Number Systems and Codes
Computers store everything as binary (0 and 1) because electronic circuits have two stable states, on and off.
| System | Base | Digits used |
|---|---|---|
| Binary | 2 | 0, 1 |
| Octal | 8 | 0 to 7 |
| Decimal | 10 | 0 to 9 |
| Hexadecimal | 16 | 0 to 9 and A to F (A = 10, B = 11, C = 12, D = 13, E = 14, F = 15) |
Decimal to binary (divide by 2)
Divide repeatedly by 2 and write the remainders from bottom to top.
Example: 45 to binary. 45 ÷ 2 = 22 r 1; 22 ÷ 2 = 11 r 0; 11 ÷ 2 = 5 r 1; 5 ÷ 2 = 2 r 1; 2 ÷ 2 = 1 r 0; 1 ÷ 2 = 0 r 1. Read upward: 101101.
Check: 32 + 8 + 4 + 1 = 45.
Binary to decimal (place values)
Multiply each bit by a power of 2 and add. Example: 1101 = 8 + 4 + 0 + 1 = 13.
Fractions: the digits after the point have weights 1/2, 1/4, 1/8. So 1101.101 = 13 + 0.5 + 0 + 0.125 = 13.625. To convert a decimal fraction, multiply by 2 and note the integer parts: 0.625 × 2 = 1.25 (1), 0.25 × 2 = 0.5 (0), 0.5 × 2 = 1.0 (1), so 0.625 = 0.101.
Binary, octal and hexadecimal shortcuts
- Binary to octal: group bits in threes from the right. Binary to hex: group in fours.
- Example: 10110111 → octal groups 10 | 110 | 111 = 267; hex groups 1011 | 0111 = B7. In decimal both equal 183.
- Hex to decimal: 2F = 2 × 16 + 15 = 47. Octal to decimal: 17 = 1 × 8 + 7 = 15.
Binary arithmetic
| Rule | Result |
|---|---|
| 0 + 0 | 0 |
| 0 + 1 and 1 + 0 | 1 |
| 1 + 1 | 10 (0 carry 1) |
| 1 + 1 + 1 | 11 (1 carry 1) |
Example: 1011 + 0110 = 10001 (11 + 6 = 17).
Negative numbers
| Method | How |
|---|---|
| Sign and magnitude | First bit is the sign (0 positive, 1 negative). |
| 1's complement | Invert every bit. |
| 2's complement | 1's complement + 1. Used by computers. |
Example, -5 in 8 bits: 5 = 00000101 → invert = 11111010 → add 1 = 11111011. An 8-bit two's complement number ranges from -128 to +127.
Character codes
| Code | Facts |
|---|---|
| BCD | Binary Coded Decimal. Each decimal digit in 4 bits (8421). |
| ASCII | American Standard Code for Information Interchange. 7-bit = 128 characters; extended ASCII is 8-bit = 256. |
| EBCDIC | 8-bit code used on IBM mainframes. |
| Unicode | Covers all world languages, including Urdu, Hindi and Kashmiri scripts. UTF-8, UTF-16, UTF-32. |
| Gray code | Only one bit changes between successive values. |
ASCII values worth knowing: A = 65, Z = 90, a = 97, z = 122, 0 = 48, space = 32. Lower case is 32 more than upper case.
Keep practising
Notes are only half the work. Test yourself with questions and discuss doubts with other aspirants.
Prepared for exam revision from standard computer textbooks, Microsoft documentation and official sources. Features differ between Excel and Word versions (2016, 2019, 2021, 365) so check which version your exam follows. Found a mistake? Tell us.