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Understanding number systems is essential for OCR J277 Section 2.6 (Data Representation). Computers store and process all data using binary (base-2), but humans normally use denary (base-10, also called decimal). You must be able to convert fluently between these two systems.
The denary number system uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Each position in a denary number represents a power of 10.
| Position | Thousands (10^3) | Hundreds (10^2) | Tens (10^1) | Units (10^0) |
|---|---|---|---|---|
| Example: 4527 | 4 | 5 | 2 | 7 |
| Value | 4000 | 500 | 20 | 7 |
4527 = (4 x 1000) + (5 x 100) + (2 x 10) + (7 x 1) = 4527
This is the system you use every day, so it feels natural. Computers, however, use binary because their electronic circuits can only distinguish between two states: on (1) and off (0).
The binary number system uses only two digits: 0 and 1. Each position represents a power of 2. A single binary digit is called a bit.
| Position | 128 (2^7) | 64 (2^6) | 32 (2^5) | 16 (2^4) | 8 (2^3) | 4 (2^2) | 2 (2^1) | 1 (2^0) |
|---|---|---|---|---|---|---|---|---|
| Example: 10110011 | 1 | 0 | 1 | 1 | 0 | 0 | 1 | 1 |
To convert to denary: 128 + 32 + 16 + 2 + 1 = 179
Use the successive division method (divide by 2 and record remainders):
Example: Convert 156 to binary
| Division | Quotient | Remainder |
|---|---|---|
| 156 / 2 | 78 | 0 |
| 78 / 2 | 39 | 0 |
| 39 / 2 | 19 | 1 |
| 19 / 2 | 9 | 1 |
| 9 / 2 | 4 | 1 |
| 4 / 2 | 2 | 0 |
| 2 / 2 | 1 | 0 |
| 1 / 2 | 0 | 1 |
Read remainders from bottom to top: 156 = 10011100
Alternatively, use the place value method: find the largest power of 2 that fits and subtract:
OCR Exam Tip: The place value method is usually quicker in the exam and less prone to error. Always double-check by converting your binary answer back to denary.
Write out the place values and add up the columns where there is a 1.
Example: Convert 11010110 to denary
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
|---|---|---|---|---|---|---|---|
| 1 | 1 | 0 | 1 | 0 | 1 | 1 | 0 |
128 + 64 + 16 + 4 + 2 = 214
| Reason | Explanation |
|---|---|
| Electronic circuits | Transistors have two states: on (1) and off (0) |
| Reliability | It is easy to distinguish between two voltage levels, reducing errors |
| Simplicity | Binary arithmetic is simpler to implement in hardware |
| Boolean logic | Binary maps directly to true/false logic operations |
| Term | Definition |
|---|---|
| Bit | A single binary digit (0 or 1) |
| Nibble | 4 bits |
| Byte | 8 bits |
| Denary | Base-10 number system (0-9) |
| Binary | Base-2 number system (0, 1) |
| Place value | The value of a digit based on its position |
OCR Exam Tip: The maximum value for an n-bit binary number is 2^n - 1. For 8 bits, this is 2^8 - 1 = 255. You should memorise this.