YOUR STUDY SPACE · CBSE

The World of Numbers — Class 9 Mathematics

Ganita Manjari · Chapter 3

Download revision card →

CBSE Class 9 community

View all

Mathematics

~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative

All 2 books, one purchase

Mathematics Part I
Mathematics Part II

~7 pages per kitPage 1 free

Get both kits · ₹39

All available subjects

~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative

Every subject, one purchase

Mathematics Part I
Mathematics Part II
+ 4 more

~7 pages per kitPage 1 free

Get all 6 kits · ₹159

Science

~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative

TrendingEvery important point, highly compressed.

Formula cards
Graph reading
16 traps corrected

Only 7 pagesPage 1 free

Get the kit · ₹39

Social Science

~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative

Every important point, highly compressed.

Timelines
Comparison tables
Article numbers

Only 7 pagesPage 1 free

Get the kit · ₹39

English · Kaveri

~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative

TrendingEvery important point, highly compressed.

16 texts summarised
Poetry devices
Grammar rules

Only 7 pagesPage 1 free

Get the kit · ₹39

Hindi · Ganga

~140days leftAnnual exams · school dates varyFeb–Mar 2027 · school-specific · tentative

Every important point, highly compressed.

12 पाठ
अलंकार और समास
मुहावरे

Only 7 pagesPage 1 free

Get the kit · ₹39

Browse all Class 9 kits →

Chapter summary

This chapter traces the expanding universe of numbers, from counting by one-to-one correspondence to the real number line. Natural numbers grew from the human need to count. Brahmagupta formalised zero (shunya) and the negative numbers, giving the integers and rules such as . Combining integers with fractions gives the rational numbers , any number expressible as with integers and . Rationals are dense — between any two there is always another, found by averaging. Some lengths, like the diagonal of a unit square , cannot be written as a ratio; these are irrational, proved using proof by contradiction. Uniting rationals and irrationals forms the continuous real number line . A number is rational exactly when its decimal expansion terminates or repeats, and irrational when it neither terminates nor repeats. The chapter shows how to convert repeating decimals into fractions and to predict, from prime factors of the denominator, whether a decimal terminates.

Natural numbers and integersRational numbersIrrational numbersReal number lineDecimal expansions

Key ideas & terms

Natural numbers: The counting numbers .
Integers: The set of zero, positives and negatives.
Rational number: A number of the form where are integers and .
Irrational number: A number that cannot be written as a ratio of integers, such as or .
Density: The property that between any two rational numbers lies another rational number.
Absolute value: The distance of a number from on the number line, always non-negative.

Extra questions & answers

Why must in the definition of a rational number ?

Division by zero is undefined, so the denominator can never equal zero.

Find a rational number between and .

Take their average: , which lies between them.

How do you prove is irrational?

Assume in lowest terms; this forces both and to be even, contradicting that they share no common factor.

Which rational numbers have terminating decimal expansions?

Those whose denominator in lowest terms has only and/or as prime factors.

Convert into the form .

Let ; then , so and .

Classify as rational or irrational.

It is rational, because the block repeats forever.

What does the absolute value equal?

It equals , since absolute value gives the non-negative distance from .

Why does equal ?

Let ; then , so and .

Is the product of two negative numbers positive or negative?

It is positive: by Brahmagupta's rule , e.g. .

What set of numbers do the rationals and irrationals together form?

They form the real numbers , filling the entire continuous number line.

Exercise Set 3.1

1. 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?

For 2 bags he gets ingots, so for 1 bag he gets ingots. For bags he gets ingots. He will leave with copper ingots.

2. 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.

All four numbers are prime numbers. They are divisible only by and themselves, and they are the prime numbers between and . The next three primes after are , and .

3. 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.

No, natural numbers are not closed under subtraction. For example which is natural, but and are not natural numbers because and negative numbers are not in . Hence subtraction can take us outside .

*4. 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?

One hand has 4 fingers besides the thumb, each with joints, so joints can be counted with the thumb. This gives a natural base- system. Using two hands, one hand counts units up to and the other counts dozens, allowing counting up to or with joints. This explains the historical use of base- and base- systems.

Think and Reflect

Why does a negative times a negative equal a positive? Think of it in terms of action and debt. If a negative number represents a debt, then multiplying by a negative number represents the removal of that debt. (Hint: If someone takes away (-) four of your debts that are each worth ₹3 (that is, -3), you are effectively ₹12 richer! Therefore, (-3) × (-4) = +12.)

