In simple terms
A friendly intro before the formal notes — no formulas yet.
Flipping the Switches
Computers represent all information using binary, a system of only 0s and 1s. This is like a huge bank of light switches, where each switch is either off (0) or on (1), and patterns of these switches represent numbers, letters, and colours.
Imagine you need to count items but you can only use your hands. You could raise a finger for each item, but you'd run out at ten. A more efficient system is to assign a value to each finger: your thumb is 1, index finger is 2, middle finger is 4, ring finger is 8, and little finger is 16. By raising combinations of fingers (e.g., index and thumb for '3'), you can represent any number up to 31 on one hand. Binary works exactly like this, but with 'on' (1) and 'off' (0) states instead of raised or lowered fingers.
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First, understand that our normal numbers are 'denary' or 'base-10', with place values like 1, 10, 100 (powers of 10).
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Binary is 'base-2'. Its place values are powers of 2: 1, 2, 4, 8, 16, 32, 64, 128, and so on, from right to left.
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To convert a denary number to binary, find the largest power of 2 that fits inside it, place a '1' in that position, subtract it, and repeat with the remainder.
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To convert a binary number to denary, simply add up the place values for every position that contains a '1'.
Explore the concept
Use the live diagram, PhET or GeoGebra sim, and synced steps — play it, drag controls, or tap a step.
Place value Each bit is worth double the one to its right, so the place values that are switched on add up to the denary number.
3 more simulations for this topic — run them in the Simulations section below
Simulations
Every simulation here runs the real model — try the steps on a card, then check what you see against the notes.
3 simulations
- GeoGebraCore
8-bit binary place-value table
Tap a bit to flip it and watch the place values add up to the decimal total. Press +1 again and again to see a byte count upwards from 0.
D Stark · GeoGebra · GeoGebra Terms of Service
- GeoGebraCoreIB 2.3
RGB colour mixer, 0–255 per channel
Three sliders, R, G and B, each showing a value (128 to start), and a disc that shows the colour those three values make.
Try this
- Drag R to its maximum and G and B to 0 for pure red, then mix yellow, cyan and magenta.
- Set all three equal and move them together from lowest to highest: black, through greys, to white.
- Write the colour on the disc as three bytes in binary: that 24-bit pattern is what is stored for one pixel.
Look for A pixel in 24-bit colour is three numbers, one byte each for red, green and blue: 256 × 256 × 256, about 16.7 million colours.
Ricardo Misturini · GeoGebra · GeoGebra Terms of Service
- GeoGebraCoreIB 2.3
Character and Unicode code converter
Four boxes: type a character to get its code, a code to get its character, a word to get its list of codes, or a list of codes to get text.
Try this
- In Letter to Unicode type A, then a. Note 65 and 97: the cases differ by 32, a single bit.
- Type the digit 7. Its code is 55, not 7: the character and the number are stored differently.
- In Text to Unicode type your name and read one code per character.
Look for Text is stored as numbers. Every character has a fixed code, and the first 128 Unicode codes are the ASCII set, so ASCII text is also valid Unicode.
Steve Phelps · GeoGebra · GeoGebra Terms of Service
Key formulas
Tap any symbol to reveal exactly what it means and its units.
Tap a symbol — great for exam definitions
Full topic notes
Formal explanation with the rigour you need for the exam.
Understanding Number Systems
We are accustomed to the denary (or base-10) number system, which uses ten digits (0-9). The position of a digit determines its value based on powers of 10. For example, the number 345 is really . Computers use the binary (or base-2) system, which works on the same principle but with only two digits (0 and 1) and place values based on powers of 2.
Binary Place Values (right to left): Which are:
Converting Between Denary and Binary
Being able to convert numbers between denary and binary is a crucial skill. To convert from denary to binary, you can use the subtraction method. To convert from binary to denary, you sum the place values of all positions that hold a '1'.
