1. What is it?

A number system is a way of writing numbers using a fixed set of digits. The system you use every day is denary (also called decimal): it has ten digits, 0 to 9, so it is base 10. Computers use binary: it has only two digits, 0 and 1, so it is base 2. A third system, hexadecimal (base 16), is used by people as a short way to write binary (see topic 1.05).

2. Why do computers use binary?

Every computer is built from millions of tiny electronic switches called transistors. A switch has only two reliable states: on and off. We write on as 1 and off as 0. It is much easier and far more reliable to tell two states apart (voltage present / voltage absent) than ten different voltage levels, which would be confused by electrical noise. Everything a computer stores or processes — numbers, text, pictures, sound, video, programs — is therefore turned into patterns of 1s and 0s.

3. Place value: how a binary number is worth what it is worth

In denary each column is worth ten times the one to its right (1, 10, 100, 1000). In binary each column is worth two times the one to its right. An 8-bit number has these place values:

The bit on the far left is the most significant bit (MSB) because it is worth the most; the bit on the far right is the least significant bit (LSB). Binary numbers are normally written in groups of four bits (nibbles) to make them readable: 1001 1100.

4. Converting binary to denary

5. Converting denary to binary

There are two accepted methods. Use whichever you make fewer mistakes with, and always write your working.

Method A — subtract place values (fastest for 8-bit numbers)

Start at 128. If the place value fits into what is left, write 1 and subtract it; otherwise write 0.

Method B — repeated division by 2

Divide by 2 again and again, writing the remainder each time; the binary number is the remainders read from the bottom up.

6. How many bits are needed? What is the biggest number?

With n bits you can make 2ⁿ different patterns, so the values run from 0 up to 2ⁿ − 1.

7. Try the converter

8. Real-world context

• A calculator or car-park machine stores each number you type in a binary register before it calculates.
• A cost such as $14.98 can be stored as two 8-bit registers, one for the dollars (14 = 0000 1110) and one for the cents (98 = 0110 0010).
• Every pixel, every letter and every sound sample eventually ends up as a binary number like these.

9. How Cambridge tests it

• Convert denary to 8-bit (or 12-bit) binary and back, usually 2–3 numbers for 2–3 marks: one mark per correct number.
• State/Explain what is meant by binary: “a number system that uses only two digits, 0 and 1 / base 2” (2 marks: two digits and base 2).
• Explain why data is converted to binary: computers are made of switches/transistors with two states; only binary can be processed by the CPU; all data (text, images, sound) must be in binary to be stored and processed.

10. Check yourself

1. What is it?

Computers add binary numbers with exactly the same column method you use for denary numbers, but because there are only two digits the rules are very short. The result of a calculation must fit back into the register it is stored in. If it does not, an overflow error happens.

2. The four rules of binary addition

Always start at the right-hand (least significant) column and work left, carrying into the next column.

3. Worked example with no overflow

4. Worked example that overflows

Second overflow example: 1100 1100 + 0101 1101 = 204 + 93 = 297 = 1 0010 1001 (9 bits) → overflow.

5. What actually happens when overflow occurs?

The CPU's arithmetic logic unit (ALU) produces a 9-bit result but the destination register has only 8 bits, so the extra left-most bit is lost or a flag is set. The value left in the register (0010 1001 = 41) is wrong. Unless the program checks for it, the calculation silently gives an incorrect answer; that is why overflow matters in banking, engineering and game scores.

6. How Cambridge tests it

• “Add the two 8-bit binary numbers… Show your working” — usually 3–4 marks: marks for the correct sum, for the visible carries, and for showing the ninth bit.
• “Explain why an overflow error occurs”: the result of the calculation is larger than the largest value that can be stored in the register; the number needs more bits (a 9th bit) than are available.

7. Check yourself

1. What is it?

A logical shift moves every bit in a register a number of places to the left or right. Bits that are shifted out of the register are lost, and the empty places that appear at the other end are filled with zeros.

2. Step by step

3. The arithmetic effect (why programmers use shifts)

• A left shift of n places multiplies by 2ⁿ (each place is worth twice as much).
• A right shift of n places divides by 2ⁿ and throws away the remainder (integer division).
• Shifts are extremely fast for the CPU, so they are used in graphics, games and encryption.

