Symbols and notation

Mathematics has its own “language”: a collection of symbols and standard ways of writing things. This chapter focuses on recognizing and reading common symbols and understanding how mathematical writing is usually structured. You will meet most of these symbols again in later chapters, where they are used in more detail.

Why symbols are used

Mathematical symbols are used to:

• Make statements shorter and clearer.
• Remove ambiguity that ordinary language might have.
• Express complex ideas in a compact form.

For example, the sentence “two plus three equals five” can be written as
$$2 + 3 = 5.$$
This is both shorter and unambiguous.

In mathematics, how things are written (notation) is as important as the symbols themselves. Changing the notation can change the meaning.

Basic relational symbols

Relational symbols compare quantities or show a relationship between them.

• Equality: $=$
  Means “is equal to.”
  Example: + 1 = 5$.
• Inequality: $\neq$
  Means “is not equal to.”
  Example: \neq 5$.
• Less than: $<$
  Example: < 7$ (“2 is less than 7”).
• Greater than: $>$
  Example: > 4$ (“9 is greater than 4”).
• Less than or equal to: $\le$
  Example: $x \le 10$ (“$x$ is less than or equal to 10”).
• Greater than or equal to: $\ge$
  Example: $k \ge 0$ (“$k$ is greater than or equal to 0”).

These symbols will appear very frequently in equations, inequalities, and later in functions and limits.

Basic arithmetic symbols

These symbols represent fundamental operations on numbers. The operations themselves are discussed in detail later; here we focus on reading the notation.

• Addition: $+$
  Example: + 3$ (seven plus three).
• Subtraction: $-$
  Example: - 4$ (ten minus four).
• Multiplication: there are several notations:
• $a \times b$ (often used in early arithmetic).
• $a \cdot b$ (dot notation).
• $ab$ (juxtaposition: writing letters next to each other).

In algebra, $ab$ is preferred. All three can mean “$a$ multiplied by $b$,” but context decides which is used.

• Division:
• $a \div b$ (common in school arithmetic).
• $\dfrac{a}{b}$ or $\frac{a}{b}$ (fraction bar notation).

In most further mathematics, the fraction bar is standard: $\dfrac{a}{b}$.

• Parentheses and grouping: $(\;)$, $[\;]$, $\{\;\}$
  These symbols group expressions so it is clear what is done first.
• Example: $2 \times (3 + 4)$ means $2 \times 7$, not $2 \times 3 + 4$.
• $[\;]$ and $\{\;\}$ can also be used for grouping, though they also have other roles, such as in set notation.

Common mathematical punctuation

Mathematics uses punctuation marks with specific meanings.

• Comma: ,
  Separates items in a list, coordinates, or arguments of a function.
  Example: $(2, 5)$ (a point), $f(x, y)$ (a function of two variables).
• Colon equals: $:=$
  Means “is defined as.”
  Example: $a := 3$ means “from now on, we define $a$ to be 3.”
• Implied “and”: writing statements side by side
  Longer mathematical writing often uses punctuation like semicolons or logical symbols for “and” and “or,” which will appear in logic chapters. Here, note that a single line like
  $$x > 0,\ y < 5$$
  is often read as “$x$ is greater than 0 and $y$ is less than 5.”

The role of letters: variables and constants

Letters are used in a standardized way:

• Lowercase letters like $x, y, z, a, b, c$ often represent numbers whose specific values may not be known or may change (variables).
• Some letters, such as $\pi$ or $e$, are used as standard constants with fixed values in later topics.
• Capital letters often label objects such as points ($A, B, C$), sets ($A, B, C$), or special functions.

Reading expressions with letters is a core skill. For example:

• $x + 3$ means “a number plus three.”
• $ab$ means “the product of $a$ and $b$.”
• $f(x)$ means “the value of the function $f$ at the input $x$.”

The concepts of variables and constants themselves are treated in their own chapter; here the important thing is to recognize letters as part of standard notation.

Special symbols for common concepts

Some symbols have names and standard meanings that appear across many areas of mathematics.

The symbol $\pi$

• Written as $\pi$ (Greek letter “pi”).
• Represents a specific constant related to circles.
• Its exact meaning and properties are explained later (in geometry and number chapters), but you should recognize it as a special numeric symbol, not a variable like $p$.