Representing Other Data
Binary isn't just for numbers. Every piece of data is stored as a binary pattern. For text, computers use character sets like ASCII or Unicode, which are essentially dictionaries that map each character (like 'A', 'b', or '£') to a unique binary number. For example, in ASCII, the capital letter 'A' is represented by the denary number 65, which is in binary.
All data in a computer is stored in binary (base-2).
A bit is a single 0 or 1; a byte is a group of 8 bits.
An 8-bit byte can represent any integer from 0 to 255.
Conversion from denary to binary often uses repeated subtraction from the largest place value.
Conversion from binary to denary involves summing the place values where a '1' is present.
Character sets like ASCII and Unicode map characters to specific binary codes.
Worked examples
See the formulas applied — reveal one step at a time, like the exam.
Convert the denary number into an 8-bit binary integer. Show your working.
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Is ? Yes. Place a '1'. Remainder: .
Convert the 8-bit binary number into its denary equivalent. Show your working.
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We write down the binary number and the corresponding place values for each bit. Then, we sum the place values where the bit is '1'.
How it all connects
The big idea sits in the middle — tap a linked idea to explore the link.
Tap a linked idea to see how it connects back to the main topic — that connection is what examiners reward.
Glossary
Key terms for this topic — skim now; the Check step will test them.
- Bit
The smallest unit of data in a computer, representing a single binary value of either 0 or 1. Short for 'binary digit'.
- Byte
A group of 8 bits. A byte is the standard unit of measurement for digital information and can represent 256 different values (from 0 to 255).
- 'denary' number system
The base-10 number system we use every day. It uses ten digits (0-9) and place values that are powers of 10 (1, 10, 100, etc.).
- 'binary' number system
The base-2 number system used by computers. It uses two digits (0 and 1) and place values that are powers of 2 (1, 2, 4, 8, etc.).
- largest denary number
- This is when all bits are '1' (11111111), which equals . A common mistake is to think it's 256.
- Character set
A collection of characters and their corresponding binary codes that a computer can represent. Examples include ASCII and Unicode.
- ASCII
American Standard Code for Information Interchange. An early 7-bit (later 8-bit) character set that represents 128 (or 256) characters, mostly for English text and control characters.
- Unicode
A modern, universal character set designed to represent every character from every language. It uses variable-length encoding, often 16 or 32 bits per character, allowing for over a million unique characters.
Name it
Read the meaning, then pick which of this lesson’s terms it describes. Miss one and you see what your choice really means.
A modern, universal character set designed to represent every character from every language. It uses variable-length encoding, often 16 or 32 bits per character, allowing for over a million unique characters.
Quick check
Write your answer first, then compare it with the model one — the gap is what you would have lost.
Teach it back
If you can explain it simply, you own it — gaps here are marks you’d lose.
Teach it back
Explain this topic as if teaching a friend. We name the gaps an examiner would still dock.
Revision flashcards
Guess first, then flip — retrieval beats re-reading.
Key takeaways
Review these before you close the topic — retrieval beats re-reading.
All data in a computer is stored in binary (base-2).
A bit is a single 0 or 1; a byte is a group of 8 bits.
An 8-bit byte can represent any integer from 0 to 255.
Conversion from denary to binary often uses repeated subtraction from the largest place value.
Conversion from binary to denary involves summing the place values where a '1' is present.
Character sets like ASCII and Unicode map characters to specific binary codes.
Practice — then mark it
The whole point: a real Cambridge question, marked mark-by-mark.
Test Your Knowledge on Binary Representation
Test Your Knowledge on Binary Representation
Extra simulations & links
PhET, GeoGebra and other curated tools — open in a new tab.
Frequently asked
Checkpoint
One marked question is worth ten re-reads — close the loop before you move on.
Reading it isn’t knowing it — prove it.
Before you move on: do Test Your Knowledge on Binary Representation on paper, snap a photo, and get examiner-style feedback on exactly where you win and lose marks.
Discuss Binary representation of data
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