4. When accuracy is lost

5. How Cambridge tests it

• Give the 8-bit result of a stated shift (usually 1 mark). Show the shifted bits and keep 8 bits.
• Describe the process: “bits are moved left/right, left-most/right-most bits are lost, 0s are added at the opposite end.”
• Give the denary value of the result, or the effect (× or ÷ 2ⁿ, with the loss of precision).

6. Check yourself

1. What is it?

Ordinary binary can only store zero and positive numbers. Two’s complement is the number system that lets a register store positive and negative whole numbers. In it the left-most bit has a negative place value.

For 8 bits the place values are:

• If the left-most bit is 0 the number is positive (0 to 127).
• If the left-most bit is 1 the number is negative (−128 to −1).
• Range of 8-bit two’s complement: −128 to +127.

2. Reading a two’s complement number (binary → denary)

Add the place values as normal, but the first column counts as −128.

3. Making a negative number (denary → two’s complement)

A second method that gives the same answer: copy the bits from the right up to and including the first 1, then invert everything to the left of it.

4. Why is it used?

The CPU can subtract by adding the negative number, using the same adder circuit, so no extra subtraction hardware is needed.

5. Special values to remember

6. Real-world context

A digital thermometer stores the temperature as a two’s complement integer so it can display −14 °C as well as 32 °C. Bank balances, game scores and audio samples also use signed representations.

7. How Cambridge tests it

• “State the method used to represent negative integers in binary” → two’s complement (1 mark).
• Convert a two’s complement number to denary (e.g. 1110 1010 → −22) or a denary number to 8-bit two’s complement (e.g. −14 → 1111 0010, 14 → 0000 1110).
• Show working: the mark scheme rewards the positive binary, the inversion and the +1.

8. Check yourself

1. What is it?

Hexadecimal is not used inside a computer (the computer only stores binary). It is used by people because one hex digit stands for exactly four bits (a nibble), which makes long binary numbers short.

2. Binary ↔ hexadecimal (the core skill)

Binary to hex

Hex to binary

Replace every hex digit by its 4-bit pattern (keep the leading zeros in each group).

3. Hex ↔ denary

Hex to denary — multiply by place values (…256, 16, 1)

Denary to hex — divide by 16, or convert to binary first

4. Why do programmers use hexadecimal instead of binary?

Give two of these in an exam (each is a separate mark):

• It is shorter — 1 hex digit replaces 4 bits, so an 8-bit number needs 2 digits, a 32-bit number only 8.
• It is easier to read, remember and write down than a long string of 0s and 1s.
• It is less error-prone to copy or type — fewer digits to get wrong.
• It is easy to convert to and from binary because each hex digit is exactly one nibble.

5. Where hexadecimal is used

6. How many bits does a hex number need?

Every hex digit = 4 bits. The hex number A4D has 3 digits, so it needs 3 × 4 = 12 bits. (The least number of bits is 12, as in the Paper 1 question on this.)

7. How Cambridge tests it

• Binary → hex and hex → binary for 8-bit and 12-bit values (1 mark each number).
• Hex → denary and denary → hex (2 marks for two numbers).
• “Give two reasons why a programmer may use hexadecimal” and “give one similarity/two differences between hexadecimal and binary”.
• Examples of use: colour codes, MAC addresses, error messages, memory locations.

8. Check yourself

1. What is it?

A computer cannot store the letter “A”; it can only store binary numbers. To store text, every character (letter, digit, punctuation mark, space, symbol) is given its own unique binary number. The full list of characters and their numbers is called a character set.

2. How does a keypress become binary? (step by step)

3. Useful ASCII values (memorise the pattern)

Two patterns you can use in the exam:

• Letters are in alphabetical order, so if ‘c’ = 99, then ‘a’ = 97 and ‘t’ = 116 (t is 17 letters after c).
• A lower-case letter is exactly 32 more than its capital: ‘T’ = 84 = 0101 0100, ‘t’ = 116 = 0111 0100. In binary you add 0010 0000 (32) — the only bit that changes is the sixth bit.

4. Why was Unicode needed?

ASCII has only 128 (or 256) characters. That is enough for English but not for Chinese, Arabic, Hindi, Amharic, mathematical symbols or emojis. Different countries created different extended sets that clashed. Unicode gives every character in every language one unique number.

5. Advantages and disadvantages of Unicode over ASCII

• Advantages: represents many more characters and every language; one standard so files display correctly on any device; includes emojis and symbols; the first 128 codes match ASCII so old text still works.
• Disadvantage: each character uses more bits, so files are larger and need more storage/transmission time.

