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Basic Maths for Computing

Computers represent and process information using the binary number system, which contains only two digits: 0 and 1.
This section explains how different number systems work, how conversions between them are performed, and how arithmetic operates in binary form.

Number Systems

A number system defines how digits represent values using a specific base (radix).
Each position represents a power of the base.

\[ N = d_n \times b^n + d_{n-1} \times b^{n-1} + \dots + d_0 \times b^0 \]
Base System Digits Used Example Common Use
2 Binary 0, 1 101101₂ Digital circuits, CPUs
8 Octal 0–7 45₈ Compact binary notation
10 Decimal 0–9 37₁₀ Everyday arithmetic
16 Hexadecimal 0–9, A–F 2AF₁₆ Memory addressing, machine code

Positional Weighting

graph LR
  subgraph Binary_Representation
  B1["1 × 2⁵ = 32"]
  B2["0 × 2⁴ = 0"]
  B3["1 × 2³ = 8"]
  B4["1 × 2² = 4"]
  B5["0 × 2¹ = 0"]
  B6["1 × 2⁰ = 1"]
  end

Each binary digit (bit) corresponds to a specific power of two.

Converting Between Number Systems

Conversion between bases is essential for understanding how computers encode values.

Decimal to Binary

Use repeated division by 2 and record remainders (bottom to top).

Example: Convert \( 37_{10} \) to binary.

Step Division Quotient Remainder
1 37 ÷ 2 18 1
2 18 ÷ 2 9 0
3 9 ÷ 2 4 1
4 4 ÷ 2 2 0
5 2 ÷ 2 1 0
6 1 ÷ 2 0 1

Reading remainders upward:

\[ 37_{10} = 100101_2 \]

Binary to Decimal

Multiply each bit by its corresponding power of 2 and sum the results.

\[ 101101_2 = (1\times2^5) + (0\times2^4) + (1\times2^3) + (1\times2^2) + (0\times2^1) + (1\times2^0) \]
\[ = 32 + 8 + 4 + 1 = 45_{10} \]

Binary to Hexadecimal

Group bits in sets of 4 from right to left.

\[ 10111101_2 = BD_{16} \]
Binary Hex
1011 B
1101 D 
graph TD
  A[Binary: 10111101]
  B1["1011 → B"]
  B2["1101 → D"]

Hexadecimal to Decimal

Each digit represents a power of 16.

\[ 2AF_{16} = (2 \times 16^2) + (10 \times 16^1) + (15 \times 16^0) \]
\[ = 512 + 160 + 15 = 687_{10} \]

Binary Arithmetic

Binary arithmetic uses the same logic as decimal arithmetic, but carries occur at base 2.

Addition

A B Carry In Sum Carry Out
0 0 0 0 0
0 1 0 1 0
1 0 0 1 0
1 1 0 0 1

For Example:

\[ 1011_2 + 0110_2 = 10001_2 \]
graph LR
  A["Input A"] --> S["Sum = A ⊕ B ⊕ Cin"]
  B["Input B"] --> S
  C["Carry In"] --> S
  A --> Cout["Carry Out = AB + AC + BC"]
  B --> Cout
  C --> Cout

Subtraction Using Two’s Complement

Subtraction is performed by adding the two’s complement of the subtrahend.

\[ A - B = A + (\lnot B + 1) \]

Example: \( 0101_2 - 1000_2 \)

  1. Invert B: \( \lnot 1000 = 0111 \)
  2. Add 1: \( 0111 + 1 = 1000 \)
  3. Add to A: \( 0101 + 1000 = 1101_2 \)
\[ 1101_2 = -3_{10} \]

Binary Multiplication

Performed similarly to long multiplication in decimal.

\[ \begin{array}{r} \phantom{\times}\ 101\\ \times\ 110\\ \hline \ \ \ 000\\ \ \ 1010\\ \ 10100\\ \hline \ 11110_2 \end{array} \]
\[ 11110_2 = 30_{10} \]

Binary Division

Binary division is repeated subtraction with shifting.

\[ 11101_2 \div 11_2 = 1001_2 \]

Quotient: \( 1001_2 \) Remainder: \( 0_2 \)

Representing Negative Numbers

Unsigned binary cannot represent negative values.
To handle signed numbers, computers use the most significant bit (MSB) as a sign bit.

Representation Range (4-bit) Method Notes
Signed Magnitude ±0 … ±7 MSB = sign Simple but has two zeros
One’s Complement ±0 … ±7 Invert bits for negative End-around carry
Two’s Complement −8 … +7 Invert bits then add 1 Modern standard

Two’s Complement Process

graph TD
  A["Start with positive value"]
  B["Invert all bits"]
  C["Add 1 to result"]
  D["Result is negative representation"]
  A --> B --> C --> D

Example: represent −6 in 4 bits

\[ +6 = 0110,\quad \lnot0110 = 1001,\quad 1001 + 1 = 1010 \]
\[ 1010_2 = -6_{10} \]

Comparison of Representations

Property Signed Magnitude One’s Complement Two’s Complement
Zero forms +0 / −0 +0 / −0 Single zero
Arithmetic simplicity Poor Partial Excellent
Range (4 bit) −7 to +7 −7 to +7 −8 to +7
Hardware usage Obsolete Rare Universal

Practice Problems

  1. Convert \( 101011_2 \) to decimal
  2. Convert \( 3A_{16} \) to binary
  3. Add \( 1010_2 + 0111_2 \)
  4. Subtract \( 1011_2 - 0010_2 \)
  5. Represent −6 in 4-bit two’s complement

Hint:
$$ +6 = 0110 \Rightarrow \lnot0110 = 1001 \Rightarrow 1001 + 1 = 1010 $$

Summary

  • Binary is the foundation of all digital systems.
  • Two’s complement allows signed arithmetic with one zero.
  • Conversions between number systems are essential for computer architecture.
  • Bit width defines numeric range and overflow behaviour.

References