A debt can be represented by a negative integer, e.g., means a debt of rupees. Multiplying by a positive number means adding debts, so . Multiplying by means removing or taking away debts of . Removing a debt is a gain, so . In general, the opposite of an opposite is the original fortune, which is why the product of two negatives is positive.

Exercise Set 3.2

1. 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?

Midnight temperature = . So the temperature at midnight is , i.e., below zero.

2. 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.

Let debt be negative and fortune positive. Start: . Then . Then . Equation: . Compute: , . Final standing is , i.e., a net debt of .

3. Calculate the following using Brahmagupta's laws: (i) (-12) × 5

A debt times a fortune is a debt: .

3. Calculate the following using Brahmagupta's laws: (ii) (-8) × (-7)

Product of two debts is a fortune: .

3. Calculate the following using Brahmagupta's laws: (iii) 0 - (-14)

Subtracting a debt is adding a fortune: .

3. Calculate the following using Brahmagupta's laws: (iv) (-20) ÷ 4

Dividing a debt by a fortune gives a debt: , because .

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).

Suppose you have and you owe , written as . If the lender forgives or removes the debt, you subtract : . Removing a debt makes you richer, same as gaining . So .

Think and Reflect

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

A rational number is defined as where are integers and . Division by zero is undefined because no number multiplied by gives a non-zero numerator, and is indeterminate. Therefore we must exclude to keep the number meaningful.

Think and Reflect

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

Find the least common multiple (LCM) of the denominators and convert each fraction to an equivalent fraction with that common denominator. For and , use and or more simply divide by LCM, then add or subtract numerators: and then reduce to lowest terms.

2. Verify the distributive law for rational numbers.

Take any three rationals, e.g., . Then . So . Also . Both sides equal, verifying . The law holds generally because multiplication and addition of fractions follow the same arithmetic rules.

Exercise Set 3.3

1. Prove that the following rational numbers are equal: (i) and

Two rationals and are equal if . Here and , so . Hence .

1. Prove that the following rational numbers are equal: (ii) and

Check and . Since , .

1. Prove that the following rational numbers are equal: (iii) and

and . The products are equal, so .

1. Prove that the following rational numbers are equal: (iv) and

Write as . Then and . So .

2. Find the sum: (i)

Common denominator : . So .

2. Find the sum: (ii)

LCM of and is . , . Sum = .

2. Find the sum: (iii)

. So .

3. Find the difference: (i)

LCM : , . Difference = .

3. Find the difference: (ii)

. So .

3. Find the difference: (iii)

Subtracting a negative is adding: .

4. Find the product: (i)

4. Find the product: (ii)

4. Find the product: (iii)

5. Find the quotient: (i)

5. Find the quotient: (ii)

5. Find the quotient: (iii)

6. Show that: .

Left side: . Then . Right side: , . Sum = . Both sides equal, verifying distributivity.

7. Simplify the following using the distributive property: .

8. Find the rational number such that: .

Expand left: . This identity holds for every rational . So any rational number satisfies it; the equation is an identity demonstrating the distributive law.

Think and Reflect

Try and represent and on a number line.

To represent , divide the interval between and into equal parts and go parts to the right of . For , divide the interval between and into equal parts and go part to the left of (or parts to the right of ). Both lie between integers as shown on the number line.

Exercise Set 3.4

1. Represent the rational numbers , and on a single number line.

lies between and , at the second of three equal divisions from . lies one quarter left of . lies halfway between and . All three can be marked by dividing each unit interval into the required number of equal parts.

2. Find three distinct rational numbers that lie strictly between and .

One method is taking averages. Average of and is . Another between and is , and between and is . So three examples are , and , all lie strictly between the given numbers. Many answers are possible.

3. Simplify the expression: .

Absolute value gives distance from zero, so and . Sum = . If the intended expression was , then it equals .

4. A tailor has metres of fine silk. If making one kurta requires metres of silk, exactly how many kurtas can he make?

Convert to improper fractions: , . Number of kurtas = . He can make exactly kurtas.

5. Find three rational numbers between 3.1415 and 3.1416.

Any numbers with more decimal places between them work, e.g., , , . All lie strictly between and .

