Class 9 Maths Part 1 Chapter 3 – The World of Numbers from Ganita Manjari with easy Notes and complete Book Solutions. Get clear explanations, important concepts, questions and answers, and revision material for Chapter 3 to help Class 9 students learn, practise and prepare effectively for examinations.
📘 The World of Numbers
Grade 9 Mathematics | Ganita Manjari | Part I
Easy Notes • Concepts • Complete Book Solutions
🌟 Easy Notes & Concepts
1. Natural Numbers
Natural numbers are the counting numbers:
N = {1, 2, 3, 4, 5, …}
They are closed under addition and multiplication, but not always under subtraction or division.
2. Zero — Śhūnya
Zero represents nothing and is also a number on which arithmetic operations can be performed.
a + 0 = a a − 0 = a a × 0 = 0
3. Integers
Integers include negative numbers, zero and positive numbers.
Z = {…, −3, −2, −1, 0, 1, 2, 3, …}
Think of positive numbers as fortunes and negative numbers as debts.
4. Rational Numbers
A rational number can be written as:
p/q, where p and q are integers and q ≠ 0
Examples: 1/2, −3/4, 5, 0.25 and 0.333…
5. Irrational Numbers
Irrational numbers cannot be written as a ratio of two integers.
Examples:
√2, √5, π
Their decimal expansions are non-terminating and non-repeating.
6. Real Numbers
Rational numbers and irrational numbers together form the real numbers.
Real Numbers = Rational Numbers ∪ Irrational Numbers
7. Decimal Expansions
- Terminating decimal → rational.
- Non-terminating repeating decimal → rational.
- Non-terminating non-repeating decimal → irrational.
8. Important Number-Line Idea
Between any two distinct rational numbers, there is always another rational number.
A simple way is to take their average:
(a + b) / 2
9. Absolute Value
The absolute value of a number is its distance from 0.
|5/3| = 5/3 |−5/3| = 5/3 |0| = 0
10. Remember
- Natural numbers are contained in integers.
- Integers are contained in rational numbers.
- Rational + irrational numbers make real numbers.
- Every rational number has a terminating or repeating decimal expansion.
- Irrational decimal expansions never terminate and never repeat.
🧠 Think and Reflect — Chapter 3
Think and Reflect
Book p. 42
Why are Natural Numbers called ‘Natural’?
They are called Natural Numbers because they are the basic numbers we naturally use for counting objects: 1, 2, 3, 4, …
Think and Reflect
Book p. 42
Are all counting numbers Natural Numbers?
Yes. The counting numbers 1, 2, 3, … are Natural Numbers.
📗 Exercise Set 3.1
1.Book p. 43
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?
2 bags → 15 ingots
12 bags = 6 × 2 bags
So ingots = 6 × 15 = 90
✅ Answer: 90 copper ingots.
2.Book p. 43
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.
11, 13, 17 and 19 are prime numbers. The next three prime numbers are 23, 29 and 31.
3.Book p. 43
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.
5 − 2 = 3, which is natural.
2 − 5 = −3, which is not a natural number.
Therefore, subtraction does not always give a natural number.
*4.Book p. 43
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?
There are 4 fingers other than the thumb.
Each has 3 joints.
4 × 3 = 12.
So, one hand can be used to count 12. This connects naturally with a base-12 counting system.
📗 Exercise Set 3.2 — Integers
1.Book p. 46
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?
4 − 15 = −11
✅ Midnight temperature = −11 °C.
2.Book p. 46
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.
−850 + 1200 − 450 = −100
He has a final debt of ₹100.
