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?

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:

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

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

(iii) $0 - (-14)$

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

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

Prove that the following rational numbers are equal:

(i) $\frac{2}{3}$ and $\frac{4}{6}$ (ii) $\frac{5}{4}$ and $\frac{10}{8}$ (iii) $-\frac{3}{5}$ and $-\frac{6}{10}$ (iv) $\frac{9}{3}$ and $3$

Find the sum:

(i) $\frac{2}{5} + \frac{3}{10}$ (ii) $\frac{7}{12} + \frac{5}{8}$ (iii) $-\frac{4}{7} + \frac{3}{14}$

Find the difference:

(i) $\frac{5}{6} - \frac{1}{4}$ (ii) $\frac{11}{8} - \frac{3}{4}$ (iii) $-\frac{7}{9} - \left(-\frac{2}{3}\right)$

Find the product:

(i) $\frac{2}{3} \times \frac{3}{10}$ (ii) $\frac{7}{11} \times \frac{5}{8}$ (iii) $-\frac{4}{7} \times \frac{5}{14}$

Find the quotient:

(i) $\frac{2}{3} \div \frac{3}{10}$ (ii) $\frac{7}{11} \div \frac{5}{8}$ (iii) $-\frac{4}{7} \div \frac{5}{14}$

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:

$\frac{5}{6}\left(x + \frac{3}{5}\right) = \frac{5}{6}x + \frac{1}{2}$

Represent the rational numbers $\frac{2}{3}$, $-\frac{5}{4}$ and $1\frac{1}{2}$ on a single number line.

Find three distinct rational numbers that lie strictly between $-\frac{1}{2}$ and $\frac{1}{4}$.

Simplify the expression: $\left(-\frac{1}{4}\right) + \left(\frac{5}{12}\right)$.

A tailor has $15\frac{3}{4}$ metres of fine silk. If making one kurta requires $2\frac{1}{4}$ metres of silk, exactly how many kurtas can he make?

Find three rational numbers between 3.1415 and 3.1416.

Can you think of other way(s) to find a rational number between any two rational numbers?

Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: $\frac{7}{20}$, $\frac{4}{15}$ and $\frac{13}{250}$. Then check your answers by explicitly performing the long divisions and expressing these rational numbers as decimals.

Perform the long division for $\frac{1}{13}$. Identify the repeating block of digits. Does it show cyclic properties if you evaluate $\frac{2}{13}$? Now compute $\frac{3}{13}$, $\frac{4}{13}$, etc. What do you notice?

Classify the following numbers as rational or irrational:

(i) $\sqrt{81}$

(ii) $\sqrt{12}$

(iii) $0.33333 \dots$

(iv) $0.123451234512345 \dots$

(v) $1.01001000100001 \dots$ (Notice the pattern: Is it repeating a single block?)

(vi) $23.560185612239874790120$

Find the explicit fractions in case they are rational.

The number $0.\overline{9}$ (which means $0.99999\dots$) is a rational number. Using algebra (let $x = 0.\overline{9}$, multiply by 10, and subtract), explain why $0.\overline{9}$ is exactly equal to 1.

*5. We have seen that the repeating block of $\frac{1}{7}$ is a cyclic number. Try to find more numbers ($n$) whose reciprocals ($\frac{1}{n}$) produce decimals with repeating blocks that are cyclic.

Convert the following rational numbers in the form of a terminating decimal or non-terminating and repeating decimal, whichever the case may be, by the process of long division:

(i) $\frac{3}{50}$ (ii) $\frac{2}{9}$

Prove that $\sqrt{5}$ is an irrational number.

Convert the following decimal numbers in the form of $\frac{p}{q}$.

(i) $12.6$ (ii) $0.0120$ (iii) $3.05\bar{2}$ (iv) $1.2\overline{35}$ (v) $0.\overline{23}$ (vi) $2.0\bar{5}$ (vii) $2.12\bar{5}$ (viii) $3.12\bar{5}$ (ix) $2.\overline{1625}$

Locate the following rational numbers on the number line.

(i) $0.532$ (ii) $1.1\bar{5}$

Find 6 rational numbers between 3 and 4.

Find 5 rational numbers between $\frac{2}{5}$ and $\frac{3}{5}$.

Find 5 rational numbers between $\frac{1}{6}$ and $\frac{2}{5}$.

If $\frac{x}{3} + \frac{x}{5} = \frac{16}{15}$, find the rational number $x$.

Let $a$ and $b$ be two non-zero rational numbers such that $a + \frac{1}{b} = 0$. Without assigning any numerical values, determine whether $ab$ is positive or negative. Justify your answer.

A rational number has a terminating decimal expansion whose last non-zero digit occurs in the 4th decimal place. Show that such a number can be written in the form $\frac{p}{10^4}$, where $p$ is an integer not divisible by 10. Is it necessary that the denominator of this rational number, when written in the lowest form, is divisible by $2^4$ or $5^4$? Give reasons.

Without performing division, determine whether the decimal expansion of $\frac{18}{125}$ is terminating or non-terminating. If it terminates, state the number of decimal places.

A rational number in its lowest form has denominator $2^3 \times 5$. How many decimal places will its decimal expansion have? Explain your answer.

Let $a = \frac{7}{12}$ and $b = \frac{5}{6}$. Express both $a$ and $b$ in the form $\frac{k_1}{m}$ and $\frac{k_2}{m}$ where $k_1$, $k_2$ and $m$ are integers and $k_2 - k_1 > 6$. Using the same denominator $m$, write exactly five distinct rational numbers lying between $a$ and $b$ keeping an integer numerator. Explain why the condition $k_2 - k_1 > n + 1$ is necessary to find $n$ such rational numbers between the two rational numbers $a$ and $b$ using this method.

Three rational numbers $x$, $y$, $z$ satisfy $x + y + z = 0$ and $xy + yz + zx = 0$. Show that all the rational numbers $x, y, z$ must be simultaneously zero.

Show that the rational number $\frac{(a+b)}{2}$ lies between the rational numbers $a$ and $b$.

Find the lengths of the hypotenuses of all the right triangles in Fig. 3.14 which is referred to as the square root spiral.