Problem 56 - Algebra 1: Concepts and Skills

Question

Determine whether the following statement is true or false. If it is true, give two examples. If it is false, give a counter example. $$ \begin{aligned} &\text {The opposite of the sum of two numbers is equal to the sum of the}\\\ &\text { opposites of the numbers.} \end{aligned} $$

Step-by-Step Solution

Answer

The statement 'The opposite of the sum of two numbers is equal to the sum of the opposites of the numbers.' is True

Step 1: Understanding the problem statement

A given statement is: 'The opposite of the sum of two numbers is equal to the sum of the opposites of the numbers.' Let's denote two numbers as \(a\) and \(b\). Then 'the opposite of the sum of two numbers' can be represented as \(-1 * (a + b)\) or \(-(a+b)\), and 'the sum of the opposites of the numbers' can be written as \((-a) + (-b)\), which is also the same as \(-a - b\). The goal is to verify whether these two expressions are equal.

Step 2: Examples verification

Consider \(a=2\) and \(b=3\). Then, '-(a+b)' gives \(-5\) and '(-a)+(-b)' results in \(-5\). Hence, for these values, the statement is true. Another example, \(a=5\), \(b=7\), then '-(a+b)' is \(-12\) and '(-a)+(-b)' is also \(-12\). Thus, the statement is also true for these values.

Step 3: Generalization and counter-example search

Considering the general formula derived, '\(-(a+b)\)' is equal to '\(-a - b\)', the opposite of sum of two numbers a and b, is indeed equal to the sum of the opposites of the numbers a and b. Hence, there can't be a counter-example, as the statement holds true for all real numbers a and b.

Key Concepts

Opposite of a Sum

In algebra, finding the opposite of a sum involves combining numbers and then taking the inverse, or opposite, of that total. Imagine adding two numbers, say \(a\) and \(b\), to get a sum \((a+b)\). Taking the opposite of this sum turns the expression into \(-(a+b)\). This approach is significant because it helps demonstrate the distributive property when dealing with negative signs and ensures that calculations involving subtraction follow the correct order of operations.
Interestingly, the opposite of a sum is not just about rewriting an expression. It's about understanding how different quantities interact in terms of positives and negatives. If you "flip" the sign of a sum, it changes the nature or direction of that value on the number line. Abruptly changing from a positive sum to a negative indicates a complete reversal, which can simplify complex algebraic expressions.

Properties of Addition

Addition has several key properties that are fundamental in algebra. Understanding these properties can significantly aid in simplifying expressions and solving equations. Here are the basic properties relevant to our discussion:

• Commutative Property: This property states that the order of numbers does not affect the sum. Mathematically, \(a + b = b + a\).
• Associative Property: This states that the grouping of numbers does not affect the sum. This can be shown as \((a + b) + c = a + (b + c)\).
• Identity Property: The identity property implies that adding zero to any number leaves the number unchanged. Thus, \(a + 0 = a\).

Each of these properties plays a crucial role in verifying expressions like "the opposite of the sum". When combining negative and positive numbers, understanding these properties ensures that the calculations are consistent and structurally sound across different problems.

Real Numbers

Real numbers encompass all the numbers on the number line, including rational and irrational numbers. They are critical in algebra because they give us a complete system for performing arithmetic calculations.
Any number you can think of, whether it's an integer like \(7\), a fraction like \(\frac{3}{4}\), or an irrational number like \(\sqrt{2}\), is a real number. Real numbers follow the standard operations like addition, subtraction, multiplication, and division (except by zero), and they extend the properties of these operations.
In the context of our statement, all real numbers \(a\) and \(b\) make \(-(a+b) = -a-b\) a universally applicable truth. This is because real numbers maintain the integrity of arithmetic operations involving negatives across all calculations, showing the extensive utility of algebraic expressions. Knowing how these numbers interact keeps everything sensible and results in consistent outcomes.