6. Real-world context

An email address containing accented or non-Latin letters (for example a name written in Cyrillic or Arabic script) needs Unicode. Phones, websites and social media all use Unicode so people can mix languages and emojis in one message.

7. How Cambridge tests it

• Explain how text is converted to binary: each character has a code in a character set (e.g. ASCII/Unicode); the code is converted to binary; the binary codes are stored in sequence.
• Identify another character set besides ASCII → Unicode (or extended ASCII).
• Differences between ASCII and Unicode: bits per character, number of characters, languages supported.
• Calculations from a given code: “‘c’ is 99, give the codes for ‘a’ and ‘t’”, or “explain how to get lower-case ‘t’ from ‘T’”.

8. Check yourself

1. What is it?

Sound is made of vibrations in the air. The strength of the vibration at any moment is called its amplitude, and the wave changes smoothly and continuously, so it is an analogue signal. A computer can only store numbers, so the wave must be converted into a digital form. This is done by sampling.

2. Step by step: how a recording is made

3. Sample rate and sample resolution

• Sample rate: 44 100 Hz (CD quality) means 44 100 measurements every second. A telephone call uses about 8 000 Hz, which is why phone audio sounds thin.
• Sample resolution: with n bits each sample can be one of 2ⁿ values. 8 bits = 256 levels; 16 bits = 65 536 levels.

4. Sound file size

File size (bits) = sample rate × sample resolution × duration (seconds) (× number of channels for stereo). Divide by 8 for bytes, then by 1 024 for KiB.

5. Real-world context

Music streaming services choose a balance: high sample rate and resolution sound better but cost more data. Voice-message apps use lower values to save mobile data. A “lossless” music file keeps every sample; an MP3 removes detail humans are unlikely to hear (see 1.11).

6. How Cambridge tests it

• Explain how analogue sound is converted to digital: the wave is sampled at regular intervals; the amplitude of each sample is measured; each value is stored as a binary number.
• State two ways the accuracy could be increased: increase the sample rate; increase the sample resolution.
• Define sample rate and sample resolution (see key terms above — number of samples per second / number of bits per sample).
• Benefit and drawback: benefit — better quality/more accurate; drawback — bigger file, more storage, longer to transmit.

7. Check yourself

1. What is it?

A bitmap image is a grid of tiny squares called pixels. Each pixel stores one binary number that says what colour it is. The computer draws the picture by colouring every pixel in the grid.

2. How it works

3. Colour depth: how many colours?

With n bits per pixel you can have 2ⁿ different colours.

4. Effect of changing resolution and colour depth

5. Image file size

File size (bits) = width × height × colour depth. Divide by 8 for bytes. (Metadata is ignored unless the question says to include it.)

More calculations are in topic 1.10.

6. Real-world context

A phone camera photo is millions of pixels at 24-bit colour, so a raw photo is several MiB. Websites reduce resolution and use compression so pictures load quickly on mobile data.

7. How Cambridge tests it

• State what is meant by resolution / colour depth / pixel / metadata (1 mark each).
• “Give one benefit of increasing the colour depth”: more colours can be represented, so the image looks more realistic/higher quality.
• “Explain the effect on the image of reducing the colour depth to 8 bits”: fewer colours (256) are available → less accurate colours/banding, but a smaller file size.
• Complete a table of image terms (pixel, resolution, colour depth, metadata).

8. Check yourself

1. What is it?

All data is stored as bits, but we group them into bigger units so that sizes are easy to talk about. This course uses the IEC binary units (kibibyte, mebibyte…), in which each step is 1 024 (2¹⁰) times bigger than the one before.

The order, smallest to largest, is bit < nibble < byte < KiB < MiB < GiB < TiB < PiB < EiB.

2. Why 1 024 and not 1 000?

Computers count in powers of 2, and 2¹⁰ = 1 024, which is very close to 1 000. Manufacturers and the SI system use the decimal prefixes kilo/mega/giga (1 000-based), which is why a “500 GB” disk shows as about 465 GiB in the operating system. The names with “bi” in them (kibi, mebi, gibi) were invented to make the 1 024 meaning unambiguous.