3.Book p. 46
Calculate the following using Brahmagupta’s laws:
(i) (–12) × 5
(ii) (–8) × (–7)
(iii) 0 – (–14)
(iv) (–20) ÷ 4
(i) (−12) × 5 = −60
(ii) (−8) × (−7) = 56
(iii) 0 − (−14) = 14
(iv) (−20) ÷ 4 = −5
(ii) (−8) × (−7) = 56
(iii) 0 − (−14) = 14
(iv) (−20) ÷ 4 = −5
4.Book p. 46
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 owe ₹10. If someone removes a debt of ₹5 from your account, your debt decreases by ₹5. That makes you ₹5 richer compared with before.
So:
So:
10 − (−5) = 10 + 5 = 15
🧠 Think and Reflect — Rational Numbers
1.Book p. 48
While adding or subtracting two rational numbers having different denominators, how will you make the denominators equal?
Find a common denominator, usually the LCM of the denominators. Convert both fractions to equivalent fractions having that denominator.
2.Book p. 48
Verify the distributive law for rational numbers.
Take p = 1/2, q = 1/3 and r = 1/6.
p(q+r) = 1/2 × (1/3 + 1/6) = 1/2 × 1/2 = 1/4.
pq + pr = 1/2×1/3 + 1/2×1/6 = 1/6 + 1/12 = 1/4.
Both sides are equal. Therefore the distributive law is verified.
📗 Exercise Set 3.3 — Rational Numbers
1.Book p. 49
Prove that the following rational numbers are equal:
(i) 2/3 and 4/6
(ii) 5/4 and 10/8
(iii) −3/5 and −6/10
(iv) 9/3 and 3/2.
(i) 2/3 = 2×2 / 3×2 = 4/6 ✓
(ii) 5/4 = 5×2 / 4×2 = 10/8 ✓
(iii) −3/5 = −3×2 / 5×2 = −6/10 ✓
(iv) 9/3 = 3, whereas 3/2 = 1.5.
(ii) 5/4 = 5×2 / 4×2 = 10/8 ✓
(iii) −3/5 = −3×2 / 5×2 = −6/10 ✓
(iv) 9/3 = 3, whereas 3/2 = 1.5.
The expressions in part (iv), as extracted from the uploaded PDF, are not equal. Therefore they cannot be proved equal as written.
2.Book p. 49
Find the sum:
(i) 2/5 + 3/10
(ii) 7/12 + 5/8
(iii) −4/7 + 3/14
(i) 2/5 = 4/10, so 4/10 + 3/10 = 7/10.
(ii) LCM of 12 and 8 = 24.
7/12 = 14/24 and 5/8 = 15/24.
Sum = 29/24 = 1 5/24.
(iii) −4/7 = −8/14.
−8/14 + 3/14 = −5/14.
(ii) LCM of 12 and 8 = 24.
7/12 = 14/24 and 5/8 = 15/24.
Sum = 29/24 = 1 5/24.
(iii) −4/7 = −8/14.
−8/14 + 3/14 = −5/14.
3.Book p. 49
Find the difference:
(i) 5/6 − 1/4
(ii) 11/8 − 3/4
(iii) −7/9 − (−2/3)
(i) LCM 12: 10/12 − 3/12 = 7/12.
(ii) 3/4 = 6/8, so 11/8 − 6/8 = 5/8.
(iii) −7/9 + 2/3 = −7/9 + 6/9 = −1/9.
(ii) 3/4 = 6/8, so 11/8 − 6/8 = 5/8.
(iii) −7/9 + 2/3 = −7/9 + 6/9 = −1/9.
4.Book p. 49
Find the product:
(i) 2/3 × 3/10
(ii) 7/11 × 5/8
(iii) −4/7 × 5/14
(i) 2/3 × 3/10 = 1/5.
(ii) 7/11 × 5/8 = 35/88.
(iii) −4/7 × 5/14 = −20/98 = −10/49.
(ii) 7/11 × 5/8 = 35/88.
(iii) −4/7 × 5/14 = −20/98 = −10/49.
5.Book p. 49
Find the quotient:
(i) 2/3 ÷ 3/10
(ii) 7/11 ÷ 5/8
(iii) −4/7 ÷ 5/14
(i) 2/3 × 10/3 = 20/9.
