Number Systems and Computer Arithmetic

Vocabulary flashcards covering number base conversions, signed binary representations (sign-magnitude, 1's complement, 2's complement), binary arithmetic errors, and IEEE 754 floating-point format standard.

Radix (Base)

The number of unique digits used to represent numbers in a positional numeral system (e.g., base $8$ for octal and base $16$ for hexadecimal).

Octal System

A base-$8$ (radix-$8$) number system that uses positional values based on powers of $8$ and digits from $0$ to $7$.

Hexadecimal System

A base-$16$ (radix-$16$) number system that uses digits $0$ through $9$ and letters $A$ through $F$ to represent decimal values $10$ through $15$.

Decimal to Binary Conversion (Integer Part)

The process of repeatedly dividing an integer by $2$, recording the remainders, and writing the sequence of remainders in reverse order until the quotient becomes $0$.

Decimal to Binary Conversion (Fractional Part)

The process of repeatedly multiplying a fractional part by $2$, recording the resulting integer digits in sequence, and discarding each integer part before repeating on the remaining fraction.

Leading Zeroes

Zeroes added at the beginning (left) of an integer binary number to complete a group of four bits for hexadecimal conversion.

Trailing Zeroes

Zeroes added at the end (right) of a fractional binary number to complete a group of four bits for hexadecimal conversion.

Unsigned Binary Numbers

An $n$-bit binary representation for non-negative integers only, capable of representing values in the range from $0$ to $2^n - 1$.

Sign-Magnitude Representation

A signed number format where the most significant bit (MSB) represents the sign ($0$ for positive, $1$ for negative) and the remaining $n-1$ bits represent the magnitude. Range is $-(2^{n-1} - 1)$ to $2^{n-1} - 1$.

One's Complement Representation

A signed number format where positive numbers are represented normally and negative numbers are formed by complementing (flipping) every bit ($0$ to $1$ and $1$ to $0$). Range is $-(2^{n-1} - 1)$ to $2^{n-1} - 1$.

Two's Complement Representation

A signed number format where negative numbers are formed by complementing every bit of the positive number and adding $1$. Range is $-2^{n-1}$ to $2^{n-1} - 1$, offering a single representation for zero.

End-Around Carry

An extra step required in 1's complement addition/subtraction where, if addition produces a carry out of the most significant bit, that carry of $1$ is added back to the least significant bit of the result.

Overflow

A condition that occurs when adding two positive numbers yields a negative result, detected when the carry-in and carry-out of the MSB are different.

Underflow

A condition that occurs when adding two negative numbers yields a positive result, detected when the carry-in and carry-out of the MSB are different.

IEEE 754 Standard

The technical standard established by the Institute of Electrical and Electronics Engineers for floating-point representation and binary arithmetic.

Single Precision Floating-Point

A $32$-bit IEEE 754 floating-point format consisting of a $1$-bit sign, an $8$-bit exponent field, and a $23$-bit fraction field.

Double Precision Floating-Point

A $64$-bit IEEE 754 floating-point format consisting of a $1$-bit sign, an $11$-bit exponent field, and a $52$-bit fraction field.

Normalized Numbers (IEEE 754)

Floating-point numbers in single precision calculated using the formula $(-1)^s \times 1.f \times 2^{E-127}$.

Denormalized Numbers (IEEE 754)

Floating-point numbers in single precision where the exponent field $E = 0$, calculated using the formula $(-1)^s \times 0.f \times 2^{-126}$.

NaN (Not a Number)

A special floating-point value in IEEE 754 represented by an exponent field of all 1s ($11111111$) and any non-zero fraction field.

Floating-Point Infinity (Inf)

A special representation in IEEE 754 single precision defined by an exponent field of all 1s ($11111111$) and a fraction field of all zeroes ($00000000000000000000000$).