The World of Numbers: Class 9 Ganita Manjari · Chapter 3

From counting notches to the real number line

Ganita Manjari Part 1 · Companion notes, simulations and practice · Official NCERT chapter PDF ↗

Published 28 Sep 2026 · Updated 28 Sep 2026

110 min
1 simulation
47 practice questions
40 MCQs

What you will learn in The World of Numbers

Explain how the number system grew from counting numbers to integers, rational numbers and real numbers, and why each step was needed
Apply Brahmagupta's rules for zero and for signed numbers, and justify why the product of two negative numbers is positive
Add, subtract, multiply and divide rational numbers, test two fractions for equality and use the distributive law
Locate rational numbers on the number line, find absolute values and distances, and find rational numbers between any two given ones
Prove that the square root of 2 is irrational by contradiction, and construct lengths such as the square roots of 2 and 3 on the number line
Predict whether a fraction has a terminating or a repeating decimal, and convert repeating decimals into fractions
Recognise irrational numbers such as the square root of 2 and pi by their non-terminating, non-repeating decimals

Original companion text aligned to NCERT Mathematics Class 9, Chapter 3, pp. 41–67. Not affiliated with or endorsed by NCERT.

3.1 The Dawn of Mathematics: The Human Need to Count

Imagine a birthday party with a stack of paper plates. Are there enough plates for every child? You do not need to know how many children came. Hand one plate to each child. If the plates run out while someone is still waiting, you are short. If some plates are left over, you have extra.

Pairing every object in one collection with exactly one object in another is called one-to-one correspondence. It compares the sizes of two collections without using a single number word. Long before anyone wrote a numeral, people relied on it. A notch on a stick, a knot in a cord or a stone in a bag could stand for each animal, each day or each basket of grain.

At some point people noticed that five goats, five fingers and five pebbles all match one another. What they share is an idea, the number five. The numbers we use for counting, 1, 2, 3, 4 and so on, are the natural numbers (प्राकृत संख्याएँ). The collection of all of them is written $ℕ = {1, 2, 3, 4, \dots}$ . The three dots say that the list never ends: every natural number is followed by one that is bigger by 1.

Definition

Natural numbers

The counting numbers $1, 2, 3, 4, \dots$ The collection of all of them is written $ℕ$ .

ExampleThe number of pages in this book, or of students in your class.

3.1.1 A History Written in Bone

Try the puzzle in Fig. 3.1 before you read on. The counts on its four cards come from a real object, a bone handle about 10 cm long. It was dug up in the middle of the twentieth century at Ishango, beside Lake Edward in the Democratic Republic of the Congo, close to one of the sources of the Nile. It was carved more than 20,000 years ago, and it is one of the oldest objects that seem to record numbers.

4/4Every other number from 10 to 20 splits into smaller factors. These four do not: they are exactly the primes between 10 and 20.

Fig. 3.1The counts in one column of the Ishango bone, redrawn as a puzzle.NCERT Fig. 3.1 · p. 42

Four cards each hold a group of short carved notches arranged in two rows. In the first step the groups are uncounted and each has a question mark under it. In the second step the counts appear: 11, 13, 17 and 19. In the third step a strip of cells numbered 10 to 20 appears and dashed arrows carry each count to its cell, which is shaded. In the last step the unshaded numbers show a way to split them into smaller factors, such as 2 × 5 under 10 and 3 × 4 under 12, while the shaded cells 11, 13, 17 and 19 have no such split: they are the primes between 10 and 20.

Every number from 10 to 20 except 11, 13, 17 and 19 splits into a product of smaller whole numbers, so these four are exactly the prime numbers in that range. The bone carries two more columns of notches. In one of them, numbers seem to sit next to their doubles, such as 3 and 6 or 5 and 10. Whether the carver really knew about primes, nobody can say.

An even older find comes from a cave in the Lebombo Mountains, where South Africa meets Eswatini: a small baboon bone with 29 cuts, at least 35,000 years old. The Moon takes about 29½ days to go from one full Moon to the next, so the bone may have been a calendar. Objects like these tell us that people kept count, and perhaps noticed patterns, long before anyone could write.

3.1.2 The Indian Context: Trade and Astronomy

Excavations at Harappa, Mohenjo-daro, Dholavira and Lothal have turned up hundreds of small, polished stone cubes. They are weights, and they come in a strict series. Starting from the smallest, each one weighs twice as much as the one before: 1, 2, 4, 8, 16, 32 and 64 units. The heavier ones continue in multiples of ten. One exact set of weights was shared across the whole region. So traders from distant towns, and even merchants from Mesopotamia across the sea, could agree on how much cotton or how many beads changed hands.

The composers of the Vedas also loved large numbers. Instead of inventing a new word only now and then, they gave every power of ten a name of its own, as the table shows. The list in the Yajurveda ends at parārdha, $10^{12}$ , a million million.

Table 3.1Some names for powers of ten in the Yajurveda

Name	Value	Name	Value
eka	$1$	prayuta	$10^{6}$
daśa	$10$	arbuda	$10^{7}$
śata	$10^{2}$	samudra	$10^{9}$
sahasra	$10^{3}$	anta	$10^{11}$
ayuta	$10^{4}$	parārdha	$10^{12}$

A name for every power of ten suggests a way of writing numbers too: let the position of a digit say which power of ten it counts. The same 5 then means five, fifty or five hundred. This is place value. Its Indian form, with just ten digits, later travelled west through Arabic texts and became the system you write every day. It needs one more thing, though: a way to show that a position holds nothing. That need is where the next section begins.

Quick check

1 of 3

A teacher gives one worksheet to each student and finds three worksheets left over. What can she say without counting anyone?

There are three more students than worksheets.There are three more worksheets than students.There are exactly three students.Nothing, until she counts both.

Show answer

Answer (B) There are three more worksheets than students.

Every student got exactly one worksheet, so the students were matched one-to-one with some of the worksheets, and three worksheets had no partner. Counting is not needed: that is the power of one-to-one correspondence.

Practice · 4 questions

Practice 3.1: questions with answers

NumericalEasyNCERT Exercise Set 3.1, Q1
Astronomers in ancient India split a full day and night into 60 ghaṭikās. One way to keep track was a small copper bowl with a hole in its base, floating in a pot of water: it filled and sank once every ghaṭikā. The bowl sinks 5 times in every 2 hours. How many times does it sink between sunrise at 6 am and sunset at 6 pm?
Show solution
From 6 am to 6 pm is 12 hours, which is $12 \div 2 = 6$ blocks of 2 hours. Each block holds 5 sinkings, so the bowl sinks $6 \times 5 = 30$ times. That is half of the 60 ghaṭikās in a whole day, as it should be for 12 of the 24 hours.
Short answerEasyNCERT Exercise Set 3.1, Q2
A carved stick has groups of 23, 29, 31 and 37 notches. A friend says the carver simply picked odd numbers. Find a test that 23, 29, 31 and 37 all pass but the skipped odd numbers 25, 27, 33 and 35 all fail. Then name the next three numbers after 37 that pass your test.
Show solution
AnswerThe four carved numbers are prime and the skipped ones are not; the next three primes are 41, 43 and 47.
Try to split each number into a product of two smaller whole numbers. The skipped numbers split: $25 = 5 \times 5$ , $27 = 3 \times 9$ , $33 = 3 \times 11$ and $35 = 5 \times 7$ . None of 23, 29, 31 and 37 splits, so each is prime: the carver chose exactly the primes from 23 to 37.
After 37, the numbers 38, 39 and 40 are composite ( $2 \times 19$ , $3 \times 13$ , $2 \times 20$ ), so the next primes are 41, 43 and 47. (42, 44, 45 and 46 are all composite.)
3
Short answerMediumNCERT Exercise Set 3.1, Q3
Adding two counting numbers always gives a counting number, so we say $ℕ$ is closed under addition. Is $ℕ$ closed under each of these operations? Give an example to support each answer.
1. multiplication
  closednot closed
  Show answer
  Answer (A) closed
  The product of two counting numbers is a counting number, e.g. $6 \times 9 = 54$ .
2. subtraction
  closednot closed
  Show answer
  Answer (B) not closed
  $4 - 9 = - 5$ and $7 - 7 = 0$ are not natural numbers, so one example is enough to show $ℕ$ is not closed.
3. division
  closednot closed
  Show answer
  Answer (B) not closed
  $3 \div 4 = \frac{3}{4}$ is not a natural number.
4
ActivityMediumNCERT Exercise Set 3.1, Q4
Many vegetable sellers in Indian markets count dozens without a calculator. The thumb taps the segments of the other four fingers of the same hand (the parts between the creases), one tap for each item.
1. How many taps does one hand allow before the thumb runs out of segments?
  Show answer
  Answer 12
  4 fingers × 3 segments = 12 taps: exactly one dozen.
2. Each time one hand completes a dozen, the seller folds down one finger of the other hand, thumb included. What is the largest count the two hands can hold?
  Show answer
  Answer 60
  Five fingers record 5 dozens: $5 \times 12 = 60$ .
3. The seller has counted 40 bananas. How many fingers are folded on the tally hand, and how many segments has the thumb tapped on the counting hand?
  Show solution
  Answer3 fingers are folded and 4 segments have been tapped, because $40 = 3 \times 12 + 4$ .
  $40 = 3 \times 12 + 4$ : three full dozens (three folded fingers) and 4 more (four taps). The two hands are really writing 40 in base 12, with the dozens on one hand and the units on the other.
4. Name two things in daily life that are still counted in twelves or sixties.
  Show solution
  AnswerFor example: eggs and bananas sold by the dozen, 12 months in a year, 60 minutes in an hour and 60 seconds in a minute.
  Twelve is handy because it splits evenly into halves, thirds, quarters and sixths, and 60 splits into even more equal parts. That is one reason dozens, hours, minutes and degrees still use 12 and 60.

3.2 The Revolution of Śhūnya: When Nothing Became Something

How would you write two hundred and five if your number system had no zero? You could write 2 and 5 with an empty space between them for the missing tens. But a space is easy to miss, and 25, 205 and 250 start to look alike. Early writers of place-value numbers faced exactly this problem.

Other civilisations met the same problem. Babylonian scribes eventually pressed a special pair of wedges into the clay to mark an empty place, and Mayan astronomers drew a shell-shaped sign. Useful as they were, these signs only held a place open. Nobody wrote a sum such as 7 plus empty, or empty times 3. Fig. 3.2 traces the journey from an empty column to a number you can calculate with, a journey completed in India.

4/4Zero becomes a digit and a number we can add, subtract and multiply.