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

Yes. Besides the average , you can bring them to a common denominator and pick a numerator in between, e.g., and have between them. Scaling by a larger multiple, e.g., multiply numerator and denominator by , gives many more intermediate fractions.

Think and Reflect

Can be written as a rational number ?

No. As proved by contradiction, if in lowest terms with , then , so is even, hence is even, write . Then gives , so is also even. Both and even contradicts lowest terms. Therefore is irrational and cannot be expressed as .

Think and Reflect

Try to prove the irrationality of using the approach of proof by contradiction. Will the same approach work for , , or ?

Assume in lowest terms, then , so divides and hence . Let , then giving , so divides as well, contradiction. Same argument works for and because and are primes. For , the proof also shows irrationality, but the step needs prime factor or dividing . The method works for any non-perfect-square integer.

Think and Reflect

We have seen how to obtain a line whose length is a rational number. How do we obtain lines whose lengths are irrational?

Use the Pythagoras theorem. Construct a right triangle with legs of integer or rational length, e.g., legs and give hypotenuse . With compass, transfer that hypotenuse length onto the number line. Repeating with right triangles built on previous hypotenuses yields lengths , etc.

Think and Reflect

Try to extend this method for constructing line segments of lengths and using a ruler and a compass. Generalise this method to construct a line segment of any length of the form , where n is a positive integer.

Build the square-root spiral: Start with . Erect perpendicular , then . Next, erect perpendicular to of length , hypotenuse becomes . Continuing, each new right triangle has one leg as previous hypotenuse and other leg , giving , , after steps. With compass, each length can be marked on the number line.

Think and Reflect

Try to find the decimal expansions of and . What do you observe about the repetition of the digits after the decimal point?

– a single digit repeats immediately. – after the non-repeating , the digit repeats indefinitely. Both show eventually periodic repeating blocks, as must happen for rationals.

Think and Reflect

The decimal expansion of will be terminating precisely when the prime factors of q are only 2, only 5 or both 2 and 5. Can you explain why?

If in lowest terms has only prime factors and , we can multiply numerator and denominator by suitable powers of or to make denominator , giving a terminating decimal with places. If has any other prime factor, e.g., , the denominator can never become a power of , so long division never ends with remainder zero and must eventually repeat.

Exercise Set 3.5

1. Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: , and . Then check your answers by explicitly performing the long divisions and expressing these rational numbers as decimals.

Write denominators in lowest terms: – only so terminating: . contains prime so repeating: . – only so terminating: . Long division confirms these results.

2. Perform the long division for . Identify the repeating block of digits. Does it show cyclic properties if you evaluate ? Now compute , , etc. What do you notice?

– the 6-digit block repeats. , , etc. Each multiple is a cyclic rotation of the same digits, and . The block length is at most digits and shows cyclic permutations for multiples through .

3. Classify the following numbers as rational or irrational: (i)

Rational, because , a ratio of integers, with terminating decimal .

3. Classify the following numbers as rational or irrational: (ii)

Irrational. , so . Since is irrational, is also irrational, its decimal never terminates or repeats.

3. Classify the following numbers as rational or irrational: (iii)

Rational, it is . It is a pure repeating decimal.

3. Classify the following numbers as rational or irrational: (iv)

Rational, block repeats forever, so it equals . It is purely periodic.

3. Classify the following numbers as rational or irrational: (v) (Notice the pattern: Is it repeating a single block?)

Irrational. The number of zeros between ones grows each time, so no fixed block repeats; it is non-terminating and non-repeating.

3. Classify the following numbers as rational or irrational: (vi) Find the explicit fractions in case they are rational.

Rational and terminating. It can be written as after removing decimal, then simplified. Any terminating decimal is rational.

4. The number (which means 0.99999...) is a rational number. Using algebra (let , multiply by 10, and subtract), explain why is exactly equal to 1.

Let . Then . Subtract: , so and . Hence . This shows the decimal representation is not unique.

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

Numbers where is a full reptend prime reciprocal, such as . For example and its multiples are rotations of the same -digit block. In general primes for which is a primitive root modulo give maximal length cyclic numbers.

Think and Reflect

Consider this puzzle: What is the square root of -1? We know that . We also know that . There is no Real Number that, when multiplied by itself, results in a negative number. Thus, cannot exist on number line.

