Variables and constants

In mathematical language, we use letters and other symbols to stand for numbers and other fixed values. Two of the most important ideas are variables and constants.

This chapter explains what they are, how they are written, and how to read simple expressions that use them. More complicated uses (such as in equations and functions) will be covered in later chapters.

What is a variable?

A variable is a symbol, usually a letter, that can represent different values at different times or in different situations.

• We usually use letters like $x, y, z, a, b, c$, but any agreed symbol can be a variable.
• The same variable symbol in the same expression or problem usually stands for the same (but possibly unknown) number.
• Different variable symbols usually stand for different numbers (unless we are told they are equal).

Examples of variables in use:

• “Let $x$ be the number of apples.” Here $x$ is a variable that can take any number of apples we decide on.
• In the expression $x + 5$, we do not know what $x$ is, but we know it is some number.
• In a word problem, “A rectangle has length $L$ and width $W$.” Both $L$ and $W$ are variables.

A variable does not have to “change over time” to be called a variable. It is simply a placeholder for a value that is not fixed in advance.

Types of roles variables can play

Even at a beginner level, it is helpful to notice that variables can play different roles:

• Unknown quantity in a problem:
  In $x + 3 = 7$, $x$ is an unknown we want to find.
• General quantity in a rule or formula:
  In the rule “add 2 to any number” written as $n + 2$, $n$ is a general number, not a specific one.
• Changing quantity depending on something else:
  In “the cost $C$ depends on the number of items $n$”, both $C$ and $n$ are variables, and $C$ may change when $n$ changes.

These different uses are still based on the same basic idea: a variable is a symbol that can stand for a number (or more generally, a value) that is not fixed once and for all.

What is a constant?

A constant is a value that does not change within the situation we are considering.

There are two common kinds of constants:

1. Named constants: numbers that are always the same in all contexts.
• For example, the number $5$ is always $5$; it never changes.
• Important named numbers like $\pi$ (pi) and $e$ (Euler’s number) are also constants:
• $\pi \approx 3.14159$ is always the same value.
• $e \approx 2.71828$ is always the same value.
2. Problem-specific constants: values that are fixed in a particular problem or formula, even if we don’t know their numerical value.
• In the formula $C = 3n + 2$, the numbers $3$ and $2$ are constants. They do not change as $n$ changes.
• In $A = kx$, if $k$ is described as “a fixed constant,” then $k$ is a constant for that situation, even if we don’t know its number yet.

So, while a variable can take different values, a constant has one fixed value in the context we are working in.

Telling variables and constants apart in expressions

In written mathematics, we often mix variables and constants in the same expression. It is important to be able to recognize which is which from the context.

Consider the expression:
$$
2x + 5
$$

• $x$ is a variable: it can represent different numbers in different situations.
• $2$ and $5$ are constants: they are fixed numbers in this expression.

Another example:
$$
A = \pi r^2
$$

• $A$ and $r$ are variables: $A$ is the area, and $r$ is the radius; they can vary from one circle to another.
• $\pi$ is a constant: it has the same value for all circles.

Context tells you which symbols are variables and which are constants. If a symbol is described as “fixed,” “given,” or “constant,” then it is a constant in that situation. If it is described as something that can “vary,” “change,” or be “any number,” then it is a variable.

Notation for variables and constants

Variables and constants are both written with symbols, but there are some common habits:

• Letters for variables:
• Single letters like $x, y, z$ are common in simple algebra.
• Sometimes letters hint at meaning:
• $t$ for time,
• $n$ for a counting number (like “number of items”),
• $A$ for area,
• $V$ for volume.
• Letters or special symbols for constants:
• Fixed numbers like $2, -5, \frac{1}{2}$ are written as usual.
• Special constants use letters:
• $\pi$ for the circle constant,
• $e$ for Euler’s number.
• Sometimes a letter like $k$ or $c$ is used to represent “some fixed constant.”

The same letter may represent different things in different problems. You must always read the description in the current context.

For example:

• In one problem, $c$ might be a constant “speed of light.”
• In another, $c$ might be “the cost of one item.”

It is the surrounding text and definitions that tell you what a symbol means in that particular situation.

Reading expressions with variables and constants

To become comfortable with variables and constants, it helps to practice reading expressions in words.

Examples:

1. Expression: $x + 3$
• $x$ is a variable, $3$ is a constant.
• Read as: “$x$ plus three.”
2. Expression: $5y - 2$
• $y$ is a variable, $5$ and $2$ are constants.
• Read as: “five times $y$ minus two.”
3. Expression: $A = lw$
• $A$ is a variable (area), $l$ is length, $w$ is width; all are variables that can change from one rectangle to another.
• Read as: “$A$ equals $l$ times $w$.”
4. Expression: $y = 2x + 1$
• $x$ and $y$ are variables.
• $2$ and $1$ are constants.
• Read as: “$y$ equals two times $x$ plus one.”

Being able to say in words what each symbol stands for, and whether it is variable or constant in that context, is a key skill.

Choosing letters for variables and constants

The choice of letters is mostly a matter of convenience and habit. Some common patterns:

• General variables:
• $x, y, z$ are generic.
• $n, m$ are often used for counting numbers (like “the $n$th term”).
• Constants:
• $k$ is often used for an unspecified constant (sometimes called a “constant of proportionality” or just “a constant”).
• $c$ is sometimes used for a general constant, especially in more advanced topics.

Although these patterns are common, they are not rules. The author or teacher can choose any symbols, as long as the meaning is clearly explained.

Why use variables at all?

From the point of view of mathematical language, variables allow us to:

• Write general rules that work for many numbers at once.
  For example, “add 5 to any number” can be expressed as $x + 5$.
• Describe relationships between quantities.
  For example, “distance equals speed times time” becomes
  $$
  d = st,
  $$
  where $d, s, t$ are variables.
• Express unknowns in equations that we want to solve.
  For example, in $x + 4 = 10$, $x$ stands for an unknown number.

Constants provide the fixed “background” numbers in these rules and relationships.

Changing and fixed within a problem

It is important to see that “variable” and “constant” are relative to the situation:

• A symbol might be treated as a variable in one problem and as a constant in another.
• Whether a symbol is variable or constant depends on what is allowed to change.

Example:

• In the formula $C = 3n + 2$ for cost $C$:
• If we are interested in how cost changes when we change $n$, then $n$ and $C$ are variables, and $3$ and $2$ are constants.
• In a different context, we might say, “Now suppose $n$ is fixed at $4$,” then for that moment, $n$ is acting like a constant (it is not allowed to vary anymore), while $C$ might still vary in some larger context.

At this level, it is enough to remember:

• Variables: symbols that can take different values (depending on the context).
• Constants: values that stay the same within the context.

Summary

• A variable is a symbol that can represent different values; it may be unknown, general, or changing.
• A constant is a value that does not change within the situation, either a familiar fixed number (like $5$ or $\pi$) or a specified fixed quantity (like a given $k$).
• In expressions, both variables and constants appear together, and context tells you which is which.
• Letters like $x, y, n$ are commonly used for variables; numbers and special symbols like $\pi$ are constants, and letters like $k$ or $c$ often represent fixed but unspecified constants.

Later chapters will use these ideas in equations, functions, and more advanced topics, but the basic language of variables and constants is the same throughout mathematics.