Chapter 7: Arithmetic

Addition

a simple dyadic operation in that operates on two operands, the addends, to produce a result, a sum.

Overflow error

An error that occurs when a calculation results in a number that exceeds the maximum limit that can be represented in a given number system.

Carry Out (Cout)

the process of transferring a value from one digit to the next during addition when the sum exceeds the base value.

Carry in (Cin)

the value transferred into a digit from the previous digit during addition, particularly when summing multiple digits.

Full Adder

A digital circuit that adds binary numbers and accounts for carry-in values while producing a sum and carry-out output.

Behavior of Sum (S)

expressed as odd parity (XOR) which is defined as “true when the number of high inputs is odd

Fixed point arithmetic

is a method of representing real numbers that have a fixed number of digits after the decimal point, allowing for a consistent precision in calculations.

Two’s Complement overflows for addition

occur when two positives produce a negative sum, or two negatives produce a positive sum. This can be expressed as a boolean expression, where the “m” subscript indicates the most significant bit

Subtraction Hardware

refers to the circuits and techniques used to perform subtraction operations in digital systems, often utilizing the two's complement method for negative numbers.

Negation in Two’s Complement

Just invert (complement) each bit and add one

multi-bit adder/subtracter

can employ the adder hardware to perform addition and subtraction by using an additional input to determine the operation, allowing for flexible arithmetic operations.

How are Binary Addition and Subtraction implemented?

Binary addition and subtraction are implemented using logic circuits that perform bitwise operations, often using adders to achieve both operations through techniques like two's complement for subtraction and the using of XOR and NAND gates