Fig. 3.2From an empty place to the number zero.NCERT p. 43

Three boxes labelled hundreds, tens and ones hold two pebbles, nothing and five pebbles. In the first step we only see the boxes and the question of how to write the number. In the second step it is written as 2, an empty dashed box, then 5, with the question whether it means 25, 205 or 250. In the third step a solid dot, the bindu, fills the empty place. In the last step the dot has become the digit 0, giving 205, and three small sums show zero acting as a number: 5 − 5 = 0, 7 + 0 = 7 and 9 × 0 = 0.

3.2.1 From Philosophy to Mathematics: The Concept of Śhūnyatā

In Sanskrit, śhūnya (शून्य) means empty, and in Hindi it is still an everyday word for zero. Long before it named a digit, the idea behind it mattered in philosophy. Buddhist teachers, the Upanishads and Patañjali's Yoga Sūtras all speak of śhūnyatā (शून्यता), emptiness. For someone who meditates, it is something to reach, a quiet mind free of restless thoughts, and not a loss. Many historians think that a culture which prized emptiness found it natural to give nothing a name and a symbol of its own.

Astronomers who wrote long numbers in verse used several such words, śhūnya, kha and ākāśa (empty and sky), to stand for the digit 0. Piṅgala, in his ancient study of Sanskrit metre, already used śhūnya as a working mark in a counting method. Around 499 CE Āryabhaṭa described place value as each place ten times the one before, which only works when empty places can be shown. Brahmagupta then took the last step and gave zero rules of its own.

3.2.2 The Bakhśhālī Manuscript and Brahmagupta’s Rules

In 1881 a farmer digging near the village of Bakhśhālī, now in Pakistan, found an old manuscript written on birch bark. It is full of arithmetic problems, and it marks an empty place with a solid dot, the bindu. Carbon dating suggests that parts of it go back to the early centuries CE.

Can you add a dot to 7, or multiply 5 by it? Questions like these were settled by Brahmagupta, an astronomer from Bhinmal in present-day Rajasthan. His Brāhmasphuṭasiddhānta, finished in 628 CE, is the oldest known book that treats zero as a number with rules of its own. His starting point was an everyday subtraction: $7 - 7$ , $5 - 5$ or any number minus itself leaves zero, so $a - a = 0$ . The other rules follow below.

PropertyBrahmagupta’s rules for zero

For every number $a$ : $a + 0 = a$ , $a - 0 = a$ and $a \times 0 = 0$ .

Notice what is missing from the list: dividing by zero. Brahmagupta tried to handle that case too, but his answers do not fit with the rest of arithmetic. Today we say that division by zero is not defined. The puzzle below shows why.

Quick check

1 of 3

What is the difference between a placeholder and the number zero?

There is no difference: they are two names for the same idea.A placeholder only marks an empty place; zero can also be added, subtracted and multiplied.A placeholder is a number, while zero is only a symbol.Placeholders came from Brahmagupta, and zero from the Babylonians.

Show answer

Answer (B) A placeholder only marks an empty place; zero can also be added, subtracted and multiplied.

The Babylonian and Mayan marks only showed that a place in a written number was empty. Brahmagupta gave zero rules such as

a + 0 = a

and

a \times 0 = 0

, which made it a number to calculate with. The history in the last option is the wrong way round.

3.3 Integers: Expanding the Horizon

Meena has ₹4 in her purse. At a shop that knows her family, she buys a notebook costing ₹6 and promises to pay the rest tomorrow. How much money does she have now? Not zero: she has less than nothing, because she owes ₹2. The natural numbers and zero cannot describe this, since $4 - 6$ has no answer among them.

Brahmagupta used exactly this language of the market. He called an amount that you own dhana (धन), a fortune, and an amount that you owe ṛiṇa (ऋण), a debt. Meena’s fortune of 4 has turned into a debt of 2, which we write as $- 2$ . So $4 - 6 = - 2$ .

Debts get their own places on the number line, to the left of zero, as mirror images of the natural numbers (Fig. 3.3). The counting numbers, zero and the debts together make up the integers (पूर्णांक). Mathematicians write this collection as $ℤ = {\dots, - 3, - 2, - 1, 0, 1, 2, 3, \dots}$ .The letter $ℤ$ comes from Zahlen, German for numbers. The counting numbers with 0 added are also called the whole numbers.

4/4Brahmagupta read the left side as debts (ṛiṇa) and the right side as fortunes (dhana).

Fig. 3.3The integers: natural numbers, zero and their mirror images.NCERT Fig. 3.2 · p. 45

In the first step a ray starts at 1 and shows the counting numbers 1 to 6 as blue dots. In the second step zero is added as an open circle at the start. In the third step the negative numbers −1 to −6 appear to the left as orange dots, and dashed arches join 3 to −3 and 5 to −5 to show that each pair is the same distance from zero. In the last step the negative side of the line sits in a dashed orange band labelled ṛiṇa (debts) and the positive side in a solid blue band labelled dhana (fortunes); −4 is marked owe ₹4 and 4 is marked have ₹4.

Each integer $a$ has a partner $- a$ at the same distance from zero on the other side, and $a + (- a) = 0$ . We call $- a$ the additive inverse of $a$ . For example, the additive inverse of 7 is $- 7$ , and the additive inverse of $- 3$ is 3, because $(- 3) + 3 = 0$ .

3.3.1 The Arithmetic of Integers

Brahmagupta also stated rules for calculating with fortunes and debts. The table gives them in modern symbols, where $a$ and $b$ stand for positive numbers.

Table 3.2Brahmagupta’s rules for fortunes and debts

Rule	In symbols	Example
Two fortunes add to a fortune	$a + b$ is positive	$6 + 9 = 15$
Two debts add to a debt	$(- a) + (- b) = - (a + b)$	$(- 6) + (- 9) = - 15$
Removing zero changes nothing	$a - 0 = a, (- a) - 0 = - a$	$(- 8) - 0 = - 8$
A fortune less a bigger fortune is a debt	if $b > a$ , $a - b = - (b - a)$	$4 - 6 = - 2$
A debt times a fortune is a debt	$(- a) \times b = - a b$	$(- 7) \times 2 = - 14$
A debt times a debt is a fortune	$(- a) \times (- b) = a b$	$(- 7) \times (- 2) = 14$

Most of these rules match common sense about money. If you owe ₹6 to one friend and ₹9 to another, you owe ₹15 in all. The last rule is the surprising one. Why should two debts multiply to give a fortune? Let us look for a pattern, as in Fig. 3.4.

3/3To keep rising by 3, the next answers must be $(- 3) \times (- 1) = 3$ and $(- 3) \times (- 2) = 6$ .

Fig. 3.4Why a negative times a negative is positive: follow the pattern.NCERT p. 46

On the left, rows of products: (−3) × 3 = −9, (−3) × 2 = −6 and (−3) × 1 = −3 in the first step, with the answers marked on a number line below and joined by hops labelled +3. In the second step the row (−3) × 0 = 0 is added and the hops reach 0. In the last step two more rows appear in blue, (−3) × (−1) = 3 and (−3) × (−2) = 6, the hops continue to 3 and 6, and a note says that the pattern forces (−3) × (−1) = 3.

Read down the list. Each time the second number drops by 1, the answer rises by 3. When the pattern reaches $(- 3) \times 0 = 0$ , the only way for it to continue is $(- 3) \times (- 1) = 3$ , then $(- 3) \times (- 2) = 6$ . So a negative times a negative must be positive, if the rules of arithmetic are to stay consistent.

Example

A cold-storage room

The temperature in a cold-storage room for apples falls steadily by 3 °C every hour once the cooling is switched on. At noon the thermometer reads 0 °C.

(a) What will it read at 4 pm? (b) What did it read at 10 am, if it had already been falling at the same rate?

Try it yourself first. Then reveal the solution one step at a time.

Quick check

1 of 4

Find

(- 8) + (- 5) - (- 6)

Show answer

Answer -7

(- 8) + (- 5) = - 13

, and subtracting

- 6

adds 6:

- 13 + 6 = - 7

Practice · 4 questions

Practice 3.2: questions with answers

NumericalEasyNCERT Exercise Set 3.2, Q1
A pearl diver near Thoothukudi is 12 m below sea level. She rises 5 m, then dives a further 9 m. Write this as a calculation with integers and find her final position.
m
Show solution
Start: $- 12$ m. Rising 5 m adds 5, diving 9 m subtracts 9: $- 12 + 5 - 9 = - 7 - 9 = - 16$ . She is 16 m below sea level.
NumericalEasyNCERT Exercise Set 3.2, Q2
A women's self-help group keeps an account book. This month it began owing a bank ₹2,400, earned ₹3,150 by selling pickles, paid ₹1,200 rent for its stall and got a refund of ₹325 from a supplier. Write one integer expression for the group's final position and work it out. Does it still owe money?
₹
Show solution
$- 2400 + 3150 - 1200 + 325 = 750 - 1200 + 325 = - 450 + 325 = - 125$ . The result is negative, so the group still owes ₹125.
3
NumericalEasyNCERT Exercise Set 3.2, Q3
Use Brahmagupta’s rules to work these out.
1. $(- 14) \times 3$
  Show answer
  Answer -42
  A debt times a fortune is a debt.
2. $(- 6) \times (- 11)$
  Show answer
  Answer 66
  A debt times a debt is a fortune.
3. $- 9 - (- 17)$
  Show answer
  Answer 8
  Subtracting $- 17$ adds 17: $- 9 + 17 = 8$ .
4. $(- 48) \div (- 6)$
  Show answer
  Answer 8
  $(- 6) \times 8 = - 48$ , so $(- 48) \div (- 6) = 8$ .
5. $(- 2) \times (- 3) \times (- 5)$
  Show answer
  Answer -30
  $(- 2) \times (- 3) = 6$ , then $6 \times (- 5) = - 30$ .
Short answerMediumNCERT Exercise Set 3.2, Q4
In a quiz, each wrong answer scores $- 2$ points. Riya’s score is 9. The quizmaster then finds that one of her answers was marked wrong by mistake and removes that $- 2$ from her record. Use this situation to explain why taking away a negative number has the same effect as adding its positive partner.
Show solution
AnswerRemoving the $- 2$ means $9 - (- 2)$ , and her score goes up to 11, which is $9 + 2$ .
Riya’s 9 points already include the wrong penalty of $- 2$ . Taking that penalty away from her record is the calculation $9 - (- 2)$ .
Without the penalty her score is 2 points higher, 11. So $9 - (- 2) = 11 = 9 + 2$ : removing a debt of 2 has the same effect as adding a fortune of 2.

