Axiomatic Justification of Negative Number Operations

1. The Starting Point: Proving a ⋅ 0 = 0

Before we can tackle negative numbers, we need to formally prove something that seems obvious: any number multiplied by zero is zero. In our axiomatic system, this is not an assumption but a theorem we can derive. The proof relies beautifully on the axioms you've already mastered.

Here's the logical flow:

1. We start with a known truth based on the Additive Identity Property: $0+0=0$.
2. We can multiply both sides of this equation by any number $a$, which gives us $a\cdot (0+0)=a\cdot 0$.
3. Using the Distributive Property on the left side, we get $a\cdot 0+a\cdot 0=a\cdot 0$.
4. Now, we have an equation. Let the term $a\cdot 0$ be called $x$. The equation is $x+x=x$.
5. We add the additive inverse of $x$ (which is $-x$) to both sides: $(x+x)+(-x)=x+(-x)$.
6. Using the Associative Property: $x+(x+(-x))=x+(-x)$.
7. Using the Additive Inverse Property: $x+0=0$.
8. Finally, using the Additive Identity Property: $x=0$.

So, we have proven that $a\cdot 0=0$.

To see this proof in action, watch the following short video.

2. Proving the Rules of Negative Multiplication

With the theorem $a\cdot 0=0$ established, we can now proceed to the main proofs. We will logically deduce the rules for multiplying with negatives.

The following video from the EpsilonDelta channel provides elegant proofs for each step. The video uses the term 'ring', which is a concept from abstract algebra for a set with addition and multiplication that follows most of the axioms we've learned. The integers, rational numbers, and real numbers are all examples of rings.

Let's summarize the logic for the final, most important proof: (-a)(-b) = ab.

1. From the first video segment, we know that for any $x$ and $y$, (-x)y = -(xy).
2. Let's apply this rule to (-a)(-b). Let $x=a$ and $y=-b$. This gives us (-a)(-b) = -(a(-b)).
3. We also know from the first segment that a(-b) = -(ab).
4. Substituting this into our equation: (-a)(-b) = -(-(ab)).
5. From the second video segment, we know that for any number $z$, -(-z) = z.
6. Therefore, -(-(ab)) = ab.

We have successfully proven (-a)(-b) = ab using only our foundational axioms.

3. An Alternative Proof: The Elegance of a Single Expression

The sequential proofs we've seen are logical and clear. However, mathematicians often seek proofs that are not just correct but also "elegant" in their construction. The following resource shows a clever way to prove (-a)(-b) = ab by constructing a single expression and simplifying it in two different ways.

This method is powerful because it collapses multiple steps into one by setting up the initial expression perfectly. It's a great example of how a deep understanding of the axioms allows for creative problem-solving.

4. What if the Rule Were Different?

Your goal is to understand why things are the way they are. A powerful way to do this is to ask, "What would happen if they were different?" What if we decided to define negative times negative as negative? Would our system of math still work?

The answer is no. If we were to declare that (-1)(-1) = -1, it would break one of our most important axioms: the distributive property.

Let's see how:

1. Consider the expression (-1) ⋅ (1 + (-1)).
2. We know from the Additive Inverse Property that 1 + (-1) = 0. So, the expression is (-1) ⋅ 0, which we've proven is 0.
3. Now, let's use the Distributive Property on the original expression: (-1) ⋅ (1 + (-1)) = (-1) ⋅ 1 + (-1) ⋅ (-1).
4. We know (-1) ⋅ 1 = -1 (from the Multiplicative Identity and the rule (-a)b = -(ab)). So we have -1 + (-1) ⋅ (-1).
5. If we were to use our hypothetical, "broken" rule that (-1) ⋅ (-1) = -1, the expression would become -1 + (-1), which equals -2.

By evaluating the same expression in two ways, we get 0 = -2, a contradiction. This shows that the only possible value for (-1)(-1) that keeps the distributive property intact is +1. The rule is not a choice; it's a logical consequence.

Summary and Next Steps

Today, we moved from defining axioms to using them as tools for proof. This is the heart of "doing mathematics."

• Key Takeaways:
  • The rules for multiplying negative numbers are not arbitrary; they are logical consequences of the Field Axioms.
  • To maintain consistency, especially with the distributive property, the product of two negative numbers must be positive.
  • We proved this by first establishing that a ⋅ 0 = 0, then using that to show (-a)b = -(ab), and finally using that result to prove (-a)(-b) = ab.
  • Elegant proofs, like evaluating a single expression in two ways, are a hallmark of mathematical thinking.

In the previous lesson, we briefly noted that subtraction is defined as adding the additive inverse, and division is multiplying by the multiplicative inverse. In our next lesson, we will explore the full implications of this, justifying the familiar procedures for subtraction and division and explaining why division by zero is undefined from an axiomatic standpoint.