Problem 10 - Algebra 1: Concepts and Skills

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

The theorem \(a - b = -b + a\) for any two real numbers \(a\) and \(b\) is proved by using the definition of subtraction \(a - b = a + (-b)\) and the commutative property of addition which states that \(a + (-b) = -b + a\).

Step 1: Understand the Definition of Subtraction

The definition of subtraction in terms of addition states: For any two real numbers \(a\) and \(b\), \(a - b\) is defined as \(a + (-b)\), where \(-b\) is the additive inverse of \(b\). Hence \(a - b = a + (-b)\).

Step 2: Apply the Commutative Property of Addition

The Commutative Property of Addition for real numbers tells us that for any two real numbers, their sum will be the same regardless of the order in which they are added. Therefore, \(a + (-b)\) is the same as \(-b + a\).

Step 3: Prove the Theorem

On piecing together Steps 1 and 2, we obtain that \(a - b = a + (-b) = -b + a\). Hence, by the definition of subtraction and the commutative property of addition, \(a - b = -b + a\) for any two real numbers \(a\) and \(b\).

Key Concepts

Commutative Property of Addition

Understanding the Commutative Property of Addition is like learning a simple dance move that works no matter who leads. At its core, this algebraic property ensures that, for any two numbers you choose to add together, the order doesn't change the sum.

In simpler terms, if you have two real numbers, let's say 3 and 5, their sum is always 8, whether you do 3 + 5 or 5 + 3. This property holds true for all real numbers, which includes the numbers we use in everyday life.

Applying it to our textbook problem, we're looking at the scenario of adding a number and its additive inverse (which we'll delve into shortly). The key takeaway here is that the commutative property allows us to flip-flop the addends. So, if you see an expression like \( a + (-b) \), you can confidently rearrange it to \( -b + a \) without changing the value, just as if you decided to dance with your partner starting with the other foot—it's the same dance!

Definition of Subtraction in Algebra

Subtracting numbers might seem straightforward. You have an amount, you take some away, and you're left with less right? But in algebra, subtraction hides a secret identity—it's actually addition in disguise! The definition of subtraction in algebra helps us understand this act of 'mathematical espionage'.

When we write \( a - b \), we're saying 'start with \( a \) and remove \( b \)'. But algebra twists the plot by defining subtraction as the addition of the additive inverse. Hence, \( a - b = a + (-b) \). Imagine you have 10 apples and you take away 3. Algebraically, you're really adding -3 apples to your 10. It sounds unusual, but this view allows us to use all of our addition tools, like the commutative property, to handle subtraction as well. That's why, in algebra, subtraction is not just 'taking away'—it's a reorganizing of sorts, inviting addition into the process.

Additive Inverse

In every superhero movie, there's a hero and their opposite— the additive inverse is that opposite in the world of numbers. It's the 'anti-number' that when added to the original number, results in zero. The nemesis of 7? It's -7. The archenemy of -9? That would be 9. The trick is to think of any number and its additive inverse as two sides of the same coin-- together they amount to nothing.

In our exercise, when we talk about \( a - b \), we are bringing into the equation the additive inverse of \( b \), which is \( -b \). To solve the problem, we add \( a \) and \( -b \) instead of thinking of it as taking away \( b \) from \( a \). This clever swap allows us to use the cooperative powers of the commutative property which is more restrictive in subtraction since it doesn't apply. Remember, additive inverses are crucial in algebra because they transform subtraction into addition, opening a world of possibilities for problem-solving.