(ii) 7/11 × 8/5 = 56/55.
(iii) −4/7 × 14/5 = −8/5.
(ii) 7/11 × 8/5 = 56/55.
(iii) −4/7 × 14/5 = −8/5.
6.Book p. 50
Show that: (1 2/3 + 3/4) × 8/3 = 1 2/3 × 8/3 + 3/4 × 8/3.
1 2/3 = 5/3.
LHS = (5/3 + 3/4) × 8/3.
5/3 + 3/4 = 20/12 + 9/12 = 29/12.
LHS = 29/12 × 8/3 = 58/9.
RHS = 5/3×8/3 + 3/4×8/3 = 40/9 + 2 = 58/9.
LHS = RHS. Hence the distributive law is verified.
7.Book p. 50
Simplify the following using the distributive property: 7/9 − (6/7 − 3/4).
6/7 − 3/4 = 24/28 − 21/28 = 3/28.
7/9 − 3/28.
LCM of 9 and 28 = 252.
196/252 − 27/252 = 169/252.
8.Book p. 50
Find the rational number x such that: 5/6 (3/5 + x) = 5/6 + 1/2.
5/6(3/5 + x) = 1/2 + 5/6.
1/2 + 5/6 = 3/6 + 5/6 = 8/6 = 4/3.
3/5 + x = (4/3) × (6/5) = 8/5.
x = 8/5 − 3/5 = 5/5.
✅ x = 1
📏 Rational Numbers on the Number Line
To represent p/q on a number line:
- Divide the unit interval into q equal parts.
- Move p parts from 0.
- Move right for a positive number and left for a negative number.
Simple Number Line
0
1/4
1/2
3/4
1
Think and ReflectBook p. 51
Try and represent 8/5 and –7/4 on a number line.
8/5 = 1 3/5, so it lies between 1 and 2.
−7/4 = −1 3/4, so it lies between −2 and −1.
−7/4 = −1 3/4, so it lies between −2 and −1.
Idea: Divide each unit interval into 5 parts for 8/5 and 4 parts for −7/4.
Think and ReflectBook p. 52
Try to explain why the average of two rational numbers a and b, which equals (a + b) / 2, is always a rational number between a and b.
Since a and b are rational, a+b is rational. Dividing a rational number by the non-zero rational number 2 gives another rational number.
Also, the average of two different numbers lies halfway between them. Therefore (a+b)/2 is a rational number between a and b.
Also, the average of two different numbers lies halfway between them. Therefore (a+b)/2 is a rational number between a and b.
📗 Exercise Set 3.4 — Rational Numbers and Density
1.Book p. 52
Represent the rational numbers 2/3, –5/4 and 11/2 on a single number line.
2/3 lies between 0 and 1.
−5/4 = −1 1/4 lies between −2 and −1.
11/2 = 5 1/2 lies between 5 and 6.
−5/4 = −1 1/4 lies between −2 and −1.
11/2 = 5 1/2 lies between 5 and 6.
📏 Mark 0 first, then divide each required unit interval into the required number of equal parts.
2.Book p. 52
Find three distinct rational numbers that lie strictly between –1/2 and 1/4.
One possible set is:
−1/4, 0, 1/8.
All three lie strictly between −1/2 and 1/4.
All three lie strictly between −1/2 and 1/4.
3.Book p. 53
Simplify the expression: −(1/4) + (5/12).
LCM of 4 and 12 = 12.
−1/4 = −3/12.
−3/12 + 5/12 = 2/12.
✅ 1/6
4.Book p. 53
A tailor has 15 3/4 metres of fine silk. If making one kurta requires 2 1/4 metres of silk, exactly how many kurtas can he make?
15 3/4 = 63/4.
2 1/4 = 9/4.
(63/4) ÷ (9/4) = 63/9 = 7.
✅ He can make 7 kurtas.
