The World of Numbers Class 9 Mathematics

20 Questions | 54 Marks

The Lebombo bone, discovered in South Africa, is believed to be used for what purpose? 1mark

Explain the concept of one-to-one correspondence using the example of a herder and his cattle. 2marks

What is the significance of the number groupings found on the Ishango bone? 2marks

Brahmagupta formally introduced negative numbers by relating them to which real-world concept? 1mark

If 'a' is any integer, what is the result of

a + 0

and

a \times 0

? 1mark

Convert the rational number

\frac{3}{4}

into its decimal form. Is it terminating or repeating? 2marks

Convert the decimal

0.75

into a fraction in its simplest form. 2marks

What are Brahmagupta's rules for adding and subtracting zero from any number? 1mark

Prove that

\sqrt{5}

is an irrational number using the method of proof by contradiction. 5marks

A merchant receives 8 copper ingots for every 3 bags of spices he sells. If he brings 15 bags of spices to the market, how many copper ingots will he receive? 2marks

Explain why the denominator

q

cannot be zero in the definition of a rational number

\frac{p}{q}

. 1mark

Find the sum of two rational numbers:

\frac{2}{5} + \frac{7}{10}

. 2marks

Find the product of two rational numbers:

\frac{- 3}{4} \times \frac{8}{9}

. 2marks

Represent the rational number

\frac{3}{5}

on the number line. 3marks

The temperature in a city at noon is

10^{\circ} C

. By midnight, it drops by

18^{\circ} C

. What is the midnight temperature? 2marks

Between any two distinct rational numbers, how many other rational numbers can be found? Justify your answer. 2marks

Without performing long division, determine if the decimal expansion of

\frac{7}{12}

will be terminating or repeating. Explain why. 3marks

Convert the repeating decimal

0.454545 \dots

into a fraction in its simplest form. 3marks

The distance between two points on a number line is given by the absolute value of their difference. Calculate the distance between

- 5

and

3

. 2marks

Explain the difference between rational and irrational numbers with examples. 3marks

Answers

The Lebombo bone is believed to be used as a lunar phase counter or a menstrual calendar.

One-to-one correspondence is a method where early humans matched one object to another. For example, a herder could place one pebble in a pot for each cow that left the settlement and remove one pebble for each cow that returned. This ensured they knew if any cows were missing.

The specific grouping of notches on the Ishango bone, such as prime numbers between 10 and 20 (11, 13, 17, 19) and doubling, suggests that early humans understood abstract mathematical concepts like prime numbers and multiplication.

Brahmagupta formally introduced negative numbers by relating them to debts (Ṛiṇa) as opposed to fortunes (Dhana).

a + 0 = a

and

a \times 0 = 0

\frac{3}{4} = 0.75

. This is a terminating decimal.

0.75 = \frac{75}{100} = \frac{3 \times 25}{4 \times 25} = \frac{3}{4}

Brahmagupta's rules for zero are:

When zero is added to a number, the number remains unchanged: $a + 0 = a$ .
When zero is subtracted from a number, the number remains unchanged: $a - 0 = a$ .
9. Proof by contradiction:
Step 1: Assume $\sqrt{5}$ is rational. Then it can be written as $\frac{p}{q}$ where $p, q$ are integers, $q \neq 0$ , and $p, q$ are co-prime.
Step 2: Square both sides: $5 = \frac{p^{2}}{q^{2}} ⟹ p^{2} = 5 q^{2}$ .
Step 3: This means $p^{2}$ is a multiple of 5, so $p$ must also be a multiple of 5. Let $p = 5 k$ for some integer $k$ .
Step 4: Substitute $p = 5 k$ into the equation: $(5 k)^{2} = 5 q^{2} ⟹ 25 k^{2} = 5 q^{2} ⟹ 5 k^{2} = q^{2}$ .
Step 5: This means $q^{2}$ is a multiple of 5, so $q$ must also be a multiple of 5.
Step 6: Contradiction: We found that both $p$ and $q$ are multiples of 5, which contradicts our initial assumption that $p$ and $q$ are co-prime. Therefore, $\sqrt{5}$ must be irrational.
10. If 3 bags of spices yield 8 copper ingots, then 15 bags of spices will yield $\frac{15 bags}{3 bags} \times 8 ingots = 5 \times 8 = 40$ copper ingots.
11. If $q = 0$ , the expression $\frac{p}{q}$ would involve division by zero, which is undefined in mathematics.
12. $\frac{2}{5} + \frac{7}{10} = \frac{2 \times 2}{5 \times 2} + \frac{7}{10} = \frac{4}{10} + \frac{7}{10} = \frac{4 + 7}{10} = \frac{11}{10}$ .
13. $\frac{- 3}{4} \times \frac{8}{9} = \frac{- 3 \times 8}{4 \times 9} = \frac{- 24}{36}$ . Simplifying by dividing numerator and denominator by 12, we get $\frac{- 2}{3}$ .
14. To represent $\frac{3}{5}$ on the number line:
Divide the unit interval between 0 and 1 into 5 equal parts.
Mark the third part from 0. This point represents $\frac{3}{5}$ .
15. The midnight temperature is $10^{\circ} C - 18^{\circ} C = - 8^{\circ} C$ .
16. Between any two distinct rational numbers, infinitely many other rational numbers can be found. This is due to the density property of rational numbers, where we can always find the average of two numbers, and that average will be a rational number lying between them.
17. The decimal expansion of $\frac{7}{12}$ will be repeating because the prime factors of the denominator $12 = 2^{2} \times 3$ include a factor other than 2 or 5 (which is 3).
18. Let $x = 0.454545 \dots$ .
Multiply by 100 (since there are 2 repeating digits): $100 x = 45.454545 \dots$ .
Subtract the first equation from the second: $100 x - x = 45.4545 \dots - 0.4545 \dots$

99 x = 45

x = \frac{45}{99} = \frac{5 \times 9}{11 \times 9} = \frac{5}{11}

The distance between

- 5

and

3

∣ - 5 - 3 ∣ = ∣ - 8 ∣ = 8

Rational numbers are numbers that can be expressed in the form

\frac{p}{q}

, where

p

and

q

are integers and

q \neq 0

. Their decimal expansions are either terminating or repeating. Examples:

\frac{1}{2} = 0.5

\frac{1}{3} = 0.333 \dots

.Irrational numbers are numbers that cannot be expressed in the form

\frac{p}{q}

. Their decimal expansions are non-terminating and non-repeating. Examples:

\sqrt{2} \approx 1.41421356 \dots

π \approx 3.14159265 \dots

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