NCERT Solutions Class 9 Maths Exercise 3.2 — The World of Numbers

NCERT Class 9 Mathematics — Ganita Manjari Part I, The World of Numbers, Exercise Set 3.2. All 4 questions solved.

Exercise Set 3.1 showed that the natural numbers break under subtraction. This exercise works with the repair: zero and the negative numbers, formalised by Brahmagupta in 628 CE, which together with the naturals make the integers $\mathbb{Z}$.

Brahmagupta stated the rules in the language of dhana (fortune) and ṛiṇa (debt), and that language is still the quickest way to see why the rules are what they are:

• a debt subtracted from nothing is a fortune;
• a debt multiplied by a debt is a fortune;
• a fortune multiplied by a debt is a debt.

In modern notation those are $0 – (-a) = a$, $(-a)(-b) = ab$ and $a(-b) = -ab$.

Key insight. Every sign rule in this exercise says the same thing: a minus sign means reverse. Subtracting is the reverse of adding, and a negative number is the reverse of a positive one. Reverse something twice and you are back where you started — which is why two minus signs make a plus, in $0 – (-14)$ just as in $(-8) \times (-7)$.

Question 1

The temperature in the high-altitude desert of Ladakh is recorded as $4\,°\mathrm{C}$ at noon. By midnight, it drops by $15\,°\mathrm{C}$. What is the midnight temperature?

Solution. A drop is a subtraction, so the midnight temperature is

$$4 – 15 = -11$$

The subtraction takes us past zero, which is exactly the situation the naturals could not describe. On a number line, start at $4$ and move $15$ units to the left: four of them reach $0$, and the remaining eleven carry on to $-11$.

$$4 – 15 = 4 – 4 – 11 = 0 – 11 = -11$$

Question 2

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.

Solution. Translate each event into a signed number — debts and losses negative, profits positive — and write them as one running total.

• a debt of ₹$850$: $\;-850$
• a profit of ₹$1200$: $\;+1200$
• a loss of ₹$450$: $\;-450$

$$-850 + 1200 – 450 = ?$$

Work left to right:

$$-850 + 1200 = 350$$ $$350 – 450 = -100$$

So the equation and its value are

$$-850 + 1200 – 450 = -100$$

Reading the answer back into the trader’s situation: the final total is negative, so he is in debt by ₹$100$ — not merely out of profit, but a hundred rupees worse off than having nothing.

Question 3

Calculate the following using Brahmagupta’s laws:

Solution. Take each in turn, naming the rule being used.

(i) $(-12) \times 5$

A debt multiplied by a fortune is a debt — twelve debts of five each, or five debts of twelve each.

$$(-12) \times 5 = -60$$

(ii) $(-8) \times (-7)$

A debt multiplied by a debt is a fortune. Removing eight debts of ₹$7$ each leaves you ₹$56$ better off.

$$(-8) \times (-7) = +56$$

(iii) $0 – (-14)$

Subtracting a debt from nothing is a fortune — this is Brahmagupta’s rule stated almost word for word.

$$0 – (-14) = +14$$

(iv) $(-20) \div 4$

Division follows the same sign rule as multiplication, since it is its reverse. A debt shared among four is a debt.

$$(-20) \div 4 = -5$$

Check by multiplying back: $(-5) \times 4 = -20$ ✓

Question 4

Explain, using a real-world example of debt, why subtracting a negative number is the same as adding a positive number (e.g., $10 – (-5) = 15$).

Solution. Think of your total wealth as a single number: positive if you own more than you owe, negative if you owe more than you own.

Suppose you have ₹$10$ in your pocket and also a note recording a debt of ₹$5$ — so the debt contributes $-5$ to your position. Now a friend cancels that debt: he tears up the note and tells you not to repay it.

Nothing has physically entered your pocket. You still have the same ₹$10$ in cash. But removing a $-5$ from your accounts leaves you $5$ better off than you were, because you no longer owe it:

$$10 – (-5) = 10 + 5 = 15$$

Taking away a debt is the same as being given money. Subtraction means “remove”; a negative number is an obligation; removing an obligation improves your position by exactly the amount of the obligation. So the two minus signs — one for the removal, one for the debt — cancel each other.

The same idea in motion: facing backwards (negative) and walking backwards (subtract) moves you forwards.

Common mistakes

• Question 1, answering $11\,°\mathrm{C}$ or $-19\,°\mathrm{C}$. The first ignores the sign — the temperature falls below zero, so the answer must be negative. The second adds the drop to $4$ with a minus attached. A drop of $15$ from $4$ is $4 – 15$, and stepping along a number line settles it.
• Question 2, adding the loss. A loss is negative, so it is subtracted. Writing $-850 + 1200 + 450 = 800$ turns a trader in debt into one comfortably in profit.
• Question 2, answering $100$. The arithmetic gives $-100$, and the sign is the whole point: he owes money rather than having it. State the sign and then say what it means.
• Question 3(ii), answering $-56$. Two negatives multiplied give a positive. Brahmagupta’s rule — a debt of a debt is a fortune — is the reason, and the debt-cancelling picture in Question 4 is the justification.
• Question 3(iii), answering $-14$. $0 – (-14)$ removes a debt of $14$ from nothing, leaving $+14$. Reading it as $0 – 14$ drops one of the two minus signs.
• Question 3(iv), assuming division has different sign rules. It does not. Division is the reverse of multiplication, so the signs behave identically, and multiplying the answer back is a free check.
• Question 4, restating the rule instead of explaining it. “Two minuses make a plus” is the thing to be explained, not the explanation. The answer needs a situation in which removing something negative visibly leaves you better off.

Practise next

• Exercise Set 3.3 — the next extension of the number system, from integers to rational numbers, where these same sign rules carry over to fractions.
• The End-of-Chapter Exercises, Question 9 — uses the sign rules in reverse: from $a + \frac{1}{b} = 0$ you must decide the sign of $ab$ without ever choosing numbers.