Number Systems and Binary Arithmetic

Back to Digital Electronics
Arjit Sharma
01 Min

Computers use binary (0s and 1s) to store and process numbers. Understanding number systems and binary math is key to grasping how data works in digital systems.

Common Number Systems

SystemBaseDigits Used
Binary20, 1
Octal80–7
Decimal100–9
Hexadecimal160–9, A–F

Computers use binary for calculations, hexadecimal for compact code, and decimal for human readability. Conversions between them are common.

Binary Arithmetic

Binary math follows rules similar to decimal but uses only 0 and 1.

Addition Rules

0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 10 (carry 1)

Subtraction Rules

0 - 0 = 0
1 - 0 = 1
1 - 1 = 0
0 - 1 = 1 (borrow 1)

Multiplication and Division

Like decimal, but simpler (e.g., 1·1 = 1, 1·0 = 0). Example: 11 (3) × 10 (2) = 110 (6).

Signed Numbers

Computers represent positive and negative numbers using different methods:

  • Unsigned: Only positive numbers
  • Sign-Magnitude: First bit is the sign (0 = +, 1 = -)
  • 1’s Complement: Invert all bits to get negative
  • 2’s Complement: Invert and add 1 (most widely used)

Why 2s Complement ?

  • Single Zero: Unlike sign-magnitude or 1’s complement, which have +0 (0000) and -0 (1000 or 1111), 2’s complement has only one zero (0000), making comparisons easier.
  • Seamless Addition: Addition works the same for positive and negative numbers, no special circuits needed for subtraction. Example: To subtract, add the 2’s complement (e.g., 3 – 2 = 3 + (–2)).
  • No End-Around Carry: 1’s complement requires adjusting carries, which complicates hardware. 2’s complement handles carries naturally.
  • Efficient Range: For n bits, it represents numbers from –2^(n-1) to +2^(n-1)–1 (e.g., 4 bits: –8 to +7), maximizing usable values.

Example (2’s complement of 5):

00000101 → Invert → 11111010 → Add 1 → 11111011 = -5

Next →