Unit 1 · Topic 01 · Integers and Rationals
Kabir claimed that subtraction works like addition: "Just swap the order, the answer's the same either way."
Zara bet him a chocolate that 5 − 3 does NOT equal 3 − 5.
Kabir checked on paper and owed her a chocolate by lunch.
Kabir and Zara were revising for a maths quiz, arguing over which number rules actually always work. Kabir was sure that order never mattered for any operation — after all, 3 + 5 and 5 + 3 both give 8. "Same for subtraction," he said confidently. "5 − 3 is 2, and 3 − 5 is... also something, right?"
Zara worked it out: 5 − 3 = 2, but 3 − 5 = −2. "Those aren't the same number," she said. "Subtraction isn't commutative — swapping the order changes the answer, sometimes even its sign." Kabir owed her the bet.
"Okay, but grouping doesn't matter, right?" Kabir tried again, writing (10 − 4) − 2. That gave 6 − 2 = 4. Then he tried 10 − (4 − 2), which gave 10 − 2 = 8. "Those are different too!" he said, surprised. "So subtraction isn't associative either."
Zara nodded. "Addition and multiplication of integers ARE commutative and associative — you can swap order and regroup freely. But subtraction and division break both rules. That's not a coincidence; it's worth remembering as a genuine difference, not a technicality."
Kabir wanted one more test: does multiplication distribute over subtraction the same way it does over addition? He checked 3 × (5 − 2). That's 3 × 3 = 9. Then he tried (3 × 5) − (3 × 2), which is 15 − 6 = 9. "Those matched!" he said. "So distributivity DOES work with subtraction, even though commutativity and associativity don't."
"Right," said Zara. "Distributivity is its own separate property — it connects multiplication with addition OR subtraction, and it holds for both. Don't lump it in with commutative and associative just because they're all 'properties'."
By the end, Kabir had a working list: addition and multiplication are commutative and associative; subtraction and division are neither; but multiplication still distributes over both addition and subtraction. "I owe you one more chocolate," he admitted, "for making me actually check instead of assuming."
"That's the whole point," Zara said. "A property is only real if it holds for every example, not just the first one you tried."
A property of integers is a rule that holds for EVERY pair (or triple) of integers, not just some — one genuine counterexample is enough to disprove it. Commutativity means order doesn't matter: a+b=b+a and a×b=b×a hold for all integers, but a−b≠b−a and a÷b≠b÷a in general.
Associativity means grouping doesn't matter: (a+b)+c=a+(b+c) and (a×b)×c=a×(b×c) hold for all integers, but (a−b)−c≠a−(b−c) and division breaks the same way.
The distributive property connects multiplication with addition and subtraction: a×(b+c)=a×b+a×c, and equally a×(b−c)=a×b−a×c. Distributivity holds even though subtraction itself isn't commutative or associative — these are separate, independent properties.
Closure means an operation on two integers always produces another integer: integers are closed under addition, subtraction, and multiplication, but NOT under division, since dividing two integers can give a fraction.
Pick any three integers of your own and test whether (a − b) − c equals a − (b − c). Try at least two different sets of numbers.
Test closure under division: pick two integers where the division does NOT give a whole number, and confirm the result is not an integer.
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