CS-107 Lecture 3: Number Systems

Vocabulary flashcards covering number sets, data representation types, positional notation, base systems, and base conversion methods.

Data representation

The process of presenting information in a meaningful way, allowing for easy understanding and analysis by organising and structuring data in a format interpretable by computers or humans.

Ariane 5 disaster (4 June 1996)

A spaceflight failure where the rocket exploded 37 seconds after launch, caused by converting a 64-bit floating-point velocity into a 16-bit signed integer that overflowed the field.

Textual data representation

Representing data using words, sentences, or paragraphs.

Numerical data representation

Representing data through numbers or mathematical values.

Visual data representation

Representing data using graphs, charts, maps, or diagrams.

Multimedia data representation

Representing data through a combination of text, images, audio, and video.

Natural Numbers ($\mathbb{N}$)

The counting numbers achieved by repeatedly adding 1, represented as the set $\{1, 2, 3, 4, \dots\}$.

Whole Numbers ($\mathbb{W}$)

The set of natural numbers including zero, represented as $\{0, 1, 2, 3, 4, \dots\}$; also referred to as non-negative integers.

Integers ($\mathbb{Z}$)

The set containing whole numbers and negative natural numbers: $\{\dots, -4, -3, -2, -1, 0, 1, 2, 3, 4, \dots\}$.

Rational Numbers ($\mathbb{Q}$)

Numbers that can be represented as an integer or as the quotient of an integer and a non-zero integer, such as $3/8$ or $-5.28$.

Real Numbers ($\mathbb{R}$)

The complete set of all non-imaginary numbers, encompassing natural numbers, whole numbers, integers, and rational numbers.

Base (Radix)

A representation scheme that defines the total number of unique symbols available to represent numbers.

Positional Notation

A numerical system where numbers are represented by digits ordered in positions from the right, with each position representing the base raised to that position's power index.

Bit

A single binary digit, represented by either a 0 or a 1.

Byte

A sequence of eight bits, which can also be represented as two hexadecimal digits.

Binary (Base 2)

A numeral system consisting of two symbols (0 and 1) with place values of powers of 2, used in hardware because physical circuits reliably hold two distinct states (off/on, low/high voltage).

Octal (Base 8)

A numeral system using eight symbols (0 to 7) with place values that are powers of 8, commonly used as a shorthand for bit patterns such as Unix file permissions.

Hexadecimal (Base 16)

A numeral system consisting of 16 symbols (0–9 and A–F, where A = 10 and F = 15) with place values of powers of 16, frequently used for memory addresses, colour codes, and byte dumps.

Division-Remainder Method

An algorithm for converting a decimal number to base $b$ by repeatedly dividing the quotient by $b$, recording the remainder, and reading the remainders in reverse order.

Power-of-Two Grouping Shortcut

A method to convert binary numbers to octal or hexadecimal by partitioning bits from the rightmost position into groups of 3 bits (for octal) or 4 bits (for hexadecimal) and translating each group directly.