3.4 Filling the Spaces: Fractions and Rational Numbers

Of the four basic operations, division is the one that integers cannot always handle. $12 \div 4 = 3$ stays inside $ℤ$ , but sharing a 2 m plank equally among 3 shelves, or 1 km among 4 relay runners, does not. The answers, $\frac{2}{3}$ m and $\frac{1}{4}$ km, are fractions. Adding them to our numbers makes division possible whenever the divisor is not zero.

Debts come in fractions too: a loss of half a rupee is $- \frac{1}{2}$ , and a dip of two and a half degrees below zero is $- \frac{5}{2}$ . Every positive fraction has such a mirror image, and the two add up to zero, for example $\frac{2}{3} + (- \frac{2}{3}) = 0$ : it is the fraction’s additive inverse. The minus sign may sit in front, on top or underneath, since flipping the sign of either the top or the bottom flips the sign of the quotient: $- \frac{2}{3} = \frac{- 2}{3} = \frac{2}{- 3}$ .

Look back over every number met so far: counting numbers, zero, debts and fractions of either sign. Each one can be written as one integer divided by another. Numbers of this kind are the rational numbers (परिमेय संख्याएँ). The name comes from ratio, and the symbol $ℚ$ from quotient.

Definition

Rational number

Any number equal to a quotient $\frac{p}{q}$ of two integers, where the denominator $q$ is not zero.

Example $\frac{5}{8}$ , $- \frac{7}{3}$ , $4 = \frac{4}{1}$ and $0 = \frac{0}{9}$ .

Three questions often come up about this definition.

Is an integer such as $- 12$ rational? Yes: $- 12 = \frac{- 12}{1}$ , a quotient of integers. The same trick works for every integer, so $ℤ$ , and with it $ℕ$ , sits entirely inside $ℚ$ .
Are $\frac{3}{4}$ and $\frac{15}{20}$ different rational numbers? No. Scaling the top and bottom by the same non-zero integer gives another name for the same number: $\frac{3}{4} = \frac{6}{8} = \frac{15}{20} = \frac{300}{400}$ . Such fractions are called equivalent.
Which name do we use? Usually the simplest one, in lowest terms. Cancel every common factor, so that the only positive whole number dividing both top and bottom is 1 (they are co-prime), and make the denominator positive. For example, $\frac{- 18}{24}$ becomes $- \frac{3}{4}$ . This is the name we use when marking a number on the number line.

These rules for calculating with fractions go back at least to Brahmagupta, who used them for debts as freely as for fortunes. In today's symbols, with $a$ , $b$ , $c$ , $d$ integers and no denominator equal to zero:

Testing two fractions for equality

\frac{a}{b} = \frac{c}{d} exactly when a d = b c

Adding and subtracting with the same denominator

\frac{a}{b} + \frac{c}{b} = \frac{a + c}{b}, \frac{a}{b} - \frac{c}{b} = \frac{a - c}{b}

Multiplying and dividing

\frac{a}{b} \times \frac{c}{d} = \frac{a c}{b d}, \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a d}{b c} (c \neq 0)

To add or subtract fractions with different denominators, first rewrite both with a common denominator. Fig. 3.5 shows why this works: we cut both wholes into pieces of the same size before counting pieces.

3/3Now the pieces match: $8 + 3 = 11$ twelfths, so $\frac{2}{3} + \frac{1}{4} = \frac{11}{12}$ .

Fig. 3.5To add fractions, first cut both wholes into pieces of the same size.NCERT p. 48

Two bars of equal length. In the first step the top bar is cut into 3 equal parts with 2 shaded, labelled two-thirds, and the lower bar into 4 equal parts with 1 hatched part, labelled one-quarter; the pieces have different sizes. In the second step dashed lines cut both bars into twelfths, and the labels change to eight-twelfths and three-twelfths. In the last step a third bar of twelve equal parts holds the 8 shaded and 3 hatched pieces together, leaving one part empty, labelled eleven-twelfths, with the sum written below.

Why is the sum of two rational numbers always rational? Look at the formula $\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}$ . Its top and bottom are made from integers by multiplying and adding, so they are integers, and $b d \neq 0$ . A similar check works for subtraction, for multiplication and for division by a non-zero number. So $ℚ$ is closed under all four operations, as long as we never divide by 0. The familiar laws carry over as well: $x + y = y + x$ and $x y = y x$ (commutative), and $x (y + z) = x y + x z$ (distributive).

Example

Calculating with rational numbers

(a) Are $\frac{- 14}{21}$ and $\frac{10}{- 15}$ equal? (b) Find $- \frac{5}{6} + \frac{7}{10}$ . (c) Find $(- \frac{9}{10}) \div \frac{3}{4}$ .

Try it yourself first. Then reveal the solution one step at a time.

Practice · 8 questions

Practice 3.3: questions with answers

1
Short answerEasyNCERT Exercise Set 3.3, Q1
Use the test $\frac{a}{b} = \frac{c}{d}$ exactly when $a d = b c$ to decide whether the two rational numbers in each pair are equal.
1. $\frac{3}{4}$ and $\frac{9}{12}$
  equalnot equal
  Show answer
  Answer (A) equal
  $3 \times 12 = 36$ and $4 \times 9 = 36$ : the products match, so they are equal.
2. $\frac{7}{5}$ and $\frac{21}{16}$
  equalnot equal
  Show answer
  Answer (B) not equal
  $7 \times 16 = 112$ but $5 \times 21 = 105$ , so they are not equal. (It is $\frac{21}{15}$ that equals $\frac{7}{5}$ .)
3. $- \frac{4}{9}$ and $\frac{12}{- 27}$
  equalnot equal
  Show answer
  Answer (A) equal
  $(- 4) \times (- 27) = 108$ and $9 \times 12 = 108$ , so they are equal.
4. $- 2$ and $\frac{14}{- 7}$
  equalnot equal
  Show answer
  Answer (A) equal
  Write $- 2 = \frac{- 2}{1}$ . Then $(- 2) \times (- 7) = 14$ and $1 \times 14 = 14$ , so they are equal.
2
NumericalEasyNCERT Exercise Set 3.3, Q2
Find each sum in lowest terms.
1. $\frac{3}{8} + \frac{5}{12}$
  Show answer
  Answer 0.7916666666666666
  $\frac{9}{24} + \frac{10}{24} = \frac{19}{24}$ .
2. $\frac{5}{6} + \frac{4}{9}$
  Show answer
  Answer 1.2777777777777777
  $\frac{15}{18} + \frac{8}{18} = \frac{23}{18}$ .
3. $- \frac{5}{6} + \frac{7}{18}$
  Show answer
  Answer -0.4444444444444444
  $- \frac{15}{18} + \frac{7}{18} = - \frac{8}{18} = - \frac{4}{9}$ .
3
NumericalEasyNCERT Exercise Set 3.3, Q3
Find each difference in lowest terms.
1. $\frac{7}{10} - \frac{2}{5}$
  Show answer
  Answer 0.3
  $\frac{7}{10} - \frac{4}{10} = \frac{3}{10}$ .
2. $\frac{9}{4} - \frac{5}{6}$
  Show answer
  Answer 1.4166666666666667
  $\frac{27}{12} - \frac{10}{12} = \frac{17}{12}$ .
3. $- \frac{3}{8} - (- \frac{5}{6})$
  Show answer
  Answer 0.4583333333333333
  $- \frac{9}{24} + \frac{20}{24} = \frac{11}{24}$ .
4
NumericalEasyNCERT Exercise Set 3.3, Q4
Find each product in lowest terms.
1. $\frac{4}{9} \times \frac{15}{8}$
  Show answer
  Answer 0.8333333333333334
  $\frac{4 \times 15}{9 \times 8} = \frac{60}{72} = \frac{5}{6}$ .
2. $\frac{5}{12} \times \frac{9}{10}$
  Show answer
  Answer 0.375
  $\frac{45}{120} = \frac{3}{8}$ .
3. $- \frac{6}{7} \times \frac{14}{15}$
  Show answer
  Answer -0.8
  $- \frac{84}{105} = - \frac{4}{5}$ .
5
NumericalEasyNCERT Exercise Set 3.3, Q5
Find each quotient in lowest terms.
1. $\frac{4}{9} \div \frac{2}{3}$
  Show answer
  Answer 0.6666666666666666
  $\frac{4}{9} \times \frac{3}{2} = \frac{12}{18} = \frac{2}{3}$ .
2. $\frac{5}{12} \div \frac{15}{8}$
  Show answer
  Answer 0.2222222222222222
  $\frac{5}{12} \times \frac{8}{15} = \frac{40}{180} = \frac{2}{9}$ .
3. $- \frac{6}{7} \div \frac{9}{14}$
  Show answer
  Answer -1.3333333333333333
  $- \frac{6}{7} \times \frac{14}{9} = - \frac{84}{63} = - \frac{4}{3}$ .
Short answerEasyNCERT Exercise Set 3.3, Q6
Show that $(\frac{2}{3} + \frac{1}{6}) \times \frac{9}{5} = \frac{2}{3} \times \frac{9}{5} + \frac{1}{6} \times \frac{9}{5}$ by working out both sides.
Show solution
AnswerBoth sides equal $\frac{3}{2}$ .
Left side: $\frac{2}{3} + \frac{1}{6} = \frac{4}{6} + \frac{1}{6} = \frac{5}{6}$ , and $\frac{5}{6} \times \frac{9}{5} = \frac{45}{30} = \frac{3}{2}$ .
Right side: $\frac{2}{3} \times \frac{9}{5} = \frac{18}{15} = \frac{6}{5}$ and $\frac{1}{6} \times \frac{9}{5} = \frac{9}{30} = \frac{3}{10}$ . Their sum is $\frac{12}{10} + \frac{3}{10} = \frac{15}{10} = \frac{3}{2}$ .
Both sides equal $\frac{3}{2}$ , as the distributive law promises.
NumericalMediumNCERT Exercise Set 3.3, Q7
Use the distributive property to simplify $\frac{5}{8} (\frac{4}{5} - \frac{2}{3})$ .
Show solution
$\frac{5}{8} (\frac{4}{5} - \frac{2}{3}) = \frac{5}{8} \times \frac{4}{5} - \frac{5}{8} \times \frac{2}{3} = \frac{20}{40} - \frac{10}{24} = \frac{1}{2} - \frac{5}{12} = \frac{1}{12}$ .
Check by working out the brackets first: $\frac{4}{5} - \frac{2}{3} = \frac{2}{15}$ , and $\frac{5}{8} \times \frac{2}{15} = \frac{10}{120} = \frac{1}{12}$ .
8
Short answerHardNCERT Exercise Set 3.3, Q8
Equations with fractions.
1. Find the rational number $x$ for which $\frac{2}{3} (x + \frac{3}{4}) = \frac{1}{2} x + 1$ .
  Show solution
  Distribute: $\frac{2}{3} x + \frac{2}{3} \times \frac{3}{4} = \frac{2}{3} x + \frac{1}{2}$ .
  So $\frac{2}{3} x + \frac{1}{2} = \frac{1}{2} x + 1$ . Subtract $\frac{1}{2} x$ and $\frac{1}{2}$ from both sides: $(\frac{2}{3} - \frac{1}{2}) x = \frac{1}{2}$ , that is $\frac{1}{6} x = \frac{1}{2}$ , so $x = 3$ .
  Check: $\frac{2}{3} (3 + \frac{3}{4}) = \frac{2}{3} \times \frac{15}{4} = \frac{5}{2}$ , and $\frac{1}{2} \times 3 + 1 = \frac{5}{2}$ .
2. Now try $\frac{3}{2} (x - \frac{1}{3}) + \frac{1}{2} = \frac{3}{2} x$ . What happens, and what does it tell you about $x$ ?
  Show solution
  AnswerBoth sides are the same expression, so every rational number $x$ works.
  The left side is $\frac{3}{2} x - \frac{3}{2} \times \frac{1}{3} + \frac{1}{2} = \frac{3}{2} x - \frac{1}{2} + \frac{1}{2} = \frac{3}{2} x$ , exactly the right side. The equation is true for every $x$ : it is an identity, not a puzzle with one answer.

