Computer Arithmetic – Notes on Computer Architecture

Computer arithmetic is a vital part of computer architecture, focusing on how a computer performs mathematical operations like addition, subtraction, multiplication, and division using binary numbers. Every digital system—from a calculator to a supercomputer—relies on these arithmetic operations.

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1. Data Representation

Computers represent all data in binary form (0s and 1s) because digital circuits operate using two voltage levels (ON and OFF).

• Binary Number System: Uses only two digits, 0 and 1.
  Example: Decimal 5 = Binary 101.

• Octal (base 8) and Hexadecimal (base 16) are also used to simplify binary notation.

• 1 byte = 8 bits (smallest storage unit).

Layman Example:
Think of “1” as the light switch ON and “0” as OFF. Complex numbers are built by combining these ON/OFF states.

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2. Provision for Decimal Position

In binary arithmetic, numbers can be integer or fractional.

• Binary Point: Similar to the decimal point in base-10.
  Example: Binary 101.1 = Decimal 5.5.

• Position after the binary point determines the fraction (each bit represents 1/2, 1/4, 1/8, etc.).

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3. Provision for Sign Representation

Computers need a method to represent positive and negative numbers.

• Sign-Magnitude: Leftmost bit is the sign bit. (0 for +, 1 for -)
  Example: +5 = 0101, -5 = 1101

• 1’s Complement: Flip all bits to represent a negative number.

• 2’s Complement (Most common): Flip bits and add 1.
  Example: +5 = 0101 → -5 = 1011

Why 2’s Complement?
It simplifies subtraction and hardware design—only one adder circuit is needed.

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4. Integer Arithmetic

Refers to mathematical operations performed on whole numbers.

• Addition/Subtraction: Done using binary addition rules with carry/borrow bits.

• Overflow: Occurs when the result exceeds the number of bits available.
  Example: 8-bit register, 127 + 1 = overflow.

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5. Unsigned Binary Multiplication Algorithm

Used when both numbers are positive (no sign bit involved).

Steps:

1. Multiply the multiplier bit by the multiplicand.

2. Shift the partial product left by one position after each step.

3. Add all partial products.

Example:
110 × 101 (6 × 5 = 30 in decimal)
The algorithm gives 11110 (which equals 30).

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6. Signed Binary Multiplication (Booth’s Algorithm)

Booth’s algorithm handles negative numbers efficiently in binary multiplication.

Key Points:

• Reduces the number of addition/subtraction operations.

• Works by encoding the multiplier bits using rules based on bit transitions.

Simplified Example:
For 3 × 4, Booth’s algorithm manipulates binary forms of both numbers and gives the correct signed result using fewer steps.

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7. Unsigned Binary Division Algorithm

Division in binary follows repeated subtraction and shifting.

Steps:

1. Align the divisor with the dividend.

2. Subtract the divisor where possible and write the quotient bits.

3. Shift the divisor right for the next bit test.

Example:
1100 ÷ 10 → 6 in decimal ÷ 2 = quotient 110.

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8. Representation of Fractions

Fractions in binary can be represented as:

• Fixed Point: The Binary point is fixed after a certain bits.

• Floating Point: Binary point “floats” depending on the exponent value.

Example:
Fixed: 1010.01 = 10.25
Floating: $1.01001 \times 2^3$

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9. Fixed Point Representation

Used when the range and precision of numbers are limited.

• The binary point is fixed at one position.

• Simpler, faster, and used in embedded systems.

Example:
Digital clocks and microcontrollers commonly use fixed-point math.

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10. Floating Point Representation

Used for large or very small numbers—similar to scientific notation.

IEEE-754 Format:

• Sign bit (1 bit)

• Exponent (8 bits)

• Mantissa (23 bits)

Example:
Decimal 10.75 = $1.01011 × 2^3$

Allows representation of both huge and tiny values with precision.

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11. Floating-Point Arithmetic

Operations include addition, subtraction, multiplication, and division using normalised floating-point formats.

Challenges:

• Rounding errors may occur.

• Normalisation ensures the mantissa fits within the format.

Example: Used in graphics rendering, physics engines, and financial calculations.

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12. BCD (Binary Coded Decimal) Arithmetic Unit

BCD represents each decimal digit separately in 4 bits.

Example:
Decimal 59 = BCD 0101 1001.

Used in calculators where exact decimal representation is critical.

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13. BCD Adder

A specialised circuit that adds two BCD digits.

Key Steps:

1. Add the binary digits.

2. If the sum exceeds 9, add 6 (0110) to correct it.

Example:
Add 7 (0111) and 8 (1000) → 1111 (15), corrected to 0001 0101 (15 decimal).

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14. Pipelining

Pipelining speeds up arithmetic operations by dividing them into stages—like an assembly line.

• Each stage performs a specific task.

• Multiple operations can overlap in execution.

Example: CPU adds one number while fetching the next instruction.

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15. Arithmetic Pipelining

Special type of pipelining applied to arithmetic operations (like floating-point addition/multiplication).

Benefits:

• Increases instruction throughput.

• Reduces overall computation time.

Used in modern processors and GPUs for high-speed arithmetic calculations.

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Key Takeaways

• All computer arithmetic starts from binary representation.

• 2’s complement simplifies the handling of negative numbers.

• Booth’s algorithm and pipelining improve performance.

• Floating-point and BCD arithmetic handle precision-critical tasks.

• Understanding these basics helps in system design, compiler optimisation, and numerical software development.