Computer Arithmetic

Arithmetic and Logic Unit (ALU)

→ ALU is a core component of CPU that handle arithmetic and logical operations.

What it is for?

— for basic math (+ , — , x, ÷) = arithmetic operation

— (AND, OR, NOT, XOR) = logical operation

— for moving data bits left/right to multiply/divide = shift operation

Data Handling:

• Data is presented to the ALU in registers.
• Results are stored in registers.
• Control unit sends instructions and manages data movement.
• Flags like overflow and sign flags are set based on operation outcomes.

Integer Representation

→ Binary Format: Computers use binary (0s and 1s) to represent data.

Types of Integer Representation:

Unsigned Integers: Only positive numbers.

Signed Integers: Positive and negative numbers, represented using:

Sign-Magnitude: Uses the most significant bit (MSB) as the sign bit.

One’s Complement: Inverts all bits of a positive number to represent a negative number.

Two’s Complement: Inverts all bits of a positive number and adds 1 to represent a negative number.

Integer Arithmetic

Negation:

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1.Take the Boolean complement of each bit.

2.Add 1 to get the negative version of a number.

3.The negative of a negative number returns the original number.

Addition:

• Overflow occurs if the result of adding two numbers has the opposite sign.

Subtraction:

• Use two’s complement negation of the subtrahend, then add it to the minuend.

Multiplication:

• Booth’s Algorithm: Efficiently handles multiplication, especially for signed numbers.

Division:

• Restoring Division Algorithm: Handles division by repeatedly subtracting the divisor from the accumulator and shifting bits.

Floating-Point Representation

Components:

1.Significand: The significant digits of the number.

2.Base: The base of the number system (usually 2 for binary).

3.Exponent: Determines the scale of the number.

IEEE 754 Standard:

• Standard for representing floating-point numbers in computers.
• Ensures consistency and precision across different systems.