Numbers, Numerals, and Properties of Integers

Terminology and mathematical properties regarding types of numbers, numeral systems, and the properties of integers and equality as presented in the lecture notes.

Number

An item that describes a magnitude or a position.

Cardinal numbers

Numbers which allow us to count the objects or ideas in a given collection, such as $1, 2, 3, \text{...}, 1000, 100000$.

Ordinal numbers

Numbers that state the position of the individual objects in a sequence, such as First, second, third.

Numerals

Symbols, or a combination of symbols, which describe a number.

Arabic numerals

A modification of the Hindu-Arabic number signs written in digits $0, 1, 2, 3, 4, 5, 6, 7, 8, 9$ and combinations like $20, 21, 22$.

Roman numerals

Numbers which are written in the Latin alphabet, for example $MCMXCIV$.

Bracket (Roman Numeral Modifier)

A symbol used to multiply a number by $100$ times.

Vinculum

A bar above a number used to multiply it by $1000$ times.

Doorframe

A symbol used to multiply a number by $1000000$ times.

Roman Numeral L

The symbol representing the Arabic number $50$.

Roman Numeral C

The symbol representing the Arabic number $100$.

Roman Numeral D

The symbol representing the Arabic number $500$.

Roman Numeral M

The symbol representing the Arabic number $1000$.

Digit

A specific symbol or symbols used alone or in combination to denote a number.

Imaginary number ($i$)

A number denoted as $i$, equal to the square root of $-1$ (\text{i} = \text{\textsqrt{-1}}).

$$i^2$$

The equivalent value of the imaginary number squared, which is $-1$.

$$i^3$$

The equivalent value of the imaginary number cubed, which is $-i$ or -\text{\textsqrt{-1}}.

Rational numbers

Numbers which can be expressed as a quotient (ratio) of two integers, including repeating decimals like $0.3333\text{...}$.

Irrational numbers

Numbers which cannot be expressed as a quotient of two integers, such as $e$, \text{\textpi}, or \text{\textsqrt{2}}, and non-repeating decimals.

Complex number

An expression of both real and imaginary numbers combined in the form $a + bi$, where $a$ and $b$ are real numbers.

Integers

A set including all the natural numbers, the negative of the natural numbers, and the number zero.

Natural numbers

Numbers considered as the "counting numbers" such as $1, 2, 3, \text{...}$.

Additive identity

The number $0$, based on the property $a + 0 = a$.

Multiplicative identity

The number $1$, based on the property $a + 1 = a$ (noting $a \times 1 = a$).

Reflexive property

An equality property stating that $a = a$.

Symmetric property

An equality property stating that if $a = b$, then $b = a$.

Transitive property

An equality property stating that if $a = b$ and $b = c$, then $a = c$.

Substitution property

An equality property stating that if $a = b$, then $a$ can be replaced by $b$ in any expression involving $a$.