3.4.1 Representation of Rational Numbers on the Number Line

Where on a number line does $\frac{7}{3}$ belong? Not at any tick of the integer line. Those ticks mark whole steps of one unit from the origin 0: to its right for the positive integers, to its left for the negative ones. A fraction lives in a gap between two ticks, and its denominator tells you how finely to cut that gap.

Think of $\frac{p}{q}$ as a set of instructions. The denominator $q$ gives the size of each step: one unit split into $q$ equal pieces. The numerator $p$ says how many steps to take from 0, and its sign says which way to walk: towards the positive integers or towards the negative ones. Fig. 3.6 follows these instructions for $\frac{7}{3}$ and $- \frac{5}{6}$ .

3/3Cut the unit from $- 1$ to $0$ into 6 parts and take 5 steps to the left of 0.

Fig. 3.6To place

\frac{p}{q}

, cut each unit into

q

equal parts and count

p

of them from 0.NCERT Fig. 3.5 and 3.6 · p. 50

A number line from −2 to 3 with the integers marked. In the second step every unit from 0 to 3 is cut into thirds, and seven blue hops of one-third each lead from 0 to a dot just past 2, labelled seven-thirds, which is 2 and one-third. In the last step the unit from −1 to 0 is cut into sixths, and five dashed orange hops of one-sixth each lead left from 0 to a dot labelled negative five-sixths, just to the right of −1.

There is a shortcut for fractions bigger than 1: split off the whole units first. $\frac{7}{3}$ is 2 whole units and one-third more, so jump straight to 2 and cut just the next unit into thirds.

Absolute value of a rational number

Sometimes only how far a number is from zero matters, not on which side it lies. An auto-rickshaw ride of 3 km costs the same fare whether it heads east or west. This size-without-direction is the absolute value (निरपेक्ष मान) of a number, written with two upright bars: $∣ x ∣$ .

Definition

Absolute value

How far a number lies from 0, whichever side it is on. In symbols, $∣ x ∣ = x$ when $x \geq 0$ , and $∣ x ∣ = - x$ when $x < 0$ .

Example $∣ - 2.5 ∣ = 2.5$ and $∣ \frac{4}{7} ∣ = \frac{4}{7}$ .

Because distances are never negative, $∣ x ∣ \geq 0$ always, with $∣ x ∣ = 0$ only at $x = 0$ . The same bars measure the gap between any two numbers: $a$ and $b$ are $∣ a - b ∣$ apart. It does not matter which one you subtract from which, since the bars remove the sign. Fig. 3.7 checks this on a pair that sits on both sides of 0.

4/5The distance from $b$ to $a$ is $∣ a - b ∣ = ∣ \frac{5}{2} + \frac{3}{2} ∣ = 4$ . Here the two pieces above add up to it.

Fig. 3.7Absolute value measures distance from 0;

∣ a - b ∣

measures the distance between

a

and

b

.NCERT Fig. 3.8 · p. 52

A number line from −3 to 3 with point b at negative three-halves and point a at five-halves. In the first step only the two points are shown with their values. In the second step a dashed bracket above the line, from b to 0, shows that the absolute value of b is three-halves. In the third step a solid bracket above the line, from 0 to a, shows that the absolute value of a is five-halves. In the fourth step a long green bracket below the line joins b to a and gives the distance between them: the absolute value of a minus b, which is 4, the two pieces above added together. In the last step the same bracket is labelled with b minus a, whose absolute value is also 4.

Example 1NCERT Example 1 · p. 51

Absolute values and distance

Find $∣ - \frac{7}{4} ∣$ , $∣ \frac{1}{2} ∣$ and $∣ 0 ∣$ . Then find the distance between $- \frac{7}{4}$ and $\frac{1}{2}$ on the number line.

Try it yourself first. Then reveal the solution one step at a time.

Quick check

1 of 4

Between which two integers does

- \frac{13}{4}

lie?

- 3

and

- 2

- 4

and

- 3

3

and

4

- 13

and

- 4

Show answer

Answer (B) $- 4$ and $- 3$

- \frac{13}{4} = - 3 \frac{1}{4} = - 3.25

, which is a quarter of a unit to the left of

- 3

, so it lies between

- 4

and

- 3

. Choosing

- 3

and

- 2

forgets that moving further left makes a number smaller.

3.4.2 The Density of Rational Numbers

Which integer comes straight after 1? Easy: 2. Now try: which rational number comes straight after 1? Suggest $1.1$ , and someone can reply with $1.01$ ; suggest $1.0001$ , and $1.00001$ is closer still. There is no next rational number. However close two rational numbers are, another one always fits between them.

The simplest way to squeeze one in is to go exactly halfway, that is, to take the average. Halfway between $\frac{1}{3}$ and $\frac{1}{2}$ is $\frac{1}{2} (\frac{1}{3} + \frac{1}{2}) = \frac{5}{12}$ ; halfway between $\frac{1}{3}$ and $\frac{5}{12}$ is $\frac{3}{8}$ . Fig. 3.8 keeps zooming in, and the process never has to stop.

3/4Between $\frac{1}{3}$ and $\frac{5}{12}$ , halfway is $\frac{3}{8}$ .

Fig. 3.8However far we zoom in between two rational numbers, their average is a new one.NCERT Fig. 3.9 · p. 52

Three number lines stacked one above another. In the first step the top line runs from 0 to 1 with one-third and one-half marked and the gap between them highlighted. In the second step a funnel stretches that gap into the middle line, from one-third to one-half, and its midpoint five-twelfths appears as an orange dot. In the third step the gap from one-third to five-twelfths is stretched into the bottom line, whose midpoint is three-eighths. In the last step the gap from one-third to three-eighths is highlighted and a note gives the next midpoints, seventeen forty-eighths and then eleven thirty-seconds, showing that the process never ends.

TheoremRational numbers are dense

If $a$ and $b$ are rational numbers with $a < b$ , then $\frac{a + b}{2}$ is a rational number and $a < \frac{a + b}{2} < b$ .

It is rational. $a + b$ is rational because $ℚ$ is closed under addition, and dividing it by the non-zero rational number 2 keeps it rational.

It lies between $a$ and $b$ . Compare it with each end:

\frac{a + b}{2} - a = \frac{b - a}{2} > 0 and b - \frac{a + b}{2} = \frac{b - a}{2} > 0

Both differences are positive because $b > a$ . So the average is bigger than $a$ and smaller than $b$ ; in fact it is exactly halfway. Using the result again and again gives infinitely many rational numbers between $a$ and $b$ .

The average is not the only way. Rewriting both numbers with a larger common denominator shows many at once. For instance $\frac{1}{3} = \frac{20}{60}$ and $\frac{1}{2} = \frac{30}{60}$ , so $\frac{21}{60}, \frac{22}{60}, \dots, \frac{29}{60}$ are nine more rational numbers between them.

Example

Four rational numbers between two fractions

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

Try it yourself first. Then reveal the solution one step at a time.

Keep squeezing and you never run out of fractions. Does that mean a fraction sits at every point of the line? Test the idea on one particular point: the length of a unit square's diagonal, laid along the line from 0. Section 3.5 shows what happens.

Quick check

1 of 3

How many rational numbers lie between

\frac{2}{7}

and

\frac{3}{7}

none, because no fraction with denominator 7 fits between themexactly oneexactly sixinfinitely many

Show answer

Answer (D) infinitely many

Fractions with other denominators fit in, for example

\frac{5}{14}

\frac{9}{28}

or the average of any two of them, and we can keep averaging for ever. The first option only looked at sevenths.