Correct, no real number squares to because the product of two same-sign numbers is always non-negative. To handle , mathematicians define the imaginary unit with . Numbers of the form form the complex numbers, extending the real line into a plane. They are essential in engineering and physics, but lie outside the real number system studied here.

End-of-Chapter Exercises

1. 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)

– terminating, because . Long division gives remainder zero after two places.

1. 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: (ii)

– non-terminating repeating, because denominator contains prime .

2. Prove that is an irrational number.

Assume in lowest terms, . Then , so divides and hence . Let . Then so , hence divides too. Both divisible by contradicts co-primality. Therefore is irrational.

3. Convert the following decimal numbers in the form of . (i)

Let . Then , subtract: , so .

3. Convert the following decimal numbers in the form of . (ii)

Here . Multiplying by 10 shifts the bar to the first decimal: . Therefore .

3. Convert the following decimal numbers in the form of . (iii)

Let . Then and ? Simpler: Let . Multiply by to shift non-repeating: , multiply by ? Standard method gives .

3. Convert the following decimal numbers in the form of . (iv)

Let . Then , subtract: , so .

3. Convert the following decimal numbers in the form of . (v)

Let . Then , subtract: , so .

3. Convert the following decimal numbers in the form of . (vi)

Let . Then and . Subtract: , so .

3. Convert the following decimal numbers in the form of . (vii)

Let . Then and . Subtract with appropriate shifts gives .

3. Convert the following decimal numbers in the form of . (viii)

Let . Then , , so .

3. Convert the following decimal numbers in the form of . (ix)

Let . Scale to separate non-repeating and repeating . The standard conversion yields .

4. Locate the following rational numbers on the number line. (i)

lies between and , slightly beyond . Divide the interval between and into equal small parts or mark approximately between and , closer to .

4. Locate the following rational numbers on the number line. (ii)

lies between and , just after . Divide into parts and mark just beyond parts from .

5. Find 6 rational numbers between 3 and 4.

Infinitely many exist, e.g., , , , , , . All satisfy .

6. Find 5 rational numbers between and .

Bring to denominator : , . Intermediate with denominator : and . Five examples: , , , , .

7. Find 5 rational numbers between and .

Common denominator : , . Numbers between: , , , , .

8. If , find the rational number .

Combine: . So and . Check: .

9. Let a and b be two non-zero rational numbers such that . Without assigning any numerical values, determine whether ab is positive or negative. Justify your answer.

If then . So . Since , , hence . Therefore is always negative. The same conclusion holds if , i.e., .

10. 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 , 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 or ? Give reasons.

A terminating decimal with 4 places equals where is the integer formed by removing the decimal point; if ends with zero we cancel factors of so we can make not divisible by . Reducing cancels only common factors and . Hence the reduced denominator divides and is of form with . It need not contain and need not contain ; e.g., has denominator .

11. Without performing division, determine whether the decimal expansion of is terminating or non-terminating. If it terminates, state the number of decimal places.

contains only prime , so the decimal terminates. Since needs to reach , write . It terminates after decimal places.

12. A rational number in its lowest form has denominator . How many decimal places will its decimal expansion have? Explain your answer.

Denominator . To get a power of , need equal exponents of and . Max exponent is , so multiply numerator and denominator by to get denominator . Hence the decimal terminates after places, not 1.

*13. Let and . Express both a and b in the form and where , and m are integers and . Using the same denominator m, write exactly five distinct rational numbers lying between a and b keeping an integer numerator. Explain why the condition is necessary to find n such rational numbers between the two rational numbers a and b using this method.

Take common denominator : , , difference , not enough. Scale by : , , now . Five integers strictly between are ; choose any five, e.g., , , , , . To have distinct integers between and , we need at least gaps, i.e., and to stay strictly inside, if counting endpoints exclusive.

*14. Three rational numbers x, y, z satisfy and . Show that all the rational numbers x, y, z must be simultaneously zero.

Square the sum: . Given both sums zero, we get , so . Sum of squares of rationals is zero only if each square is zero, because squares are non-negative. Hence , , .

*15. Show that the rational number lies between the rational numbers a and b.

Assume . Then because adding to itself gives and dividing by preserves inequality. If , the same argument gives . Hence the average is always strictly between.

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

The spiral starts with isosceles right triangle legs giving hypotenuse . Each successive triangle uses previous hypotenuse as one leg and as other leg, so lengths are for the -th triangle. Specifically the first triangle , second , third , fourth , fifth , etc.