3. Converting between units

4. How Cambridge tests it

• Tick the largest unit from a list such as exbibyte, gibibyte, mebibyte, tebibyte (answer: exbibyte).
• How many KiB in a MiB (1 024), bits in a byte (8), nibbles in 2 bytes (4).
• Convert 3 072 bytes to KiB, 8 GiB to MiB, or complete a table of conversions.
• Name the unit that is 1 024 times larger than a kibibyte (mebibyte) and then the next one (gibibyte).

5. Check yourself

1. What is it?

A file-size question asks you to work out how much storage a picture or a sound needs. Every such question has the same structure: find the number of bits, convert to bytes, then convert to the unit that the question asks for.

2. The two formulas

Then: bytes = bits ÷ 8, KiB = bytes ÷ 1 024, MiB = KiB ÷ 1 024.

3. Method

4. Worked examples

5. Try it yourself

6. Common mistakes

7. Real-world context

A 4-minute stereo song at CD quality is about 40 MiB uncompressed; a 12-megapixel 24-bit photo is about 34 MiB uncompressed. Compression (1.11) is what makes music libraries and photo albums fit on a phone.

8. How Cambridge tests it

• 2–3 mark calculations from a given image or sound description, with “show all your working”.
• “Give the size in bits” of a file quoted in bytes: multiply by 8 (1 000 bytes = 8 000 bits).
• How many bits are needed to store a value or file (see 1.01), linked to compression questions (“give two benefits of reducing file size”).

9. Check yourself

1. What is it?

Why compress? (Any of these earn marks.)

• It saves storage space, so more files fit on a device or server.
• It reduces transmission time and uses less bandwidth, so a page, email or video loads or uploads faster.
• It reduces cost (cloud storage and data usage).
• Files can be emailed within attachment-size limits.

2. Two types of compression

3. How lossy compression reduces a file

For images: reduce the resolution and/or reduce the colour depth, and merge similar neighbouring colours. For sound: reduce the sample rate and/or sample resolution, and remove sounds outside human hearing. The removed data cannot be restored.

4. Run-length encoding (RLE) — a lossless method

RLE looks for runs (repeats) of the same value and replaces each run with one count + one value.

RLE is very effective on simple images with large blocks of one colour (logos, icons) and gives little or no saving on data with few repeats such as a photograph. To decompress, expand each count/value pair back into a run — nothing is lost.

5. Which type should be used?

6. How Cambridge tests it

• State what is meant by compression; two benefits of compressing (less storage, faster transfer).
• Explain the difference between lossy and lossless (data permanently removed vs no data lost; original cannot vs can be recovered).
• Choose a type for a scenario and justify it.
• Explain how lossless compression works (RLE: repeated data is replaced by a value and a count; nothing is removed).
• Explain how a photo is reduced in size: lower resolution / colour depth (lossy).

7. Check yourself

1. What is it?

Large files are not sent across a network in one piece. The data is broken into small chunks called packets, each sent independently, and put back together at the destination. This method of sending data is called packet switching.

2. Structure of a packet

3. Step by step: sending a file with packet switching

4. Why is packet switching used?

• Efficient use of the network: the lines are shared by many users’ packets, not tied up by one long transmission.
• Resilience: if one route is broken or congested, packets go another way.
• Security: an eavesdropper who intercepts one route only gets some of the packets, not the whole message.
• Errors are cheap to fix: only the damaged packet is re-sent, not the whole file.
• Drawbacks: packets can arrive out of order and must be reassembled; each packet carries header/trailer overhead; delays and jitter can affect live calls and games.

5. The role of the router

A router does not open or read the payload; it only reads the header. It (1) examines the destination IP address, (2) chooses the next hop from its routing table (this may consider the number of hops or time-to-live, and how busy each link is), and (3) forwards the packet. If a packet has exceeded its hop limit it is discarded, which stops packets looping forever.

6. Real-world context

Web pages, email, video calls and file downloads all travel as packets. Your home router is one of many routers a packet passes through on its way to a server that might be in another country.

7. How Cambridge tests it

• Name the three parts of a packet: header, payload, trailer, and give examples of what is in each (header: destination address, packet number, originator’s address; payload: the data; trailer: end-of-packet notification, error checking).
• Identify the device that controls the path each packet takes: router.
• Explain how a file is transmitted using packet switching (splitting, different routes, router chooses route, arrive out of order, reordered when last packet arrives, missing packets re-requested).
• Complete or annotate a diagram of packet switching between two devices via routers.
• Fill in a paragraph using terms such as packets, router, destination address, packet number, reordered.

8. Check yourself