We know that Natural Numbers are closed under addition (the sum of any two natural numbers is always a natural number). Are they closed under subtraction? Provide a couple of examples to justify your answer.
Answer: No. For example, 3 − 5 = −2 and 4 − 4 = 0, and neither −2 nor 0 is a natural number.
Step-by-step solution
Idea: ‘Closed under subtraction’ would mean every difference of two natural numbers is again a natural number. One example where the answer leaves the set is enough to show it is not closed.
- Natural numbers are 1, 2, 3, 4, …. To be closed under subtraction, a − b would have to be a natural number for every pair a, b.½ mark
- 3 − 5 = −2, which is not a natural number.½ mark
- 4 − 4 = 0, which is not a natural number either (natural numbers start at 1).½ mark
- So natural numbers are not closed under subtraction. (Some subtractions, like 9 − 2 = 7, do stay natural, but closure needs all of them to.)½ mark
No. 3 − 5 = −2 and 4 − 4 = 0 are not natural numbers, so natural numbers are not closed under subtraction.
Answer to write in the exam
No.
3 − 5 = −2, and −2 is not a natural number.
4 − 4 = 0, and 0 is not a natural number.
∴ Natural numbers are not closed under subtraction.
Common mistakes that cost marks
- Giving only examples like 9 − 2 = 7 and concluding ‘yes’. Examples that work do not prove closure; one failure disproves it.
- Counting 0 as a natural number. Here natural numbers are 1, 2, 3, …, so 0 is outside the set.
- Saying ‘not closed’ without any example; the question asks for examples as justification.
How this can come in the exam
MCQ (1 mark)
The set of natural numbers is closed under
- subtraction only
- division only
- addition and multiplication
- all four operations
Show answer
(C) addition and multiplication
Sums and products of natural numbers are natural; 2 − 5 = −3 and 2 ÷ 5 = 25 are not.
Short answer (2 marks)
Are natural numbers closed under division? Justify with an example.
Show answer
No (1 mark). For example 3 ÷ 4 = 34, which is not a natural number (1 mark).Try one yourself
Are integers closed under subtraction? Give a reason.
Show answer
Yes. The difference of any two integers is an integer, e.g. (−3) − 7 = −10 and 5 − (−2) = 7.
More questions like this
- Ancient Indians used the joints of their fingers to count, a practice still seen today. Each finger has 3 joints, and the thumb is used to count them. How many can you count on one hand? How does this relate to the ancient base-12 counting systems?
- Brahmagupta did not stop at zero. He realised that if subtraction of a number from itself can result in zero (5 − 5 = 0), then what would happen if we subtracted a larger number from a smaller one (3 − 5 = □)?
- Why does a negative times a negative equal a positive? Think of it in terms of action and debt. If a negative number represents a debt, then multiplying by a negative number represents the removal of that debt.
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- A spice trader takes a loan (debt) of ₹850. The next day, he makes a profit (fortune) of ₹1,200. The following week, he incurs a loss of ₹450. Write this sequence as an equation using integers and calculate his final financial standing.