5.Book p. 53
Find three rational numbers between 3.1415 and 3.1416.
Possible answers:
3.14151, 3.14155, 3.14159.
Each is greater than 3.1415 and less than 3.1416.
3.14151, 3.14155, 3.14159.
Each is greater than 3.1415 and less than 3.1416.
*6.Book p. 53
Can you think of other way(s) to find a rational number between any two rational numbers?
Yes. Besides taking the average, we can write the two rational numbers with a common denominator and choose a numerator between their numerators, if possible.
√ Irrational Numbers
Think and ReflectBook p. 53
Can √2 be written as a rational number p/q?
No. √2 is irrational. It cannot be written as p/q where p and q are integers and q ≠ 0.
Easy idea behind √2
If a square has side 1 unit, its diagonal has length √2. This length cannot be represented exactly as a fraction.
📗 Exercise Set 3.5 — Decimal Expansions
1.Book p. 61
Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: 7/20, 4/15 and 13/250. Then check your answers by explicitly performing the long divisions and expressing these rational numbers as decimals.
| Number | Decimal | Type |
|---|---|---|
| 7/20 | 0.35 | Terminating |
| 4/15 | 0.2666… | Repeating |
| 13/250 | 0.052 | Terminating |
A fraction in lowest form has a terminating decimal when its denominator has only 2s and/or 5s as prime factors.
2.Book p. 61
Perform the long division for 1/13. Identify the repeating block of digits. Does it show cyclic properties if you evaluate 2/13? Now compute 3/13, 4/13, etc. What do you notice?
| Fraction | Decimal |
|---|---|
| 1/13 | 0.076923076923… |
| 2/13 | 0.153846153846… |
| 3/13 | 0.230769230769… |
| 4/13 | 0.307692307692… |
| 5/13 | 0.384615384615… |
| 6/13 | 0.461538461538… |
The decimal digits repeat in cycles. The repeating blocks are related to one another and show a clear pattern.
3.Book p. 62
Classify the following numbers as rational or irrational:
(i) √81
(ii) √12
(iii) 0.33333 …
(iv) 0.123451234512345 …
(v) 1.01001000100001 … (Notice the pattern: Is it repeating a single block?)
(vi) 23.560185612239874790120
Find the explicit fractions in case they are rational.
| Number | Classification | Reason |
|---|---|---|
| √81 = 9 | Rational | 9 is an integer |
| √12 | Irrational | 12 is not a perfect square |
| 0.33333… | Rational | Repeating decimal = 1/3 |
| 0.123451234512345… | Rational | Repeating block |
| 1.01001000100001… | Irrational | Pattern does not repeat as one fixed block |
| 23.560185612239874790120 | Rational | Terminating decimal |
For (vi), the fraction is the decimal written over the appropriate power of 10 and then reduced.
4.Book p. 62
The number 0.9 (which means 0.99999 …) is a rational number. Using algebra (let x = 0.9, multiply by 10, and subtract), explain why 0.9 is exactly equal to 1.
Let x = 0.9999…
10x = 9.9999…
Subtract: 10x − x = 9.9999… − 0.9999…
9x = 9
Therefore x = 1. Hence 0.999… = 1.
*5.Book p. 62
We have seen that the repeating block of 1/7 is a cyclic number. Try to find more numbers (n) whose reciprocals (1/n) produce decimals with repeating blocks that are cyclic.
Examples include denominators such as 7, 13, 17, 19, whose reciprocals have repeating decimal patterns with cyclic behaviour.
🏁 End-of-Chapter Exercises
1.Book p. 64
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) 3/50
(ii) 2/9
(i) 3/50 = 0.06 — terminating.
(ii) 2/9 = 0.2222… — non-terminating and repeating.
(ii) 2/9 = 0.2222… — non-terminating and repeating.
2.Book p. 64
Prove that √5 is an irrational number.
Suppose √5 is rational.
