The debt-eating pigeon
Erasing a debt feels like a bonus. But feelings are not yet a proof.
The formula in this story
Story 3 of 8
What you will take away
Explain the positive result using opposite numbers, not just a memorized sign rule.
Read one piece at a time. There is no timer. “Help” stays beside you throughout.
Before reading, how does it feel? (optional)
See the five reading steps
Choose a section directly
What do the words mean?
The opposite of a number a is −a: together they add to zero. Its technical name is additive inverse. Taking the opposite twice returns a.
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 bakery with missing buns
A suspicious plan
Pip eats a debt receipt. “Minus times minus is delicious!” Mika: “We need a reason that survives paper.”
A number and its opposite cancel. Distribution makes ab and a(−b) a cancelling pair.
The clue
They place b and −b together: zero. Multiplying the whole pair by a still gives zero.
Replace a by −a in that rule. The product is the opposite of (−a)b.
The twist
Distribution makes the opposite numbers appear. Pip puts down his fork: the rule follows from arithmetic, not appetite.
Take the opposite of a negative product: two reversals return to ab.
Change the evidence
Try examples to build intuition. Checking numbers is not the proof for every value.
Step 1 of 3
Distribution gives ab + a(−b) = a(b + (−b)) = a·0 = 0. Thus a(−b) = −(ab). Similarly (−a)b = −(ab).
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 and b, (−a)(−b) = ab.
- Distribution gives ab + a(−b) = a(b + (−b)) = a·0 = 0. Thus a(−b) = −(ab). Similarly (−a)b = −(ab).
- Apply the first rule with −a in place of a: (−a)(−b) = −((−a)b).
- Substitute (−a)b = −(ab). The result is −(−(ab)) = ab, because taking the opposite twice restores the number.
Pip has one more idea…
What makes the argument apply to fractions too?
Read the explanation for Fractions obey the same distribution and inverse laws.
Yes. The argument used the laws, not the size or integer status of a and b.
Read the explanation for We checked (−2)(−3), so every case is proved.
One example can check an answer, but it cannot establish a claim for every real number.
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.
How does it feel now? (optional)
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.