Practice · 6 questions

Practice 3.4: questions with answers

DiagramEasyNCERT Exercise Set 3.4, Q1
Draw one number line from $- 2$ to $3$ and place $\frac{3}{5}$ , $- \frac{4}{3}$ and $2 \frac{1}{4}$ on it.
Show solution
Answer $\frac{3}{5}$ lies between 0 and 1, $- \frac{4}{3}$ between $- 2$ and $- 1$ , and $2 \frac{1}{4}$ between 2 and 3.
$\frac{3}{5}$ : cut the unit from 0 to 1 into 5 equal parts and take 3 of them to the right of 0.
$- \frac{4}{3} = - 1 \frac{1}{3}$ : cut the unit from $- 2$ to $- 1$ into 3 equal parts and mark the point one part to the left of $- 1$ .
$2 \frac{1}{4}$ : cut the unit from 2 to 3 into 4 equal parts and mark the point one part to the right of 2.
Short answerEasyNCERT Exercise Set 3.4, Q2
Name three rational numbers that are bigger than $- \frac{2}{3}$ but smaller than $\frac{1}{6}$ .
Show solution
AnswerFor example $- \frac{1}{2}$ , $- \frac{1}{6}$ and $0$ .
With denominator 6 the ends are $- \frac{4}{6}$ and $\frac{1}{6}$ . Any numerator from $- 3$ to $0$ works: $- \frac{3}{6} = - \frac{1}{2}$ , $- \frac{2}{6} = - \frac{1}{3}$ , $- \frac{1}{6}$ and $\frac{0}{6} = 0$ . Choose any three.
NumericalEasyNCERT Exercise Set 3.4, Q3
Simplify $(- \frac{5}{6}) + \frac{3}{8} - (- \frac{1}{4})$ .
Show solution
$- \frac{5}{6} + \frac{3}{8} + \frac{1}{4} = - \frac{20}{24} + \frac{9}{24} + \frac{6}{24} = \frac{- 20 + 15}{24} = - \frac{5}{24}$ .
NumericalMediumNCERT Exercise Set 3.4, Q4
A cyclist rides $16 \frac{1}{2}$ km along a ghat road in $2 \frac{3}{4}$ hours at a steady pace. How many kilometres does she cover in each hour?
km
Show solution
$16 \frac{1}{2} = \frac{33}{2}$ and $2 \frac{3}{4} = \frac{11}{4}$ . Distance in one hour $= \frac{33}{2} \div \frac{11}{4} = \frac{33}{2} \times \frac{4}{11} = \frac{132}{22} = 6$ . She covers 6 km every hour.
Short answerMediumNCERT Exercise Set 3.4, Q5
In a school sprint, two runners are timed at 12.47 s and 12.48 s. Write three times that lie strictly between these two, and explain why each of them is a rational number.
Show solution
AnswerFor example 12.471 s, 12.475 s and 12.479 s.
Write the ends as 12.470 and 12.480. Any of 12.471, 12.472, …, 12.479 lies between them, for instance 12.471, 12.475 and 12.479. Each is a terminating decimal, so it is a fraction over 1000 and hence rational (e.g. $12.475 = \frac{12475}{1000} = \frac{499}{40}$ ).
Short answerHardNCERT Exercise Set 3.4, Q6
Here is another way to find a fraction between $\frac{a}{b}$ and $\frac{c}{d}$ (with $b, d > 0$ ): add the tops and add the bottoms to get $\frac{a + c}{b + d}$ . Try it on $\frac{1}{3}$ and $\frac{1}{2}$ . Does your answer lie between them? Try two more pairs of your own.
Show solution
Answer $\frac{1 + 1}{3 + 2} = \frac{2}{5} = 0.4$ , which lies between $0.33 \dots$ and $0.5$ .
$\frac{2}{5} = 0.4$ and $\frac{1}{3} < 0.4 < \frac{1}{2}$ , so it works here. This fraction is called the mediant.
It works for any two positive-denominator fractions with $\frac{a}{b} < \frac{c}{d}$ . For instance $\frac{3}{5}$ and $\frac{2}{3}$ give $\frac{5}{8} = 0.625$ , between $0.6$ and $0.66 \dots$ ; and $\frac{1}{4}$ and $\frac{3}{4}$ give $\frac{4}{8} = \frac{1}{2}$ . Unlike the average, the mediant is usually not exactly halfway.

3.5 Irrational Numbers

A priest-architect has a square fire altar and is asked to build a second one with exactly twice its area. How long should the side of the new square be? Problems like this fill the Śulbasūtras, manuals for laying out Vedic altars with cords and pegs. The Śulbasūtra of Baudhāyana, from around 800 BCE, gives a neat answer, and hidden inside it is a new kind of number.

Fig. 3.9 shows how to build one. The side of the doubled square is the diagonal $d$ of the first square. If the first square has side 1 unit, the Baudhāyana–Pythagoras theorem gives $d^{2} = 1^{2} + 1^{2} = 2$ . So $d$ is the positive number whose square is 2, written $\sqrt{2}$ .

5/5A square of area 2 has $d^{2} = 2$ , so $d = \sqrt{2} \approx 1.414$ .

Fig. 3.9Doubling a square, the way altar-builders did: the new side is the diagonal of the old square.NCERT Fig. 3.10 · p. 53

A two-by-two grid of unit squares drawn in dashed lines, with the top-left square shaded and each side marked 1. In the first step the note reads side 1, area 1. In the second step the diagonal of the shaded square is drawn and called d. In the third step a tilted square is built whose four sides are diagonals of the four unit squares, each marked d. In the fourth step the four triangles inside the tilted square are shaded: each is half a unit square, so its area is 4 halves, which is 2, twice the first square. In the last step the note gives d times d equals 2, so d is the square root of 2, about 1.414.

The squares come out as $1.96$ , about $2.0069$ and about $1.9988$ : closer and closer to 2, but never equal to it. In fact no fraction ever squares to exactly 2, and the next section proves it. So $\sqrt{2}$ is our first example of an irrational number (अपरिमेय संख्या): it has a definite place on the number line, yet it is not a quotient of two integers.

Definition

Irrational number

A number with a place on the number line that is not a quotient of two integers: no integers $p$ and $q \neq 0$ make $\frac{p}{q}$ equal to it.

Example $\sqrt{2}$ , $\sqrt{3}$ and $π$ .

3.5.1 The Proof of Irrationality of $\sqrt{2}$

How can anyone be sure that no fraction squares to exactly 2? There are infinitely many fractions, so checking them one by one would never finish. We need an argument of a different kind, one that a detective might use. If the thief had escaped through the garden, there would be footprints in the fresh mud. There are none. So the thief did not go that way. This is proof by contradiction: suppose something, follow it to an impossible consequence, and conclude that the supposition was false.

For $\sqrt{2}$ we suppose that it is a fraction and look for footprints. Tradition credits this proof to Hippasus, one of the followers of Pythagoras in ancient Greece, where the discovery of irrational lengths came as a shock a few centuries after Baudhāyana. First we need one small fact about even and odd numbers.

LemmaEven squares come from even numbers

If $p$ is an integer and $p^{2}$ is even, then $p$ is even.

Theorem

\sqrt{2}

is irrational

There are no integers $p$ and $q$ , with $q \neq 0$ , such that $\frac{p}{q} = \sqrt{2}$ .

Assume the opposite. Suppose $\sqrt{2} = \frac{p}{q}$ for some integers $p$ and $q$ with $q \neq 0$ . Cancel all common factors first, so that the fraction is in lowest terms: no whole number bigger than 1 divides both $p$ and $q$ .

p^{2} = 2 q^{2}

So $p$ is even. The right-hand side $2 q^{2}$ is 2 times an integer, so $p^{2}$ is even. By the lemma, $p$ itself is even: $p = 2 k$ for some integer $k$ .

q^{2} = 2 k^{2}

So $q$ is even too. Now $q^{2}$ is 2 times an integer, so it is even, and by the lemma again $q$ is even.

The contradiction. Both $p$ and $q$ are even, so 2 is a common factor. But we chose $\frac{p}{q}$ so that no whole number bigger than 1 divides both. Both cannot be true. Every step after the assumption was correct, so the assumption itself was false: $\sqrt{2}$ cannot be written as $\frac{p}{q}$ . It is irrational.

A proof like this settles the question for every fraction at once, including ones nobody will ever write down. That is the power of reasoning over checking examples.

3.5.2 Construction of Length $\sqrt{n}$

The construction in Fig. 3.10 needs only two moves, used again and again. Stand up: at a point on the number line, raise a segment of length 1 at right angles to the line. Swing: join the top of that segment to 0 and, with the compass point on 0, swing this slanting length down onto the line. The Baudhāyana–Pythagoras theorem tells you exactly how long the slanting segment is, even though no ruler mark sits there.

Start with the unit segment from 0 to 1. One stand-and-swing turns it into a segment of length $\sqrt{1 + 1} = \sqrt{2}$ lying along the line, so its end marks the point $\sqrt{2}$ . A second stand-and-swing, made at that point, gives $\sqrt{2 + 1} = \sqrt{3}$ . Step through the figure to watch both.

5/5Swing $U$ down onto the line: the arc marks the point $\sqrt{3}$ .

Fig. 3.10Right triangles and a pair of compasses place

\sqrt{2}

and

\sqrt{3}

exactly on the number line.NCERT Fig. 3.11 · p. 56

A number line from 0 to 2 with the origin O at 0 and the unit segment from 0 to 1 highlighted in the first step. In the second step a segment of length 1 stands straight up at 1, with a right-angle mark, and the blue slanting segment from O to its top T is labelled root 2. In the third step a dashed compass arc centred at O swings T down onto the number line, where a dot is labelled root 2, a little less than halfway from 1 to 2. In the fourth step a second segment of length 1 stands up at root 2, and the green slanting segment from O to its top U is labelled root 3. In the last step a second arc swings U down to a dot labelled root 3, a little to the right of the first.

Each round adds 1 to the square of the length, so the same two moves give $\sqrt{4} = 2$ , then $\sqrt{5}$ , $\sqrt{6}$ and so on. Sometimes there is a shortcut: a right triangle with legs 2 and 1 has hypotenuse $\sqrt{2^{2} + 1^{2}} = \sqrt{5}$ . Use the squared paper in the simulation to hunt for such triangles.

SimulationLengths on squared paper

A square grid with a fixed point O at the origin and a point B that you can drag to any grid point. A segment joins O to B, the horizontal and vertical legs of its right triangle are drawn, and a panel shows the exact length of OB, such as root 5 or 5, together with its decimal value.

Fig. 3.11Drag

B

to any grid point. The panel gives the exact length of

O B

and draws its right triangle. Some lengths appear simplified:

\sqrt{8}

shows as

2 \sqrt{2}

Try this

Drag $B$ to $(2, 1)$ . What exact length does the panel show for $O B$ ?
Find a grid point that makes $O B = \sqrt{10}$ , and one that makes $O B$ exactly 5.
Try to make $O B = \sqrt{3}$ . Can any grid point do it?

3.5.3 The Story of Pi (π) and Madhava’s Infinite Series

Try this with a steel tumbler and a thread. Wrap the thread once around the rim, mark it, and measure the length; then measure the rim’s width straight across. Divide the first by the second. Whatever the size of the tumbler, you will get a little more than 3. For every circle, circumference ÷ diameter is the same number, called $π$ (पाई).

Āryabhaṭa, in 499 CE, gave a recipe for it in verse: add 4 to 100, multiply by 8, and add 62,000; this is roughly the circumference of a circle whose diameter is 20,000. That makes $π \approx \frac{62832}{20000} = 3.1416$ . He called the value āsanna, approaching: close, but not exact. He was right to be careful. In 1761 Johann Lambert proved that $π$ is irrational, so no fraction at all, however large its numerator and denominator, equals it.

