and its denary value.

Hexadecimal to binary

Reverse the process: turn each hex digit into its 4-bit binary nibble and join them together.

Hexadecimal to denary and back

To go from hex to denary, multiply each digit by its place value (powers of 16: ..., 256, 16, 1). To go from denary to hex, the easiest route is usually denary to binary, then binary to hex.

Adding binary works just like adding denary, except you carry whenever a column total reaches 2 (not 10). The four rules are:

Overflow

An 8-bit register can only hold values from 0 to 255 (that is 11111111). If an addition produces a result larger than 255, there is no room for the extra bit. This is called an overflow error: a 9th bit is generated but cannot be stored, so it is lost and the stored answer is wrong.

A logical shift moves all the bits in a register left or right by a given number of places. Empty positions are filled with zeros, and any bits pushed off the end are lost permanently.

So far every binary number has been positive. To store negative numbers in 8 bits, we use two's complement. The trick: the most significant bit (the leftmost, normally worth 128) is given a negative place value of −128.

Converting a positive number to negative

Three steps: write the positive number in 8-bit binary, flip every bit (this is called the one's complement), then add 1.

Reading a two's complement number

Add up the place values as normal, but remember the leftmost column is worth −128.

Computers store text by giving every character a unique number, called a character code. A list of characters and their codes is a character set.

ASCII

• Uses 7 bits per character, giving 128 different characters (codes 0 to 127)
• Extended ASCII uses 8 bits, giving 256 characters
• Covers the English alphabet, digits, punctuation and control characters
• Useful codes to know: 'A' = 65, 'a' = 97, '0' = 48

Unicode

• Uses more bits per character than ASCII
• Can represent over 100,000 characters
• Supports the alphabets of all world languages, plus symbols and emojis
• Is backward compatible with ASCII (the first 128 codes are identical)

Sound in the real world is an analogue wave, a smooth continuous signal. Computers can only store discrete binary numbers, so the wave must be converted. This is done by sampling: measuring the height (amplitude) of the wave at regular intervals and storing each measurement as a binary number.

A digital image (a bitmap) is made of a grid of tiny squares called pixels. The colour of each pixel is stored as a binary number.

Data size is measured in bytes, and each larger unit is 1024 times the one before it (not 1000). These are the formal "binary" units the syllabus requires.

These calculations come up almost every year. Learn the two formulas and always show your working.

Image file size

Sound file size

Compression reduces the size of a file. Smaller files need less storage space, use less bandwidth, and transfer faster across a network. There are two types.

Run length encoding (RLE)

RLE is a lossless method that replaces runs of repeated values with a single value plus a count. For example, a row of pixels WWWWWWBBB becomes 6W 3B, which takes far less space while losing nothing.

Try each question before reading the answer. These mirror the style and mark allocations of the real paper.