Appendix B: Mathematical notation
An ellipsis, \ldots, can be read as “and so on.” For example, 1, 2, 3, \ldots, 100 denotes the list of all numbers from 1 to 100, and 1 + 2 + 3 + \dots + 100 denotes the sum of all integers from 1 to 100 (inclusive).
Braces are used to denote sets, for example, \{4, 12, 31\} is the set containing the elements 4, 12, and 31.
\in denotes membership in a set. For example, 4 \in \{4, 12, 31\}.
\notin is used to indicate that some object is not an element of some set. For example, 7 \notin \{4, 12, 31\}.
\mathbb{N} denotes the set of all natural numbers, that is, 0, 1, 2, 3, \ldots
\mathbb{Z} denotes the set of all integers, that is, \ldots -2, -1, 0, 1, 2, \ldots
\mathbb{R} denotes the set of all real numbers. Unlike the integers, real numbers can be used to measure continuous quantities.
Sometimes we describe a set by stating the properties that must hold for its members. In such cases, we use the vertical bar, |, which can be read “such that.” For example, \{x \in \mathbb{R} \; | \; x \geq 0 \} is the set of all real numbers greater than or equal to zero.
\mathbb{Q} is the set of all rational numbers, that is, \mathbb{Q} = \{\frac{a}{b} \; | \; a, b \in \mathbb{Z}, b \neq 0 \}.
\equiv denotes congruence. We write a \equiv b \pmod{m} to indicate that a is congruent to b {modulo} m. For example, 5 \equiv 1 \pmod{2}, and 72 \equiv 18 \pmod{9}.
\circ is the composition operator. For example, f \circ g is the composition of functions f and g. With this notation, composition is performed right-to-left, so it may be helpful to read f \circ g as “f applied after g.”
Square brackets denote closed intervals, the elements of the interval usually determined by context. For example, [0, 12] = \{n \in \mathbb{Z} \; | \; 0 \leq n \leq 12\} and [\pi, 2\pi] = \{x \in \mathbb{R} \; | \; \pi \leq x \leq 2\pi\}.
Parentheses denote open intervals—those in which the endpoints are not included. For example, (0, 12) = \{n \in \mathbb{Z} \; | \; 0 < n < 12\} and (\pi, 2\pi) = \{x \in \mathbb{R} \; | \; \pi < x < 2\pi\} (note the strict inequalities).
Half-open intervals are denoted with a square bracket on one side and a parenthesis on the other. For example, [0, 12) = \{n \in \mathbb{Z} \; | \; 0 \leq n < 12\} and (\pi, 2\pi] = \{x \in \mathbb{R} \; | \; \pi < x \leq 2\pi\}.
Subscripts are used to denote individual elements within a set or sequence. For example, x_i is the ith element of the sequence X, and we call i the index of the element. Note: In this text, indices start with 0.
\Sigma denotes a summation. For example, given the set X = \{2, 9, 3, 5, 1\}, \Sigma \; x_i is the sum of all elements of X, that is 2 + 9 + 3 + 5 + 1 = 20. Sometimes, an operation or operations are applied to the elements of a summation. For example, given the set X = \{1, 2, 3\}, \Sigma \; x_i^2 is the sum of the squares of all elements of X, that is 1^2 + 2^2 + 3^2 = 1 + 4 + 9 = 14.
\Pi (upper-case) denotes a repeated product. For example, given the set X = \{2, 9, 3, 5, 1\}, \Pi \; x_i is the product of all elements of X, that is 2 \times 9 \times 3 \times 5 \times 1 = 270.
\pi (lower-case) is a mathematical constant, the ratio of a circle’s circumference to its diameter. This is approximately equal to 3.141592653589793.
In statistics, \mu is used to denote the mean of a sample, population, or distribution. You may have seen \bar{x} in other texts. These are different notations for the same thing.
In statistics, \sigma denotes the standard deviation of a sample, population, or distribution (\sigma^2 denotes the variance).
\pm denotes “plus or minus” for example, \mu \pm 2.5\sigma or -b \pm \sqrt{b^2 - 4ac}.
Original author: Clayton Cafiero < [given name] DOT [surname] AT uvm DOT edu >
No generative AI was used in producing this material. This was written the old-fashioned way.
This material is for free use under either the GNU Free Documentation License or the Creative Commons Attribution-ShareAlike 3.0 United States License (take your pick).