Proof: Multiplying Denominators

Let's prove the following theorem:

For clarity, here is the conclusion in fraction notation:

$\frac{1}{a}\cdot \frac{1}{b}=\frac{1}{a\cdot b}$

This theorem states that when we are multiplying fractions, and the numerator is 1, then we can just multiply the denominators.

Before you read the proof, we encourage you to try to prove this theorem on your own.

This proof multiplies both sides of = by a ⋅ b and shows that both sides become 1. This means that the two sides are equal. The proof then divides both sides by a ⋅ b to reach the conclusion.

Proof:

Proof Table

#	Claim	Reason
1	(a ⋅ b) ⋅ (1 / (a ⋅ b)) = 1	if not (a ⋅ b = 0), then (a ⋅ b) ⋅ (1 / (a ⋅ b)) = 1 (Algebra 17)
2	(a ⋅ b) ⋅ ((1 / a) ⋅ (1 / b)) = 1	if not (a = 0) and not (b = 0), then (a ⋅ b) ⋅ ((1 / a) ⋅ (1 / b)) = 1 (Product is One 3)
3	(a ⋅ b) ⋅ ((1 / a) ⋅ (1 / b)) = (a ⋅ b) ⋅ (1 / (a ⋅ b))	if (a ⋅ b) ⋅ ((1 / a) ⋅ (1 / b)) = 1 and (a ⋅ b) ⋅ (1 / (a ⋅ b)) = 1, then (a ⋅ b) ⋅ ((1 / a) ⋅ (1 / b)) = (a ⋅ b) ⋅ (1 / (a ⋅ b)) (Transitive Property of Equality Variation 1)
4	(1 / a) ⋅ (1 / b) = 1 / (a ⋅ b)	if (a ⋅ b) ⋅ ((1 / a) ⋅ (1 / b)) = (a ⋅ b) ⋅ (1 / (a ⋅ b)) and not (a ⋅ b = 0), then (1 / a) ⋅ (1 / b) = 1 / (a ⋅ b) (Removing the Common Term)

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