Mādhava of Saṅgamagrāma lived in Kerala more than eight centuries after Āryabhaṭa, and the teachers who carried on his work are known as the Kerala school. He found an expression that gives $π$ exactly, using nothing but the odd numbers:

Mādhava’s series for π

π = 4 (1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \dots)

The pattern is easy to read: odd numbers in the denominators, with signs that alternate. An endless sum can never be finished, so its value is defined differently. Add the terms one at a time and keep a running total, called a partial sum. The number that these totals settle down to is the value of the whole sum. Fig. 3.12 shows the totals swinging above and below $π$ with smaller and smaller swings, closing in on $π$ itself.

4/5After eight terms the total is about $3.02$ ; the swings keep shrinking.

Fig. 3.12Adding one more term at a time, the running totals zigzag around

π

with smaller and smaller swings.NCERT p. 56

A number line from 2.5 to 4 runs along the bottom, with a dashed vertical line rising from pi, about 3.14. Each running total of Mādhava's series is a dot on its own row, working down the figure. In the first step there is one dot, at 4. In the second step a second dot appears at 2.67, below pi. In the third step the totals 3.47 and 2.90 are added. In the fourth step the zigzag reaches eight totals, 3.34, 2.98, 3.28 and 3.02, swinging less and less on either side of the pi line. Filled dots are above pi and open dots below it. In the last step a note says that after 1000 terms the total is 3.1406, still closing in.

3.6 Real Numbers: Decimals and Cyclic Patterns

Every point of the number line can be given an address, its decimal expansion. The integer part picks out a unit. The first digit after the point picks one of that unit’s ten tenths, the next digit one of the ten hundredths inside it, and so on, homing in on the point. The zoom tool below builds such an address one digit at a time.

InteractiveNumber-line zoom

An interactive number line. You choose 13/8, 2/7, negative 7/6, root 2 or pi, or type your own fraction, decimal or square root. The line first shows the integers from −5 to 5 with the number marked as a dot. Each press of Zoom in, or a tap on the highlighted gap, stretches the gap that holds the dot into a new line cut into ten parts, stacked below the old one. A readout builds the decimal digit by digit, cards show the window, the tick spacing and, for fractions, the long-division remainder, and a message says when the dot lands exactly on a tick, when the picture starts to repeat, or that it never will.

Fig. 3.13Pick a number, then zoom in. Each zoom cuts the highlighted piece into ten and reveals one more decimal digit. For a fraction, the card shows the long-division remainder.

Try this

Three kinds of address

Predict first

Before you zoom: for which number will the dot land exactly on a tick after a few zooms?

\frac{2}{7}

\sqrt{2}

\frac{13}{8}

Choose $\frac{13}{8}$ and zoom in until the dot sits exactly on a tick. How many zooms does it take?
Choose $\frac{2}{7}$ and zoom in seven times, watching the remainder card. At which zoom does the remainder come back to a value you have already seen?
Choose $\sqrt{2}$ and zoom in ten times. Does the dot ever land on a tick? Does the picture ever repeat?

$\frac{13}{8} = 1.625$ : after three zooms (tenths, hundredths, thousandths) the dot sits exactly on the tick 1.625, and the division ends with remainder 0.

For $\frac{2}{7}$ the remainder is 2 at zoom 1 and comes back to 2 at zoom 7. From then on each picture matches the one six zooms earlier, so the digits $285714$ repeat for ever.

For $\sqrt{2}$ the dot never lands and no picture repeats: $1.414213562 \dots$ goes on with no repeating block.

Rational points and irrational points alike get addresses of this kind. Taken together, all the points of the line, that is, all the rational and all the irrational numbers, are the real numbers (वास्तविक संख्याएँ), written $ℝ$ . The experiment also suggests a test for telling the two kinds apart: the address of a rational number stops or loops, while the address of an irrational number does neither. The next three sections explain why.

3.6.1 Rational Decimals: Terminating and Repeating

In the zoom tool, each zoom is one step of long division, and the remainder decides where the dot sits in the next window. If a remainder of 0 appears, the dot lands on a tick and the address ends: the fraction has a terminating decimal (सांत दशमलव). If 0 never appears, the address goes on for ever, with a block of digits that repeats: a repeating decimal (असांत आवर्ती दशमलव). A bar over the block marks the repeat: $0.2 \overline{3} = 0.2333 \dots$ and $0. \overline{27} = 0.2727 \dots$

Example 2NCERT Example 2 · p. 57

A decimal that stops

Write $\frac{7}{16}$ as a decimal.

Try it yourself first. Then reveal the solution one step at a time.

Example 3NCERT Example 3 · p. 57

A decimal that repeats

Write $\frac{5}{27}$ as a decimal.

Try it yourself first. Then reveal the solution one step at a time.

Fig. 3.14 turns long division into a walk around a wheel, using $\frac{2}{7}$ . Each possible remainder is a stop on the wheel. The next stop depends only on the stop you are at: multiply by 10, divide by 7, and keep what is left. Dividing by 7 leaves at most six non-zero remainders. So a walk that never reaches the 0 in the middle must, within seven steps, arrive at a stop it has visited before. From then on it can only retrace its own path.

4/5Remainder 2 has come back, so the same six digits repeat: $\frac{2}{7} = 0. \overline{285714}$ .

Fig. 3.14Long division by 7 goes round a loop of remainders. Where you start decides the digits.NCERT p. 58

Six circles arranged in a ring hold the non-zero remainders 2, 6, 4, 5, 1 and 3; a dashed circle in the middle holds 0, marked stop. In the first step a panel says that dividing by 7 can only leave remainders 0 to 6. In the second step a ring marks remainder 2 as the start of 2 divided by 7, and three blue arrows go from 2 to 6 to 4 to 5, labelled with the digits 2, 8 and 5, giving 0.285. In the third step the other three arrows complete the ring from 5 to 1 to 3 and back to 2, with digits 7, 1 and 4, giving 0.285714. In the fourth step remainder 2 is shaded because it has come back, so the digits 285714 repeat. In the last step the start ring moves to remainder 1: going round the same loop from there gives 1 divided by 7, 0.142857 repeating.

The same argument works for any denominator $q$ . There are at most $q - 1$ non-zero remainders, so the repeating block of $\frac{p}{q}$ is at most $q - 1$ digits long. The zoom tool shows the same thing: the remainder fixes where the dot sits inside each window, so a repeated remainder means a repeated picture.

Predicting the type of decimal

Here is a way to see the answer coming before doing any division. A terminating decimal is really a fraction over a power of ten, such as $0.175 = \frac{175}{1000}$ . The only prime factors of a power of ten are 2 and 5, because $10 = 2 \times 5$ . So the question is whether the denominator can be turned into a power of ten. For $\frac{7}{40}$ we can: $40 = 2^{3} \times 5$ , and multiplying top and bottom by $5^{2}$ gives $2^{3} \times 5^{3} = 1000$ .

\frac{7}{40} = \frac{7}{2^{3} \times 5} = \frac{7 \times 5^{2}}{2^{3} \times 5^{3}} = \frac{175}{1000} = 0.175

TheoremWhen does a decimal terminate?

Let $\frac{p}{q}$ be a rational number in lowest terms with $q > 0$ . Its decimal expansion terminates exactly when the only prime factors of $q$ are 2 and 5, that is, when $q = 2^{m} \times 5^{n}$ for some whole numbers $m$ and $n$ .

If $q = 2^{m} 5^{n}$ , the decimal terminates. Say $m \geq n$ (the other case is the same with 2 and 5 swapped). Multiplying the top and bottom by $5^{m - n}$ turns the denominator into $2^{m} 5^{m} = 10^{m}$ , and a fraction over $10^{m}$ is a decimal with at most $m$ digits after the point.

If the decimal terminates, $q$ has no other prime factor. A decimal with $k$ digits after the point equals $\frac{N}{10^{k}}$ for some integer $N$ . Reducing $\frac{N}{10^{k}}$ to lowest terms only cancels factors, so the reduced denominator divides $10^{k} = 2^{k} 5^{k}$ and cannot contain any prime except 2 and 5. The lowest-terms form of a number is unique, so that reduced denominator is $q$ .

Example

Predicting before dividing

Without long division, decide which of these have terminating decimals: $\frac{9}{80}$ , $\frac{11}{24}$ , $\frac{21}{35}$ and $\frac{9}{75}$ .

Try it yourself first. Then reveal the solution one step at a time.

Converting decimals into the form $\frac{p}{q}$

A terminating decimal is already a fraction over a power of ten, which we then simplify.

Example 4NCERT Example 4 · p. 59

A reading from a scale

A shop’s digital scale shows $2.64$ kg of onions. Write this mass as a fraction of a kilogram in lowest terms.

Try it yourself first. Then reveal the solution one step at a time.

A repeating decimal never stops, so we cannot put it over a power of ten directly. The trick is to make two copies of the number whose endless tails match exactly, and subtract them: the tails cancel. A decimal like $0. \overline{7}$ , whose repeating block starts straight after the point, is a pure repeating decimal.

Example 5NCERT Example 5 · p. 59

Using ninths

Write $0. \overline{7}$ in the form $\frac{p}{q}$ .

Try it yourself first. Then reveal the solution one step at a time.

Example 6NCERT Example 6 · p. 59

Shifting a whole block

Write $2. \overline{63}$ in the form $\frac{p}{q}$ .

Try it yourself first. Then reveal the solution one step at a time.

When some digits come before the repeating block, as in $0.3 \overline{1}$ , the decimal is a general repeating decimal. First multiply to move those digits in front of the point; then the block starts straight after the point, and we carry on as before. Fig. 3.15 shows the two copies lining up.

5/5 $90 x = 28$ , so $x = \frac{28}{90} = \frac{14}{45}$ .

Fig. 3.15Multiply until two copies have exactly the same tail, then subtract.NCERT p. 59

Digits are lined up in columns under one another, with the decimal points in one column. In the first step x equals 0.31111 and so on, where the 1s repeat for ever. In the second step 10x equals 3.1111 and so on, and the repeating 1s now start right after the point; they are shaded. In the third step 100x equals 31.1111 and so on, with the same shaded tail of 1s directly below. In the fourth step a line is drawn and the note says to subtract: 100x minus 10x equals 31 minus 3, which is 28, because the tails cancel. In the last step this gives 90x equals 28, so x is 28 over 90, which is 14 over 45.

Example 7NCERT Example 7 · p. 59

Splitting off the first digit

Write $0.3 \overline{1}$ in the form $\frac{p}{q}$ .

Try it yourself first. Then reveal the solution one step at a time.

Example 8NCERT Example 8 · p. 60

What did the calculator divide by?

Arjun divides 1 by a whole number on his calculator, and the screen shows 0.0416666667. The calculator has rounded the last digit; the 6s really go on for ever. Find the whole number.

