Digital Logic Design: Binary Arithmetic, Character Codes, and Karnaugh Maps

Vocabulary flashcards covering fixed-precision arithmetic overflow, signed binary systems, character encodings (BCD and EBCDIC), and Karnaugh map minimization.

Overflow (Binary Addition)

A condition occurring in digital systems with fixed storage capacity when an addition produces a result that is too large to fit in the available number of bits, causing the highest bit to be discarded and yielding an incorrect result.

Overflow Detection

The technique of identifying arithmetic overflow in binary addition by checking for a carry out of the most significant column.

Ariane 5 Rocket Failure (1996)

A catastrophic failure on June 4, 1996, where a rocket costing $\$7\text{ billion}$ veered off course 40 seconds after launch and exploded because its control computer overflowed its 16-bit range due to larger values produced by a faster engine.

Sign/Magnitude Representation

A signed binary system where an $N$-bit number designates the most significant bit as the sign ($0$ for positive and $1$ for negative) and the remaining $N - 1$ bits as the magnitude (absolute value).

Two's Complement Representation

A signed binary numbering system where zero is represented as all zeros ($00\dots000_2$), the most positive number is $01\dots111_2 = 2^{N-1} - 1$, the most negative number is $10\dots000_2 = -2^{N-1}$, and $-1$ is represented as all ones ($11\dots111_2$).

Taking the Two's Complement

The process of reversing the algebraic sign of a two's complement number by inverting all of its bits and adding $1$ to the least significant bit position.

Two's Complement Subtraction

The procedure of performing subtraction by taking the two's complement of the subtrahend (the second number) and adding it to the first number.

Binary-Coded Decimal (BCD)

A numeric coding system used primarily in IBM mainframe and midrange systems that encodes each decimal digit into a 4-bit binary equivalent.

Zone (in BCD)

The upper 4-bit nibble of an 8-bit BCD byte, which indicates the sign: $1111$ for unsigned, $1100$ for positive, and $1101$ for negative.

Digit (in BCD)

The lower 4-bit nibble of an 8-bit BCD byte, which holds the binary value of the decimal digit.

Character Set

A defined collection of letters, digits, and other symbols frequently engraved onto keys and used by digital devices to represent nonnumeric data items.

EBCDIC (Extended Binary Coded Decimal Interchange Code)

An 8-bit character encoding system created by IBM for mainframe and midrange computers that represents characters by appending digit bits to zone bits.

EBCDIC Case Translation

The method of converting between uppercase and lowercase characters in EBCDIC by flipping a single bit at bit position 2.

Karnaugh Map (K-map)

A graphical method for simplifying Boolean expressions, arranged as a grid where rows and columns correspond to input values and each cell represents the output.

Minterm

A Boolean product term that includes all variables of a function exactly once, in either complemented or uncomplemented form.

K-map Grouping Rules

The set of rules for Boolean simplification on a K-map: groups contain only 1s, contain adjacent cells only (no diagonals), have a size equal to a power of 2, are as large as possible, cover all 1s, allow overlaps and wrap-around, and use the minimum number of groups.

Wrap Around (K-map)

A property of Karnaugh maps allowing cells along opposing outer edges (leftmost and rightmost columns, or top and bottom rows) to be treated as logically adjacent when forming groups of 1s.