Numerical Expressions: Definition, Method and Examples

What is a numerical expression?

A numerical expression is a combination of numbers and operations (such as addition, subtraction, multiplication, and division) that can be evaluated to find a specific value.

Unlike a full sentence, an expression is just a phrase. It states a quantity but does not make a complete mathematical statement on its own. For instance, $20 - 5$ is a numerical expression that represents the quantity $15$.

Parts of a numerical expression

A numerical expression contains different components that work together.

• Numbers: The mathematical values being operated on.
• Operations: The symbols that tell you what to do, such as addition ($+$) or multiplication ($\times$).
• Grouping symbols: Parentheses $()$, brackets $[]$, or braces $\{\}$ that group parts of the expression and change the standard order of calculation.
• Powers: Exponents that indicate repeated multiplication.

Expression versus equation

It is common to confuse an expression with an equation, but they have different mathematical rules and components.

• Numerical expression: Contains only numbers and operations. It represents a single quantity and has no equals sign. Example: $10 - 4$.
• Equation: Contains an equals sign ($=$) and shows that two expressions have the exact same value. Example: $10 - 4 = 6$.
• Algebraic expression: Contains variables (letters representing unknown numbers) along with numbers and operations. Example: $x - 4$.

Write expressions from words

To translate word problems or verbal descriptions into mathematical phrases, match the action words to their corresponding mathematical operators.

• Addition ($+$): sum, increased by, more than, combined.
• Subtraction ($-$): difference, decreased by, less than, minus.
• Multiplication ($\times$): product, times, twice, doubled.
• Division ($\div$): quotient, divided by, half of, shared equally.

Here are three verbal situations translated into numerical expressions:

1. "The sum of $8$ and $5$, multiplied by $2$" translates to $(8 + 5) \times 2$.
2. "Twenty decreased by the product of $3$ and $4$" translates to $20 - (3 \times 4)$.
3. "Half of $18$, increased by $6$" translates to $\dfrac{18}{2} + 6$.

Evaluate numerical expressions

To evaluate an expression, perform the calculations to find its final single value. Because expressions often contain multiple operations, you must always follow the order of operations (often remembered as PEMDAS or BODMAS):

1. Parentheses (and other grouping symbols).
2. Exponents (powers).
3. Multiplication and Division (from left to right).
4. Addition and Subtraction (from left to right).
   

A numerical expression simplifies to a single number when evaluated correctly using the standard order of operations.

Worked examples

Example 1: Identifying expressions

Question: Which of the following is a numerical expression: $x + 5$, $12 - 4 = 8$, or $7 \times (3 + 2)$?

Method:

1. Check for variables. The phrase $x + 5$ has a variable, so it is an algebraic expression.
2. Check for an equals sign. The phrase $12 - 4 = 8$ has an equals sign, so it is an equation.
3. Verify the remaining option contains only numbers and operators. The phrase $7 \times (3 + 2)$ fits.

Answer: $7 \times (3 + 2)$ is a numerical expression.

Check: The expression contains no equal signs and no letters, confirming it is purely numerical.

Example 2: Translating words into numbers

Question: Write a numerical expression for "the difference between $15$ and $7$, divided by $2$".

Method:

1. Identify the first operation: "difference between $15$ and $7$" means subtraction, written as $15 - 7$.
2. Because the entire difference must be divided, wrap it in grouping symbols: $(15 - 7)$.
3. Identify the second operation: "divided by $2$" means division, written as $\div 2$.

Answer: $(15 - 7) \div 2$.

Check: Translating the mathematical expression back gives "the difference of $15$ and $7$, divided by $2$", which matches the original problem perfectly.

Example 3: Evaluating an expression

Question: A baker starts with $50$ cookies. He sells $2$ boxes of $12$ cookies each and gives $5$ cookies away. Write and evaluate a numerical expression for the number of cookies left.

Method:

1. Start with the initial amount: $50$.
2. Subtract the amount sold: $2 \times 12$.
3. Subtract the amount given away: $5$.
4. Combine these steps into one expression: $50 - (2 \times 12) - 5$.
5. Evaluate using the order of operations. First, multiply: $2 \times 12 = 24$.
6. Substitute the product back into the expression: $50 - 24 - 5$.
7. Subtract from left to right. $50 - 24 = 26$, then $26 - 5 = 21$.

Answer: The expression evaluates to $21$.

Check: This uses the logic from addition and subtraction word problems to confirm that each subtraction correctly reduces the starting total.

Common mistakes

• Treating an expression like an equation: Expressions do not have two sides and cannot be "solved" for a variable. They are "evaluated" or simplified to a final number.
• Ignoring the order of operations: Calculating strictly from left to right without prioritizing multiplication or division changes the value. For example, in $4 + 3 \times 2$, adding first gives $14$, but multiplying first gives the correct value of $10$.
• Forgetting grouping symbols: When a verbal description requires an operation out of the normal order, parentheses are required. "Double the sum of $4$ and $3$" must be written as $2 \times (4 + 3)$, not $2 \times 4 + 3$.

Frequently asked questions

Can a numerical expression be just one number?

Yes. A single number like $7$ is the simplest numerical expression. It requires no operations to evaluate.

Do numerical expressions have variables?

No. If an expression contains a letter like $x$ or $y$, it becomes an algebraic expression instead.

Can an expression have exponents and powers?

Yes. A number raised to a power represents repeated multiplication, so it is a valid part of a numerical expression.

What is the difference between simplifying and evaluating numerical expressions?

In mathematics, they usually mean the same thing. Both instruct you to perform the operations in the correct order to find the final numerical value.