Try it yourself first. Then reveal the solution one step at a time.

Example 9NCERT Example 9 · p. 60

A two-digit block after one digit

Write $3.1 \overline{27}$ in the form $\frac{p}{q}$ .

Try it yourself first. Then reveal the solution one step at a time.

Table 3.3Turning a repeating decimal

x

into a fraction

Kind of decimal	Multiply $x$ by	Then
Pure repeating: block of $n$ digits (e.g. $2. \overline{63}$ )	$10^{n}$	subtract $x$ and solve
General repeating: $m$ digits before a block of $n$ digits (e.g. $0.041 \overline{6}$ )	$10^{m}$ , and separately $10^{m + n}$	subtract these two and solve

Quick check

1 of 4

Which of these fractions have terminating decimals?

Select all that apply

Show answer

Answer

(A) $\frac{3}{64}$
(C) $\frac{14}{35}$
(E) $\frac{13}{125}$

64 = 2^{6}

and

125 = 5^{3}

\frac{14}{35}

reduces to

\frac{2}{5}

. But

\frac{7}{30}

and

\frac{5}{18}

are in lowest terms with a factor 3 in the denominator, so they repeat.

3.6.2 The Magic of Cyclic Numbers

Go back to the wheel in Fig. 3.14. Dividing by 7 visits every non-zero remainder, 1 to 6, in one big loop. The numerator only decides where you start on the loop. So the six fractions $\frac{1}{7}, \frac{2}{7}, \dots, \frac{6}{7}$ share the same six repeating digits, each starting at a different place. The table lists them in the order the loop visits their starting remainders.

Table 3.4The sevenths, in the order of the remainder loop

Fraction	Starts at remainder	Decimal
$\frac{1}{7}$	1	$0. \overline{142857}$
$\frac{3}{7}$	3	$0. \overline{428571}$
$\frac{2}{7}$	2	$0. \overline{285714}$
$\frac{6}{7}$	6	$0. \overline{857142}$
$\frac{4}{7}$	4	$0. \overline{571428}$
$\frac{5}{7}$	5	$0. \overline{714285}$

Each row is the row above with its first digit moved to the end. Since $\frac{k}{7} = k \times \frac{1}{7}$ , the block of $\frac{k}{7}$ is $k \times 142857$ . So multiplying 142857 by any whole number from 1 to 6 only rotates its digits. A number with this property is called a cyclic number (चक्रीय संख्या).

142857 \times 3 = 428571, 142857 \times 5 = 714285, 142857 \times 7 = 999999

The last product is a bonus. It says that $7 \times 0. \overline{142857} = 0. \overline{999999}$ , and of course $7 \times \frac{1}{7} = 1$ . We return to this in Two decimals for one number.

3.6.3 Irrational Decimals: Chaos and Infinity

An irrational number cannot have a terminating decimal, since that would make it a fraction over a power of ten. It cannot have a repeating decimal either, because every repeating decimal converts to a fraction, as we have just seen. So its digits go on for ever with no block that repeats: they are non-terminating and non-repeating.

\sqrt{3} = 1.7320508075688772935 \dots π = 3.1415926535897932384626 \dots

A decimal can follow a pattern without having a repeating block. In $0.12112111211112 \dots$ each 2 is followed by one more 1 than before. You can easily write the next digits, yet no fixed block repeats, so the number is irrational. Stop-or-repeat means rational; neither means irrational.

Two decimals for one number

Here is a puzzle. You know that $\frac{1}{3} = 0.333 \dots$ , with 3s for ever. Multiply both sides by 3. The left side becomes 1 and the right side becomes $0.999 \dots$ So $0. \overline{9}$ is not just below 1; it is 1. There is no gap between them into which any other number could fit.

The same trick gives every terminating decimal a twin that ends in endless 9s: $3.8 = 3.7 \overline{9}$ and $0.25 = 0.24 \overline{9}$ . A number can therefore have two decimal addresses. By custom we use the one that stops.

3.7 Conclusion: The Never-Ending Journey

Every family of numbers in this chapter was introduced to answer a question that the older numbers could not.

How many? The counting numbers, $ℕ$ .
How do we show an empty place, and what is left when nothing is left? Zero.
What is $4 - 6$ ? The negative integers, completing $ℤ$ .
What is $2 \div 3$ ? Fractions, giving $ℚ$ .
How long is the diagonal of a unit square? The irrational numbers, which complete the real numbers $ℝ$ .

Fig. 3.16 replays these steps on a single number line.

5/5 $ℝ$ fills every hole: each point of the line is a real number.

Fig. 3.16The number line fills up:

ℕ \subset ℤ \subset ℚ \subset ℝ

.NCERT Fig. 3.13 · p. 63

One number line from −4 to 4. In the first step blue dots mark the natural numbers 1 to 4. In the second step orange squares add 0 and the negative integers. In the third step a crowd of small green ticks appears at fractions with denominators up to 6, so that rational numbers seem to be everywhere. In the fourth step open rings mark negative root 7, root 2 and root 6, points that are not rational at all. In the last step a thick shaded band covers the whole line: the real numbers, with no gaps, and the chain N inside Z inside Q inside R is shown above.

Each family sits inside the next: $ℕ \subset ℤ \subset ℚ \subset ℝ$ , where $\subset$ means is contained in. The irrational numbers are inside $ℝ$ but outside $ℚ$ . The table sums up what each step added.

Table 3.5The family of real numbers

Numbers	Symbol	What they add	Examples
Natural numbers	$ℕ$	counting	1, 7, 2026
Integers	$ℤ$	zero and the negatives	$- 15$ , 0, 8
Rational numbers	$ℚ$	all quotients $\frac{p}{q}$ ; decimals that stop or repeat	$- \frac{3}{4}$ , 0.125, $0. \overline{6}$
Irrational numbers	$𝕀$ (in some books)	decimals that never stop or repeat	$\sqrt{3}$ , $- \sqrt{7}$ , $π$
Real numbers	$ℝ$	every point of the number line	all of the above

Timeline

Milestones in the story of numbers

35,000+ years ago
Lebombo bone
29 notches, perhaps counting the days of a Moon cycle.
20,000+ years ago
Ishango bone
Groups of 11, 13, 17 and 19: the primes between 10 and 20.
c. 2600–1900 BCE
Indus Valley weights
Standard weights that double, then grow in tens.
Vedic period
Names for powers of ten
Up to parārdha, $10^{12}$ .
c. 800 BCE
Baudhāyana’s Śulbasūtra
Doubling a square leads to $\sqrt{2}$ .
Early centuries CE
Bakhśhālī manuscript
A dot marks an empty place.
499 CE
Āryabhaṭa
$π \approx 3.1416$ , called āsanna, approximate.
628 CE
Brahmagupta
Rules for zero and for negative numbers.
14th century
Mādhava
An infinite series for $π$ .
1761
Johann Lambert
Proof that $π$ is irrational.

The pattern of this chapter suggests what to do. Each time an equation had no answer, a new number was declared: $x + 6 = 4$ led to $- 2$ , $3 x = 2$ led to $\frac{2}{3}$ , and $x^{2} = 2$ led to $\sqrt{2}$ . For $x^{2} = - 1$ , mathematicians declared a new number $i$ whose square is $- 1$ . It has no place on the real line, so it points away from it at right angles, and the numbers spread out to fill a whole plane. These complex numbers, which you will study in Class 11, turn out to be the natural language for anything that rotates or vibrates.

Quick check

1 of 3

Match each number with the smallest family it belongs to.

Tap an item on the left, then its match on the right.

Show answer

Answer

$- 9$ : integers $ℤ$
$12$ : natural numbers $ℕ$
$- \frac{7}{3}$ : rational numbers $ℚ$
$\sqrt{10}$ : irrational numbers

12 is a counting number.

- 9

is an integer but not a natural number.

- \frac{7}{3}

is a quotient of integers but not an integer.

\sqrt{10}

is not a quotient of integers at all, since 10 is not a perfect square.

Recap

The World of Numbers: key points and summary

Counting grew out of one-to-one correspondence; the natural numbers are $ℕ = {1, 2, 3, \dots}$ (Section 3.1).
Notched bones such as the Lebombo and Ishango bones, Indus Valley weights and Vedic names for powers of ten show how old number ideas are (Section 3.1.1, Section 3.1.2).
Zero began as a placeholder and became a number with Brahmagupta’s rules $a + 0 = a$ , $a - 0 = a$ and $a \times 0 = 0$ ; division by zero is not defined (Section 3.2.2).
The integers $ℤ$ add zero and the negative numbers; Brahmagupta’s fortunes and debts give the sign rules, including $(- a) \times (- b) = a b$ (Section 3.3.1).
A rational number can be written as $\frac{p}{q}$ with $p$ , $q$ integers and $q \neq 0$ ; $ℚ$ is closed under $+$ , $-$ , $\times$ and division by non-zero numbers (Section 3.4).
To place $\frac{p}{q}$ on the number line, cut each unit into $q$ parts and count $p$ of them from 0; $∣ x ∣$ is the distance from 0 and $∣ a - b ∣$ the distance between $a$ and $b$ (Section 3.4.1).
Rational numbers are dense: the average $\frac{a + b}{2}$ always lies between $a$ and $b$ (Section 3.4.2).
Numbers such as $\sqrt{2}$ and $π$ are irrational; $\sqrt{2}$ is proved irrational by contradiction (Section 3.5.1).
Right triangles and compasses construct $\sqrt{2}, \sqrt{3}, \dots, \sqrt{n}$ on the number line (Section 3.5.2); Mādhava’s endless series gives $π$ exactly (Section 3.5.3).
A rational number’s decimal terminates or repeats; in lowest terms it terminates exactly when the denominator is $2^{m} 5^{n}$ (Section 3.6.1).
Repeating decimals convert to fractions by multiplying by powers of 10 and subtracting; also $0. \overline{9} = 1$ (Converting decimals into the form $\tfrac{p}{q}$, Two decimals for one number).
The block of $\frac{1}{7}$ , 142857, is a cyclic number because division by 7 loops through every non-zero remainder (Section 3.6.2).
Irrational decimals never end and never repeat; rational and irrational numbers together form the real numbers $ℝ$ (Section 3.6.3, Section 3.7).

Flash cards · 12

Key terms

1 / 12

End-of-chapter exercises

The questions below practise every idea in the chapter. Question 16 uses the square-root spiral drawn here.

