Proof: Multiply Denominators
Let's prove the following theorem:
if the following are true:
- not (a = 0)
- not (b = 0)
- not (a ⋅ b = 0)
then (1 / a) ⋅ (1 / b) = 1 / (a ⋅ b)
Before you read the proof, we encourage you to try to prove this theorem on your own.
This proof multiplies both sides by (a ⋅ b), then shows that both sides are 1. It then divides both sides by (a ⋅ b).
Proof:
Given
| 1 | not (a = 0) |
|---|---|
| 2 | not (b = 0) |
| 3 | not (a ⋅ b = 0) |
| # | Claim | Reason |
|---|---|---|
| 1 | (a ⋅ b) ⋅ (1 / (a ⋅ b)) = 1 | if not (a ⋅ b = 0), then (a ⋅ b) ⋅ (1 / (a ⋅ b)) = 1 |
| 2 | (a ⋅ b) ⋅ ((1 / a) ⋅ (1 / b)) = 1 | if not (a = 0) and not (b = 0), then (a ⋅ b) ⋅ ((1 / a) ⋅ (1 / b)) = 1 |
| 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)) |
| 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) |
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