Problem 15 - Algebra 1

Question

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

Step-by-Step Solution

Answer

Through the application of the additive inverse theorem, the definition of subtraction and the commutative property, we've proven that \(a - b = -b + a\).

Step 1: Write down the axiom of additive inverse

According to the field axioms of real numbers, for all real numbers \( a \), there exists a number \( -a \) such that \( a + (-a) = 0 \). This \( -a \) is the additive inverse of \( a \).

Step 2: Rewrite \( b \) as \( -(-b) \)

The task is to prove that \( a - b = -b + a \). In the first step, rewrite \( b \) as \( -(-b) \) because we know from step 1 that -b is an additive inverse of b.

Step 3: Write subtraction as the addition of additive inverse

The subtraction in algebra can be written as the addition of an additive inverse. Hence \( a - b = a + (-b) \).

Step 4: Use commutative property of addition

Using the commutative property of addition (which states that for any real numbers \( a \) and \( b \), \( a + b = b + a \)), we can write \( a + (-b) \) as \( -b + a \).

Key Concepts

Additive Inverse

The concept of the additive inverse is fundamental in algebra. It states that for every real number \( a \), there is a corresponding number \(-a\), such that \( a + (-a) = 0 \). This means \(-a\) is the additive inverse of \( a \).

This principle allows us to transition between numbers smoothly. It's important to note that the additive inverse is not just about negative numbers. For instance, the additive inverse of \(-b\) is \(b\) itself. This idea helps lay the foundation for understanding subtraction in algebra using addition.

Commutative Property

The commutative property of addition is a simple yet powerful rule in algebra. It states that for any real numbers \( a \) and \( b \), the sum \( a + b = b + a \).

This means that the order in which you add two numbers does not affect the result. In the context of the given problem, this property allows us to rearrange terms when proving that \( a - b = -b + a \). By writing subtraction as addition (using the additive inverse concept), we can apply the commutative property and rearrange the terms to complete the proof.

Subtraction in Algebra

In algebra, subtraction can be conveniently expressed using addition and the notion of additive inverse. Instead of subtracting \( b \) from \( a \), mathematicians prefer to write it as \( a + (-b) \).

This method might seem a bit unusual at first, but it simplifies many operations and calculations, making them more intuitive. It allows us to apply properties like commutativity and associativity which are well-defined for addition. Rewriting subtraction this way is particularly useful in solving equations and simplifying expressions, as seen in proving \( a-b = -b+a \). Here, understanding subtraction in terms of addition makes the proof straightforward and seamless.