Then √5 = p/q, where p and q have no common factor and q ≠ 0.
Squaring: 5 = p²/q².
So p² = 5q².
Therefore 5 divides p², so 5 divides p.
Let p = 5k.
Then 25k² = 5q², so q² = 5k².
Therefore 5 divides q.
Thus both p and q are divisible by 5, contradicting that p/q was in lowest form. Therefore √5 is irrational.
3.Book p. 64
Convert the following decimal numbers in the form of p/q.
(i) 12.6
(ii) 0.0120
(iii) 3.052
(iv) 1.235
(v) 0.23
(vi) 2.05
(vii) 2.125
(viii) 3.125
(ix) 2.1625
| Decimal | Fraction |
|---|---|
| 12.6 | 63/5 |
| 0.0120 | 3/250 |
| 3.052 | 763/250 |
| 1.235 | 247/200 |
| 0.23 | 23/100 |
| 2.05 | 41/20 |
| 2.125 | 17/8 |
| 3.125 | 25/8 |
| 2.1625 | 173/80 |
4.Book p. 64
Locate the following rational numbers on the number line.
(i) 0.532
(ii) 1.15
0.532 lies between 0 and 1, very close to 0.5.
1.15 lies between 1 and 2, just to the right of 1.
1.15 lies between 1 and 2, just to the right of 1.
📏 Number-line idea: Divide the required interval into 1000 equal parts for 0.532 and 100 equal parts for 1.15.
5.Book p. 65
Find 6 rational numbers between 3 and 4.
One possible answer is:
3.1, 3.2, 3.3, 3.4, 3.5, 3.6.
6.Book p. 65
Find 5 rational numbers between 2/5 and 3/5.
Convert to denominator 30:
2/5 = 12/30 and 3/5 = 18/30.
So five answers are: 13/30, 14/30, 15/30, 16/30, 17/30.
So five answers are: 13/30, 14/30, 15/30, 16/30, 17/30.
7.Book p. 65
Find 5 rational numbers between 1/6 and 2/5.
Use denominator 30:
1/6 = 5/30 and 2/5 = 12/30.
Five possible numbers are: 6/30, 7/30, 8/30, 9/30, 10/30.
Five possible numbers are: 6/30, 7/30, 8/30, 9/30, 10/30.
8.Book p. 65
If x/3 + x/5 = 16/15, find the rational number x.
x/3 + x/5 = 16/15
LCM of 3 and 5 = 15.
5x/15 + 3x/15 = 16/15.
8x/15 = 16/15.
✅ x = 2
9.Book p. 65
Let a and b be two non-zero rational numbers such that 1/a + 1/b = 0. Without assigning any numerical values, determine whether ab is positive or negative. Justify your answer.
1/a + 1/b = 0
(a+b)/ab = 0.
Since ab ≠ 0, we must have a+b=0.
Therefore b = −a.
So ab = a(−a) = −a², which is negative.
10.Book p. 65
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 p/10⁴, 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 2⁴ or 5⁴? Give reasons.
If the last non-zero digit is in the 4th decimal place, write the decimal over 10⁴.
For example, 0.1234 = 1234/10000.
The numerator is not divisible by 10 because the last digit is non-zero.
No. After reducing the fraction, factors of 2 or 5 may cancel. Therefore the lowest-form denominator need not contain both 2⁴ and 5⁴.
11.Book p. 65
Without performing division, determine whether the decimal expansion of 18/125 is terminating or non-terminating. If it terminates, state the number of decimal places.
125 = 5³.
The denominator has only the prime factor 5.
Therefore the decimal expansion is terminating.
18/125 = 0.144, so it has 3 decimal places.
18/125 = 0.144, so it has 3 decimal places.
12.Book p. 65
A rational number in its lowest form has denominator 2³ × 5. How many decimal places will its decimal expansion have? Explain your answer.
The denominator is 2³ × 5.
