The World of Numbers

A merchant in the port city of Lothal is exchanging bags of spices for copper ingots. He receives 15 ingots for every 2 bags of spices. If he brings 12 bags of spices to the market, how many copper ingots will he leave with?

Look at the sequence of numbers on one column of the Ishango bone: 11, 13, 17, 19. What do these numbers have in common? List the next three numbers that fit this pattern.

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.

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?

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.

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

A spice trader takes a loan (debt) of $\text{₹}850$. The next day, he makes a profit (fortune) of $\text{₹}1,200$. The following week, he incurs a loss of $\text{₹}450$. 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., $10 - (-5) = 15$).

Can you explain why we need $q \neq 0$ in the definition of a rational number?

While adding or subtracting two rational numbers having different denominators, how will you make the denominators equal?

Verify the distributive law for rational numbers.

Prove that the following rational numbers are equal:

Find the sum:

Find the difference:

Find the product:

Find the quotient:

Show that: $\left(\frac{1}{2} + \frac{3}{4}\right) \times \frac{8}{3} = \frac{1}{2} \times \frac{8}{3} + \frac{3}{4} \times \frac{8}{3}$.

Simplify the following using the distributive property: $\frac{7}{9}\left(\frac{6}{7} - \frac{3}{4}\right).$

Find the rational number $x$ such that: $\frac{5}{6}\left(x + \frac{3}{5}\right) = \frac{5}{6}x + \frac{1}{2}$.