NCERT Solutions · Class 9 Mathematics The World of Numbers

A drop is the addition of a negative number."Drops by \(\displaystyle 15^\circ\)C" means the change is \(\displaystyle -15\), so the midnight temperature is\[4 + (-15) = 4 - 15. \]Working it out on the number line. Start at \(\displaystyle 4\) and move \(\displaystyle 15\) steps to the left. The first \(\displaystyle 4\) steps bring you down to \(\displaystyle 0\). That uses up \(\displaystyle 4\) of the \(\displaystyle 15\) steps, leaving\[15 - 4 = 11 \]steps still to take, and those \(\displaystyle 11\) steps go below zero:\[4 - 15 = -11. \]Check. Reverse the change: if it is \(\displaystyle -11\,^\circ\)C at midnight and the day was \(\displaystyle 15^\circ\) warmer, then \(\displaystyle -11 + 15 = 4\,^\circ\)C. Correct.The midnight temperature is \(\displaystyle -11\,^\circ\)C ($\displaystyle 11$ degrees below zero).

Debt is negative, fortune is positive.This is exactly Brahmagupta's language: a debt (rina) is written with a minus sign, a fortune (dhana) with a plus sign. Each event becomes one signed number:

Event	In Brahmagupta's words	As an integer (rupees)
Loan taken	a debt of \(\displaystyle 850\)	\(\displaystyle -850\)
Profit made	a fortune of \(\displaystyle 1200\)	\(\displaystyle +1200\)
Loss incurred	a debt of \(\displaystyle 450\)	\(\displaystyle -450\)

His final standing is the sum of all three:\[(-850) + 1200 + (-450) \]Step 1. \(\displaystyle (-850) + 1200 = 350\). (The fortune is larger than the debt, so what is left is a fortune of \(\displaystyle 1200 - 850 = 350\).)Step 2. \(\displaystyle 350 + (-450) = -100\). (Now the debt is larger, so what is left is a debt of \(\displaystyle 450 - 350 = 100\).)\[(-850) + 1200 + (-450) = -100 \]Check by grouping differently. Add the two debts first: \(\displaystyle (-850) + (-450) = -1300\); then \(\displaystyle -1300 + 1200 = -100\). Same answer, as it must be, since addition of integers is associative.His final standing is \(\displaystyle -100\); that is, he ends up ₹$\displaystyle 100$ in debt.

Brahmagupta's sign laws. In the Brahmasphutasiddhanta ($\displaystyle 628$ CE) Brahmagupta stated the rules we still use, in terms of fortunes (positive) and debts (negative): (i) \(\displaystyle (-12) \times 5\) — a debt times a fortune is a debt. Multiply the sizes, \(\displaystyle 12 \times 5 = 60\), then attach the minus sign:\[(-12) \times 5 = -60 \](Sense check: taking on a debt of \(\displaystyle 12\) five times over is a debt of \(\displaystyle 60\).)(ii) \(\displaystyle (-8) \times (-7)\) — a debt times a debt is a fortune. Multiply the sizes, \(\displaystyle 8 \times 7 = 56\):\[(-8) \times (-7) = +56 \](iii) \(\displaystyle 0 - (-14)\) — a debt subtracted from zero becomes a fortune. Removing a debt of \(\displaystyle 14\) leaves you \(\displaystyle 14\) better off:\[0 - (-14) = +14 \](iv) \(\displaystyle (-20) \div 4\) — a debt shared into \(\displaystyle 4\) equal parts gives debts. Divide the sizes, \(\displaystyle 20 \div 4 = 5\), and keep the minus sign:\[(-20) \div 4 = -5 \](Check by multiplying back: \(\displaystyle (-5) \times 4 = -20\).)(i) \(\displaystyle -60\) (ii) \(\displaystyle 56\) (iii) \(\displaystyle 14\) (iv) \(\displaystyle -5\).

Subtracting means removing — and removing a debt makes you richer.A ledger example. Suppose your accounts read like this:

Item	Value in rupees
Cash in hand	\(\displaystyle +15\)
Debt owed to a friend	\(\displaystyle -5\)
Net worth	\(\displaystyle +15 + (-5) = 10\)

So your net worth today is ₹10.Now your friend says, "Forget the loan." That entry of \(\displaystyle -5\) is removed from your books — in symbols, it is subtracted from your net worth:\[10 - (-5). \]Nothing was handed to you, but the burden dragging your total down is gone. Your accounts now hold only the cash:\[\text{new net worth} = 15. \]Comparing the two lines, \(\displaystyle 10 - (-5) = 15 = 10 + 5\). Cancelling a debt of \(\displaystyle 5\) has exactly the same effect on you as being given a gift of \(\displaystyle 5\). That is the whole meaning of the rule.A second way to see it — continue the pattern. Watch what happens to \(\displaystyle 10 - n\) as \(\displaystyle n\) walks down through zero:\[10 - 3 = 7,\quad 10 - 2 = 8,\quad 10 - 1 = 9,\quad 10 - 0 = 10, \] \[10 - (-1) = 11,\quad 10 - (-2) = 12,\quad \ldots,\quad 10 - (-5) = 15. \]Each time the number being subtracted goes down by \(\displaystyle 1\), the answer goes up by \(\displaystyle 1\). For the pattern to stay unbroken as we cross zero, \(\displaystyle 10 - (-5)\) has to be \(\displaystyle 15\).A third way — subtraction as "what must be added". By definition, \(\displaystyle 10 - (-5)\) is the number that must be added to \(\displaystyle -5\) to get back to \(\displaystyle 10\):\[(-5) + \square = 10 \quad \Longrightarrow \quad \square = 15, \]since \(\displaystyle -5 + 15 = 10\).(Your own real-world example may be different — a cancelled fine, a refunded penalty or a reversed bank charge all illustrate the same rule, and any of them is an equally good answer.)All three arguments agree: \(\displaystyle 10 - (-5) = 10 + 5 = 15\). Subtracting a negative number is the same as adding the corresponding positive number, because removing a debt is a gain.