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 = □)?
Answer: 3 − 5 = −2. Subtracting a larger number from a smaller one goes below zero and gives a negative number.
Step-by-step solution
Idea: On the number line, subtracting 5 means moving 5 steps to the left. From 3, the first 3 steps reach 0 and the last 2 steps go past zero, to −2. Brahmagupta called such numbers debts.
- Start at 3 on the number line and move 5 steps left: 3 → 2 → 1 → 0 → −1 → −2.½ mark
- So 3 − 5 = −2, a number less than zero: a negative number. In money terms, if you have ₹3 and must pay ₹5, you end with a debt of ₹2.½ mark
3 − 5 = −2: the answer is a negative number (a debt of 2).
Check: −2 + 5 = 3 ✓.
Answer to write in the exam
3 − 5: move 5 steps left from 3 → 2, 1, 0, −1, −2
∴ 3 − 5 = −2, a negative number (a debt of 2).
Common mistakes that cost marks
- Writing 3 − 5 = 2 by subtracting the smaller from the larger. The order matters.
- Saying 3 − 5 ‘cannot be done’. It cannot be done in natural numbers, but it can in integers.
- Writing 3 − 5 = −8 by adding the numbers and putting a minus sign.
How this can come in the exam
MCQ (1 mark)
7 − 12 =
- 5
- −5
- 19
- −19
Show answer
(B) −5
Moving 12 steps left from 7 reaches −5.
Short answer (2 marks)
Riya has ₹40 and buys a book for ₹65, borrowing the rest. Write the subtraction and interpret the answer.
Show answer
40 − 65 = −25 (1 mark). The negative answer means Riya owes ₹25: a debt of ₹25 (1 mark).Try one yourself
Find 6 − 14 and say what the sign tells you.
Show answer
6 − 14 = −8. It is negative because a larger number was taken from a smaller one.
More questions like this
- 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.
(Hint: If someone takes away (−) four of your debts that are each worth ₹3 (that is, −3), you are effectively ₹12 richer! Therefore, (−3) × (−4) = +12.) - The temperature in the high-altitude desert of Ladakh is recorded as 4 °C at noon. By midnight, it drops by 15 °C. What is the midnight temperature?
- 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.
- Calculate the following using Brahmagupta’s laws:
- Explain, using a real-world example of debt, why subtracting a negative number is the same as adding a positive number (e.g., 10 − (−5) = 15).