The symbol $\infty$

• Written as $\infty$ (“infinity”).
• Used to indicate unbounded growth or “going on without end.”
• It is not an ordinary number; it is a symbol that appears in limits, intervals, and some integrals later.

At this stage, you only need to recognize $\pi$ and $\infty$ and know they have special roles.

Function notation basics

A very common kind of notation in mathematics describes functions. You will study functions in detail later; here we focus on reading the standard symbols.

• Function name: usually a letter, often $f, g, h$.
• Input variable: usually written in parentheses.

Example:

• $f(x)$ is read “$f$ of $x$.” It represents a rule that takes input $x$ and produces an output.
• A definition might be written
  $$f(x) = x + 2.$$
  This means: “the function $f$ takes a number $x$ and returns the number $x + 2$.”

Multiple inputs can be written as
$$g(x, y) = x^2 + y^2,$$
read as “$g$ of $x$ and $y$ is $x$ squared plus $y$ squared.”

Power and root notation (symbol-focused)

Powers and roots are studied in detail later; here, focus on how they are written.

• Exponent or power: $a^n$
  Example: ^3$ is read “two to the power of three.”
• The small raised number $n$ is the exponent.
• Square: $a^2$
  Example: ^2$ read “five squared.”
• Cube: $a^3$
  Example: ^3$ read “two cubed.”
• Square root: $\sqrt{a}$
  Example: $\sqrt{9}$ is read “the square root of nine.”
• Other roots: $\sqrt[n]{a}$
  Example: $\sqrt[3]{8}$ is “the cube root of eight.”

The important point here is recognizing the notation: small raised numbers for powers, and the radical symbol $\sqrt{\;}$ for roots.

Set-related symbols (notation only)

The detailed study of sets comes later; here we highlight a few notations so they are not unfamiliar when you see them.

• Curly braces: $\{\;\}$
  Used to list elements of a set.
• Example: $\{1, 2, 3\}$ is the set containing $1$, $2$, and $3$.
• Membership: $\in$
• Example: $2 \in \{1,2,3\}$ means “2 is an element of the set $\{1,2,3\}$.”
• Non-membership: $\notin$
• Example: $5 \notin \{1,2,3\}$ means “5 is not an element of this set.”
• Empty set: $\emptyset$ or $\{\}$
  Represents a set with no elements.

You will later learn operations like union and intersection, with their own symbols, but their meanings belong to the sets chapter.

Interval notation (symbol-focused)

Intervals describe ranges of numbers on the number line. Their full meaning appears in number and function chapters; here we introduce the symbols.

• Parentheses $(\;)$ mean the endpoint is not included (open interval).
• Brackets $[\;]$ mean the endpoint is included (closed interval).

Examples:

• $(0, 1)$: all numbers strictly between 0 and 1.
• $[0, 1]$: all numbers from 0 to 1, including 0 and 1.
• $[2, \infty)$: all numbers greater than or equal to 2, with no upper bound.

Read them as “from 0 to 1, not including the endpoints” or “including” when brackets are used.

Using notation in simple statements

Here are some short statements written in standard mathematical notation, with their plain-language readings:

• $x + 5 = 9$
  “$x$ plus 5 equals 9.”
• $n \ge 1$
  “$n$ is greater than or equal to 1.”
• $f(x) = x^2 - 3$
  “$f$ of $x$ equals $x$ squared minus 3.”
• $a \in \{1, 2, 3\}$
  “$a$ is an element of the set $\{1,2,3\}$.”
• $0 < x < 1$
  “$x$ is between 0 and 1” (strictly greater than 0 and strictly less than 1).

Becoming comfortable reading and writing such expressions is a central goal of learning mathematical language.

Conventions and good habits with notation

Finally, some general principles about how notation is used:

• Consistency: A symbol should keep the same meaning within one problem or discussion.
• Clarity: Grouping symbols like parentheses should be used to avoid ambiguity.
• Economy: Mathematicians choose notation that is as short as possible without losing clarity.
• Context: The same symbol can mean different things in different areas, but within a given context its meaning should be clear.

As you proceed, each new topic will introduce some of its own symbols. This chapter provides a foundation so those symbols will feel like extensions of a familiar system rather than completely new inventions.