Fig. 3.17A square-root spiral: every outer side is 1 unit long.NCERT Fig. 3.14 · p. 66

Eight right triangles are fitted together around a common corner O, turning clockwise to form a spiral. The first triangle has two sides of length 1. Each new triangle uses the longest side of the one before as one of its shorter sides, and its other short side, on the outside of the spiral, is 1 unit long. Right-angle marks sit where each outer side meets the previous long side, and the triangles are numbered 1 to 8.

Find your NCERT exercise question

Our practice questions are written for ncert.ai and follow the NCERT exercises. Find the one that matches the question in your textbook; its answer is with it.

In your NCERT textbook	Practice question here
NCERT Exercise Set 3.1, Q1	Practice 3.1 · Q1
NCERT Exercise Set 3.1, Q2	Practice 3.1 · Q2
NCERT Exercise Set 3.1, Q3	Practice 3.1 · Q3
NCERT Exercise Set 3.1, Q4	Practice 3.1 · Q4
NCERT Exercise Set 3.2, Q1	Practice 3.2 · Q1
NCERT Exercise Set 3.2, Q2	Practice 3.2 · Q2
NCERT Exercise Set 3.2, Q3	Practice 3.2 · Q3
NCERT Exercise Set 3.2, Q4	Practice 3.2 · Q4
NCERT Exercise Set 3.3, Q1	Practice 3.3 · Q1
NCERT Exercise Set 3.3, Q2	Practice 3.3 · Q2
NCERT Exercise Set 3.3, Q3	Practice 3.3 · Q3
NCERT Exercise Set 3.3, Q4	Practice 3.3 · Q4
NCERT Exercise Set 3.3, Q5	Practice 3.3 · Q5
NCERT Exercise Set 3.3, Q6	Practice 3.3 · Q6
NCERT Exercise Set 3.3, Q7	Practice 3.3 · Q7
NCERT Exercise Set 3.3, Q8	Practice 3.3 · Q8
NCERT Exercise Set 3.4, Q1	Practice 3.4 · Q1
NCERT Exercise Set 3.4, Q2	Practice 3.4 · Q2
NCERT Exercise Set 3.4, Q3	Practice 3.4 · Q3
NCERT Exercise Set 3.4, Q4	Practice 3.4 · Q4
NCERT Exercise Set 3.4, Q5	Practice 3.4 · Q5
NCERT Exercise Set 3.4, Q6	Practice 3.4 · Q6
NCERT Exercise Set 3.5, Q1	Practice 3.5 · Q1
NCERT Exercise Set 3.5, Q2	Practice 3.5 · Q2
NCERT Exercise Set 3.5, Q3	Practice 3.5 · Q3
NCERT Exercise Set 3.5, Q4	Practice 3.5 · Q4
NCERT Exercise Set 3.5, Q5	Practice 3.5 · Q5
NCERT End-of-chapter Exercises, Q1	End-of-chapter exercises · Q1
NCERT End-of-chapter Exercises, Q2	End-of-chapter exercises · Q2
NCERT End-of-chapter Exercises, Q3	End-of-chapter exercises · Q3
NCERT End-of-chapter Exercises, Q4	End-of-chapter exercises · Q4
NCERT End-of-chapter Exercises, Q5	End-of-chapter exercises · Q5
NCERT End-of-chapter Exercises, Q6	End-of-chapter exercises · Q6
NCERT End-of-chapter Exercises, Q7	End-of-chapter exercises · Q7
NCERT End-of-chapter Exercises, Q8	End-of-chapter exercises · Q8
NCERT End-of-chapter Exercises, Q9	End-of-chapter exercises · Q9
NCERT End-of-chapter Exercises, Q10	End-of-chapter exercises · Q10
NCERT End-of-chapter Exercises, Q11	End-of-chapter exercises · Q11
NCERT End-of-chapter Exercises, Q12	End-of-chapter exercises · Q12
NCERT End-of-chapter Exercises, Q13	End-of-chapter exercises · Q13
NCERT End-of-chapter Exercises, Q14	End-of-chapter exercises · Q14
NCERT End-of-chapter Exercises, Q15	End-of-chapter exercises · Q15
NCERT End-of-chapter Exercises, Q16	End-of-chapter exercises · Q16

Quick check: MCQs on The World of Numbers

3.1.2 The Indian Context: Trade and Astronomy3 questions
3.2.2 The Bakhśhālī Manuscript and Brahmagupta’s Rules3 questions
3.3.1 The Arithmetic of Integers4 questions
3.4 Filling the Spaces: Fractions and Rational Numbers4 questions
Absolute value of a rational number4 questions
3.4.2 The Density of Rational Numbers3 questions
3.5.1 The Proof of Irrationality of $\sqrt{2}$4 questions
3.5.3 The Story of Pi (π) and Madhava’s Infinite Series4 questions
Converting decimals into the form $\tfrac{p}{q}$4 questions
Two decimals for one number4 questions
3.7 Conclusion: The Never-Ending Journey3 questions

Glossary · 14

Key terms (English–Hindi)

absolute valueनिरपेक्ष मानnirpeksh maan: The distance of a number from 0 on the number line, written $∣ x ∣$ ; it is never negative.First seen in the chapter →
additive inverseयोज्य प्रतिलोमyojya pratilom: The number $- a$ that gives 0 when added to $a$ .First seen in the chapter →

closedसंवृतsamvrit: A collection of numbers is closed under an operation when doing that operation on any two of its members always gives a member of the collection.First seen in the chapter →
cyclic numberचक्रीय संख्याchakriya sankhya: A number such as 142857 whose multiples by 1, 2, 3, … up to a limit only rotate its digits.First seen in the chapter →

integerपूर्णांकpurnank: A natural number, zero, or the negative of a natural number. The collection is written $ℤ$ .First seen in the chapter →
irrational numberअपरिमेय संख्याaparimeya sankhya: A real number that is not a ratio $\frac{p}{q}$ of integers with $q \neq 0$ , such as $\sqrt{2}$ or $π$ ; its decimal neither stops nor repeats.First seen in the chapter →

lowest termsसरलतम रूपsaraltam roop: A fraction $\frac{p}{q}$ is in lowest terms when 1 is the only positive whole number dividing both $p$ and $q$ , and $q > 0$ .First seen in the chapter →

natural numberप्राकृत संख्याprakrit sankhya: A counting number: $1, 2, 3, \dots$ The collection is written $ℕ$ .First seen in the chapter →

one-to-one correspondenceएकैकी संगतताekaiki sangatata: Pairing each object of one collection with exactly one object of another, so the two collections can be compared without counting.First seen in the chapter →

proof by contradictionविरोधाभास द्वारा उपपत्तिvirodhabhas dwara upapatti: A proof that assumes the opposite of the statement and reasons correctly to something impossible, so the assumption must be false.First seen in the chapter →

rational numberपरिमेय संख्याparimeya sankhya: A number that can be written as $\frac{p}{q}$ with $p$ , $q$ integers and $q \neq 0$ . The collection is written $ℚ$ .First seen in the chapter →
real numberवास्तविक संख्याvastavik sankhya: Any rational or irrational number; each point of the number line. The collection is written $ℝ$ .First seen in the chapter →
repeating decimalअसांत आवर्ती दशमलवasaant aavarti dashamlav: A decimal that never stops but in which a block of digits repeats for ever, such as $0. \overline{185}$ .First seen in the chapter →

terminating decimalसांत दशमलवsaant dashamlav: A decimal that stops after a finite number of digits, such as $0.4375$ .First seen in the chapter →

The World of Numbers: Class 9 Ganita Manjari · Chapter 3

What you will learn in The World of Numbers

3.1 The Dawn of Mathematics: The Human Need to Count

3.1.1 A History Written in Bone

3.1.2 The Indian Context: Trade and Astronomy

Practice 3.1: questions with answers

3.2 The Revolution of Śhūnya: When Nothing Became Something

3.2.1 From Philosophy to Mathematics: The Concept of Śhūnyatā

3.2.2 The Bakhśhālī Manuscript and Brahmagupta’s Rules

3.3 Integers: Expanding the Horizon

3.3.1 The Arithmetic of Integers

Practice 3.2: questions with answers

3.4 Filling the Spaces: Fractions and Rational Numbers

Practice 3.3: questions with answers

3.4.1 Representation of Rational Numbers on the Number Line

Absolute value of a rational number

3.4.2 The Density of Rational Numbers

Practice 3.4: questions with answers

3.5 Irrational Numbers

3.5.1 The Proof of Irrationality of $\sqrt{2}$

3.5.2 Construction of Length $\sqrt{n}$

3.5.3 The Story of Pi (π) and Madhava’s Infinite Series

3.6 Real Numbers: Decimals and Cyclic Patterns

3.6.1 Rational Decimals: Terminating and Repeating

Predicting the type of decimal

Converting decimals into the form $\frac{p}{q}$

3.6.2 The Magic of Cyclic Numbers

3.6.3 Irrational Decimals: Chaos and Infinity

Two decimals for one number

Practice 3.5: questions with answers

3.7 Conclusion: The Never-Ending Journey

The World of Numbers: key points and summary

End-of-chapter exercises

Practice questions with answers (NCERT “End-of-chapter exercises”)

Find your NCERT exercise question

Quick check: MCQs on The World of Numbers

Key terms (English–Hindi)

You reached the end

3.1 The Dawn of Mathematics: The Human Need to Count

3.1.1 A History Written in Bone

3.1.2 The Indian Context: Trade and Astronomy

3.2 The Revolution of Śhūnya: When Nothing Became Something

3.2.1 From Philosophy to Mathematics: The Concept of Śhūnyatā

3.2.2 The Bakhśhālī Manuscript and Brahmagupta’s Rules

3.3 Integers: Expanding the Horizon

3.3.1 The Arithmetic of Integers

3.4 Filling the Spaces: Fractions and Rational Numbers

3.4.1 Representation of Rational Numbers on the Number Line

Absolute value of a rational number

3.4.2 The Density of Rational Numbers

3.5 Irrational Numbers

3.5.1 The Proof of Irrationality of 2

3.5.2 Construction of Length n

3.5.3 The Story of Pi (π) and Madhava’s Infinite Series

3.6 Real Numbers: Decimals and Cyclic Patterns

3.6.1 Rational Decimals: Terminating and Repeating

Predicting the type of decimal

Converting decimals into the form pq

3.6.2 The Magic of Cyclic Numbers

3.6.3 Irrational Decimals: Chaos and Infinity

Two decimals for one number

3.7 Conclusion: The Never-Ending Journey

End-of-chapter exercises

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Quick check: MCQs on The World of Numbers

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3.5.1 The Proof of Irrationality of $\sqrt{2}$

3.5.2 Construction of Length $\sqrt{n}$

Converting decimals into the form $\frac{p}{q}$