The vanishing corridor
Two opposite corridors cancel. Pip has already booked both.
The formula in this story
Story 5 of 8
What you will take away
Explain why the two middle terms cancel in the difference-of-squares formula.
Read one piece at a time. There is no timer. “Help” stays beside you throughout.
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See the five reading steps
Choose a section directly
What do the words mean?
An identity holds for every permitted value, unlike an equation we solve for particular values. Factorization rewrites a sum as a product.
What are real numbers?
Real numbers are the numbers on the number line: integers, fractions and numbers such as √2. Our formulas apply to these numbers, even when the cake picture uses only positive lengths.
Want a hand with the prerequisites?The debt-eating pigeon · The missing two cake strips
A suspicious plan
Pip orders a rectangle with sides a + b and a − b. “Two corridors, please.”
Expand the product into four signed terms. The area picture is available only when a ≥ b ≥ 0.
The clue
Mika expands the bill: a² − ab + ba − b². The corridors carry opposite signs.
The middle terms are opposites: −ab + ba = 0.
The twist
Since ba = ab, the middle terms cancel. The hotel keeps a² − b². Pip asks where to put his suitcase.
Read the same equality backwards. With nonnegative lengths, rearrange the L-shaped area into a rectangle.
Change the evidence
Try examples to build intuition. Checking numbers is not the proof for every value.
Step 1 of 3
Write a − b as a + (−b), then distribute: a² − ab + ba − b².
The diagram illustrates the displayed argument. Numerical examples do not replace its hypotheses or prove every case.
The proof, with all its steps
Claim and conditions
For all real a, b, (a + b)(a − b) = a² − b². A rectangle interpretation additionally needs a ≥ b ≥ 0.
- Write a − b as a + (−b), then distribute: a² − ab + ba − b².
- Commutativity gives −ab + ba = 0. The remaining terms are a² − b².
- Read the equality backwards to factor a² − b². No division was used, so a = b is allowed too.
Pip has one more idea…
Can we cancel a − b from both sides when a = b?
Read the explanation for Yes, equal symbols may always be cancelled.
When a = b, the factor is zero. The identity remains true, but that division is undefined.
Read the explanation for No, that would divide by zero.
Exactly. An equality can be valid while a proposed step needs an extra nonzero condition.
What makes more sense now?
Two small checks, then your own reflection. You can use the hints and try again. This records these questions, not a score for all of mathematics.
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Your record for this story
Your first answers and your latest answers will appear here.
Where does your curiosity go next?
No score, no locked doors. You can return to any definition.
These are educational arguments and numeric experiments, not live Lean verification. The linked knowledge pages distinguish formal source excerpts from explanations.