Problem 16 - Algebra 1

Question

Prove the theorem. (Use the basic axioms of algebra and the definition of subtraction given in Example \(1 .\) ) If \(a, b,\) and \(c\) are real numbers, then \((a-b)-c=a-(b+c)\)

Step-by-Step Solution

Answer

The theorem \((a-b)-c = a-(b+c)\) is true, as evidenced by the usage of the associative law on both sides of the equation.

Step 1: Understanding the Problem

In this exercise, we want to prove the equation \((a-b)-c=a-(b+c)\). This involves the operations of subtraction, which is equivalent to adding a negative, i.e., \(a-b\) is the same as \(a+(-b)\). The same applies for \(a-(b+c)\) which translates to \(a+[-(b-c)]\).

Step 2: Apply Associative Law to the Left Side of The Equality

We start with the left side of the equation \((a-b)-c=a-(b+c)\). Using the associative law of addition, we can rewrite this as \((a+(-b))+(-c)\) or \(a+[(–b) + (–c)]\).

Step 3: Apply Associative Law to the Right Side of The Equality

Next, we turn to the right side of the equation. It can be rewritten as \(a+[-(b-c)]\) or \(a + [(-b) + c]\).

Step 4: Comparing both sides

On comparing the simplified forms of the left and right sides, we have \(a + [(-b) + (–c)]\) = \(a + [(-b) + c]\). This shows that the original statement is true, as both sides of the equality are equal.

Key Concepts

Real Numbers

Real numbers are the set of numbers that include both rational and irrational numbers. We use real numbers in various mathematical operations such as addition, subtraction, multiplication, and division.
Rational numbers include fractions and integers, whereas irrational numbers have non-repeating, non-terminating decimal expansions, like \( \pi \) or \( \sqrt{2} \).
This broad category allows us to perform operations like subtraction because it includes negative numbers.

• They can be located on the number line, providing a sense of magnitude and direction.
• They follow certain algebraic properties such as the closure, associative, and commutative laws.

Understanding these numbers is essential to proving theorems involving arithmetic operations because it allows us to apply these core algebraic properties across all real numbers.

Subtraction

Subtraction, in its simplest form, involves taking away one quantity from another. Algebraically, it is often helpful to reframe subtraction as addition to help simplify complex expressions.
For example, subtracting a number is algebraically equivalent to adding its negative:

• \(a - b\) can be rewritten as \(a + (-b)\).
• In expressions, this property aids in maintaining consistency when applying laws such as the associative law.

By converting subtraction into addition, subtraction becomes uniform with other binary operations like addition. This view allows us to easily apply algebraic properties and manipulate expressions, especially when using laws like the associative law.

Proof by Algebraic Properties

When proving statements in algebra, such as the equivalence of the expressions \((a-b)-c = a-(b+c)\), we often rely on fundamental algebraic properties.
These can be thought of as the building blocks that make arithmetic valid over real numbers.

Use of Associative Law

This property allows us to regroup numbers without affecting their sum or difference:

• The associative property of addition states that \((a + b) + c = a + (b + c)\).
• This property ensures that regardless of how the numbers are grouped, the sum remains the same.

In the original problem, applying this property allows us to prove that both sides of the equation are equal by rearranging and regrouping differences as sums.This approach displays how abstraction and rules in algebra guide solving and reasoning processes, easing complex proofs and ensuring consistency across calculations.