Question

Tell whether each rule is \(\bullet\) a convention, or \(\bullet\) a rule we can't change In an expression involving only addition and multiplication and no parentheses, such as \(2 \cdot 3+4 \cdot 5+6,\) do the multiplication first.

Short answer

It is a convention.

Step-by-step solution

Identify the Type of Mathematical Rule

Understand whether the rule applied in the problem is a convention or a fundamental rule that we cannot change. Conventions are agreed-upon methods, while fundamental rules are based on the properties of numbers.

Examine the Example Expression

Consider the given expression: \(2 \cdot 3 + 4 \cdot 5 + 6\). We need to apply the rule that says we should perform the multiplication before addition.

Apply the Order of Operations

Use the rule to solve the expression as follows: 1. First, perform the multiplications: \(2 \cdot 3 = 6\) \(4 \cdot 5 = 20\) 2. Then, add the results along with the remaining terms: 6 + 20 + 6.

Final Calculation

Add the numbers together to get the final result: 6 + 20 + 6 = 32.

Conclusion on the Rule Type

Recognize that the rule to perform multiplication before addition is not based on an innate property of numbers but rather on a widely accepted mathematical convention known as the order of operations.

Key concepts

mathematical conventions

Mathematical conventions are agreed-upon methods or standards that help us solve problems consistently. These conventions aren't inherent properties of numbers but are rules that mathematicians agree upon to avoid confusion. For instance, the order of operations is a convention that dictates the sequence in which we should perform mathematical operations to ensure that everyone arrives at the same result. Without such conventions, different people might solve the same problem in various ways and get different answers. This would lead to a lot of confusion and inconsistency in mathematical problem-solving.
Imagine you have the expression written as:
In the exercise expression 2 ⋅ 3 + 4 ⋅ 5 + 6, following the convention of the order of operations lets us solve this step-by-step methodically and ensure everyone gets the same answer (32 in this case). Always remember, even though conventions can theoretically be changed if everyone agrees upon the new standards, they are critical in maintaining clarity and uniformity in mathematical communication.

multiplication before addition

One of the key conventions in mathematics is performing multiplication before addition. This rule is part of the broader set of guidelines known as the order of operations. This rule ensures a consistent and clear method of solving expressions that contain multiple types of operations.
When you encounter an expression like 2 ⋅ 3 + 4 ⋅ 5 + 6, following this convention makes sure that we all understand to perform the multiplications (2 ⋅ 3 and 4 ⋅ 5) before performing any additions. Here’s how it works step by step:

• First, multiply 2 by 3 to get 6.
• Next, multiply 4 by 5 to get 20.
• Then add these results along with the remaining term: 6 + 20 + 6.

Following 'multiplication before addition' can make solving even complex expressions much simpler and avoid errors. If the rule were not followed, we might interpret the order differently, leading to incorrect results. This convention underpins the clarity and reliability of mathematical procedures.

expression evaluation

Evaluating mathematical expressions involves using specific rules and conventions to find the value of the expression. Let’s break down how we evaluate an expression using the exercise example 2 ⋅ 3 + 4 ⋅ 5 + 6. By following the established order of operations, we solve the expression correctly and systematically.
The steps involved in evaluating expressions are:

• Identify and perform all multiplications first. In our example, calculate 2 ⋅ 3 and 4 ⋅ 5 first, resulting in 6 and 20 respectively.
• After completing the multiplications, perform the additions. Next, add 6 + 20 + 6 together.

By assessing expressions in this logical order, we ensure the solution is accurate. It’s crucial to follow this sequence to avoid mistakes and misunderstandings. If we ignored the conventions, different sequences of operations could yield various outcomes, which would lead to confusion. This is why evaluating expressions accurately by adhering to these rules is fundamental in mathematics.