Glossary: Number Systems

Key terms and definitions for Unit 1. Definitions follow ISO 11179 metadata registry standards.

A

Analog Signal — A continuous signal that can take any value within a range, such as a smooth audio waveform from a microphone, in contrast to the discrete values used in digital systems.

B

Base (Radix) — The number of unique symbols used in a positional number system, which determines the multiplier for each position (e.g., 2 for binary, 8 for octal, 10 for decimal, 16 for hexadecimal).

Binary Addition — The process of adding two binary numbers using the rules \(0+0=0\), \(0+1=1\), \(1+0=1\), and \(1+1=10\) (0 with a carry of 1), following the same columnar approach as decimal addition.

Binary Division — The process of dividing binary numbers using the long division algorithm, simplified by the fact that only 0 and 1 are possible quotient digits at each step.

Binary Multiplication — The process of multiplying binary numbers where each partial product is either zero or the multiplicand itself (shifted appropriately), since the only possible multiplier digits are 0 and 1.

Binary Number System — A base-2 positional number system using only two symbols (0 and 1) that maps directly to the two voltage states in digital circuits, making it the native language of computers.

Binary Subtraction — The process of subtracting one binary number from another using the rules \(0-0=0\), \(1-0=1\), \(1-1=0\), and \(0-1=1\) (with a borrow from the next higher position).

Bit (Binary Digit) — The fundamental unit of digital information, representing a single 0 or 1; the term is a portmanteau of "binary" and "digit."

Borrow — A value taken from the next higher bit position during binary subtraction when a 0 must be subtracted by a 1, analogous to borrowing in decimal subtraction.

Byte — A unit of digital data comprising 8 bits (2 nibbles) that can represent 256 distinct values (0–255 unsigned) and serves as the standard addressable unit of memory in most computer architectures.

C

Carry — A value generated during binary addition when the sum of bits in a column equals or exceeds 2, which is propagated to the next higher bit position.

Conversion (Base Conversion) — The process of translating a number from one base (radix) to another, using techniques such as repeated division, weighted positional expansion, or direct binary grouping shortcuts.

D

Decimal Number System — The base-10 positional number system using ten symbols (0 through 9), where each position represents a power of 10; the familiar number system used in everyday human arithmetic.

Digital Signal — A discrete signal that represents information using only two distinct states (typically high voltage for logic 1 and low voltage for logic 0), offering superior noise immunity over analog signals.

Digital System — An electronic circuit that processes information using discrete signal levels rather than continuous values, forming the basis of all modern computing devices.

E

End-Around Carry — A correction technique used in one's complement addition where the carry-out from the most significant bit is added back to the least significant bit to obtain the correct result.

H

Hexadecimal Number System — A base-16 positional number system using sixteen symbols (0–9 and A–F, where A=10 through F=15), where each hex digit corresponds to exactly four binary digits, making it the preferred shorthand for binary data.

L

Least Significant Bit (LSB) — The rightmost bit in a binary number, which contributes the smallest positional value (either 0 or 1) to the overall number.

M

Most Significant Bit (MSB) — The leftmost bit in a binary number, which contributes the largest positional value; in signed number representations, the MSB typically indicates the sign of the number.

N

Nibble — A unit of digital data consisting of 4 bits that can represent 16 distinct values (0–15 in decimal, or 0–F in hexadecimal); each hexadecimal digit corresponds to exactly one nibble.

Noise Immunity — The ability of a digital signal to maintain its correct logical interpretation despite small variations in voltage, a key advantage of digital over analog signal representation.

O

Octal Number System — A base-8 positional number system using eight symbols (0 through 7), where each octal digit corresponds to exactly three binary digits; used in Unix file permissions and some assembly language contexts.

One's Complement — A signed number representation in which negative numbers are formed by inverting (flipping) all bits of the corresponding positive number; it has the drawback of two representations for zero.

Overflow — An arithmetic error condition that occurs when the result of an operation is too large (positive) to be represented within the available bit width, detected in two's complement when the carry into the MSB differs from the carry out of the MSB.

P

Positional Notation — A numeral system in which the value of each digit is determined by both the digit itself and its position within the number, with each position representing a successive power of the base.

R

Radix Point — The symbol (such as a decimal point in base 10 or a binary point in base 2) that separates the integer portion of a number from its fractional portion in positional notation.

Repeated Division Method — A conversion technique for translating a decimal integer to another base by repeatedly dividing by the target base and recording remainders, which are then read in reverse order.

S

Sign Extension — The technique of preserving the numerical value of a signed binary number when increasing its bit width by replicating the sign bit (0s for positive, 1s for negative) into the additional higher-order positions.

Sign-Magnitude Representation — A signed number encoding where the most significant bit indicates the sign (0 for positive, 1 for negative) and the remaining bits represent the magnitude, mirroring how humans write signed numbers but having two representations of zero.

Signed Number — A binary number encoding that can represent both positive and negative values using schemes such as sign-magnitude, one's complement, or two's complement.

T

Two's Complement — The dominant signed number representation in modern digital systems, formed by inverting all bits of a number and adding 1, which provides a unique zero and enables addition and subtraction to use the same hardware.

Two's Complement Addition — The process of adding two signed numbers in two's complement representation, which works identically to unsigned binary addition with the carry out of the MSB discarded.

Two's Complement Subtraction — The process of subtracting one signed number from another by adding the two's complement (bitwise inversion plus 1) of the subtrahend, converting subtraction into addition.

U

Underflow — An arithmetic error condition that occurs when the result of an operation is too small (too negative) to be represented within the available bit width for the given signed number format.

Unsigned Number — A binary number representation that encodes only non-negative values, where all bits contribute to the magnitude, giving an \(n\)-bit number the range 0 to \(2^n - 1\).

W

Weighted Positional Expansion — A conversion technique for translating a number from another base to decimal by multiplying each digit by its positional power of the base and summing the results.

Word — A processor-specific unit of data representing the natural data size for a particular architecture (e.g., 32 bits or 64 bits), affecting memory addressing, register width, and arithmetic precision.

Z

Zero Extension — The technique of increasing the bit width of an unsigned binary number by padding higher-order positions with zeros, preserving the original value.