Numerical Expressions

A numerical expression is a sequence of numbers connected by operations like addition, subtraction, multiplication, and division. Parentheses are often included to specify the exact order in which calculations should be carried out.

Numerical expressions are essential in mathematics and problem-solving because they allow you to represent and solve mathematical operations in an organized and unambiguous way.

Here’s a practical example of a numerical expression:

$$  3 + \{ 2 \cdot [4 + (6 - 3)] \} $$

Let's explore why this concept is so fundamental and what rules govern it.

The Order of Operations

To ensure numerical expressions are solved consistently, mathematics follows a specific order of precedence for operations:

So, when working with a numerical expression, keep in mind that certain operations take priority and must be completed before others.

An Example

For instance, imagine you need to solve this expression:

$$ 2 + 3 \cdot 4 $$

First, handle the multiplication: $ 3 \cdot 4 = 12 $

$$ 2 + 3 \cdot 4  = 2 + 12 $$

Then, perform the addition.

$$ 2 + 3 \cdot 4  = 2 + 12 = 14 $$

The final result of the expression is 14.

Without these rules, numerical expressions could be ambiguous. For instance, if you performed addition before multiplication in the previous example, you’d get a different result. $$ 2 + 3 \cdot 4 = 6 + 4 = 10 $$ So, even if it seems straightforward, without specific rules, results could vary. Which answer is correct, 10 or 14? To avoid this kind of ambiguity, operations have an assigned priority: first powers, then multiplication and division, and finally addition and subtraction. Following these rules, the correct answer is clearly 14. $$ 2 + 3 \cdot 4 = 2 + 12 = 14  $$

The Use of Parentheses

Parentheses can adjust the order of operations, allowing you to prioritize certain calculations.

There are different types of parentheses: round `()`, square `[]`, and curly `{}`, which help organize calculations visually when the expression gets complex.

A Practical Example

For example, consider this expression:

$$  3 + \{ 2 \cdot [4 + (6 - 3)] \} $$

In this case, the calculations follow a specific order: first solve the expression inside the round parentheses, then the square brackets, and finally the outer multiplication. 

Start with the operation in the round parentheses: \(6 - 3 = 3\).

$$  3 + \{ 2 \cdot [4 + \underbrace{(6 - 3)}_{3}] \} $$

$$  3 + \{ 2 \cdot [4 + 3] \} $$

Replace and solve within the square brackets: \(4 + 3 = 7\).

$$  3 + \{ 2 \cdot \underbrace{ [4 + 3] }_{7} \} $$

$$ 3 + \{ 2 \cdot 7 \} $$

Move to the curly braces and calculate the multiplication: \(2 \cdot 7 = 14\).

$$ 3 + \underbrace{ \{ 2 \cdot 7 \}}_{14} $$

$$  3 + 14   $$

Finally, add the result to the number outside: \(3 + 14 = 17\).

$$ 17 $$

So, the result of the expression is 17.

Using Only Round Parentheses in Complex Expressions

Round, square, and curly brackets are useful for organizing the order of operations, but in particularly complex expressions, this may not be enough to keep calculations clear.

In these cases, it’s common to rely on only round parentheses, always solving the innermost ones first and working outward.

This approach, though visually simpler, still ensures the correct order of operations as long as you respect the hierarchy from the inside out.

For instance, the previous expression can also be written this way: $$  3 + ( 2 \cdot (4 + (6 - 3)) ) $$ First, solve the innermost parentheses: (6 - 3) = 3. $$  3 + ( 2 \cdot (4 + 3) ) $$ Next, resolve the next innermost pair: (4 + 3) = 7 $$  3 + ( 2 \cdot 7 ) $$ Finally, calculate the product and add the final sum. $$  3 + 14 = $$ The final result is still 17 even when using only round parentheses. This is because the priority rules remain intact, eliminating the need for different types of parentheses.

Following these straightforward rules helps you correctly evaluate even complex numerical expressions.