To make the denominator a power of 10, multiply by 5²:
2³ × 5 × 5² = 2³ × 5³ = 10³.
Therefore the decimal expansion will have at most 3 decimal places.
2³ × 5 × 5² = 2³ × 5³ = 10³.
Therefore the decimal expansion will have at most 3 decimal places.
*13.Book p. 65
Let a = 7/12 and b = 5/6. Express both a and b in the form k₁/m and k₂/m where k₁, k₂ and m are integers and k₂ – k₁ > 6. Using the same denominator m, write exactly five distinct rational numbers lying between a and b keeping an integer numerator. Explain why the condition k₂ – k₁ > n + 1 is necessary to find n such rational numbers between the two rational numbers a and b using this method.
a = 7/12.
b = 5/6 = 10/12.
The numerator difference is only 3, so multiply both by 3:
a = 21/36 and b = 30/36.
Now there are 8 integers strictly between 21 and 30. We can choose any five, for example:
22/36, 23/36, 24/36, 25/36, 26/36.
To obtain n distinct fractions between two fractions using this common-denominator method, we need at least n integers strictly between the two numerators. Hence the numerator difference must be greater than n+1.
*14.Book p. 65
Three rational numbers x, y, z satisfy x + y + z = 0 and xy + yz + zx = 0. Show that all the rational numbers x, y, z must be simultaneously zero.
From x+y+z=0, z=−(x+y).
Substitute in xy+yz+zx=0:
xy + y(−x−y) + x(−x−y)=0.
xy−xy−y²−x²−xy=0.
x² + xy + y² = 0.
Multiply by 4:
(2x+y)² + 3y² = 0.
Both terms are non-negative. Their sum is zero only when both are zero. Hence y=0 and then x=0. Since z=−x−y, z=0.
Therefore x = y = z = 0.
Therefore x = y = z = 0.
*15.Book p. 66
Show that the rational number (a + b)/2 lies between the rational numbers a and b.
Assume a < b.
Adding a < b to itself gives: 2a < a+b.
Also, adding b > a to itself gives: a+b < 2b.
Therefore:
Adding a < b to itself gives: 2a < a+b.
Also, adding b > a to itself gives: a+b < 2b.
Therefore:
2a < a+b < 2b
Dividing by 2:a < (a+b)/2 < b
Hence (a+b)/2 lies between a and b.
16.Book p. 66
Find the lengths of the hypotenuses of all the right triangles in Fig. 3.14 which is referred to as the square root spiral.
In the square-root spiral, each new right triangle has one side of length 1 and the previous hypotenuse as the other side.
Therefore the successive hypotenuse lengths are:
Therefore the successive hypotenuse lengths are:
√2, √3, √4, √5, √6, …
The exact number of triangles to list depends on the triangles shown in Fig. 3.14. Each is found using the Baudhāyana–Pythagoras Theorem.
⭐ Quick Revision — Chapter Summary
Number System at a Glance
| Number Type | Easy Meaning | Examples |
|---|---|---|
| Natural | Counting numbers | 1, 2, 3, … |
| Integers | Negative numbers, zero and positive numbers | …, −2, −1, 0, 1, 2, … |
| Rational | Can be written p/q | 1/2, −3/4, 5 |
| Irrational | Cannot be written p/q | √2, √5, π |
| Real | Rational + irrational | All points on real number line |
N ⊂ Z ⊂ Q ⊂ R
Remember: Irrational numbers are also real numbers, but they are not rational.
🧠 Exam Tips
- Always reduce fractions when possible.
- For addition/subtraction of fractions, first make denominators equal.
- For division of fractions, multiply by the reciprocal.
- For terminating decimals, the denominator in lowest form contains only 2 and/or 5 as prime factors.
- Between any two different rational numbers, there are infinitely many rational numbers.
- Use a contradiction carefully when proving irrationality.
prepare more with the given Worksheet of chapter 3, which will help you to improve your knowledge .
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