CLASS ix maths Chapter 1
Exercise 1.1 — Solutions with Explanations
1. Is zero a rational number? Can you write it in the form , where and are integers and ?
Answer: Yes, zero is a rational number.
Explanation:
A number is called rational if it can be written in the form , where:
-
and are integers, and
-
We can write:
In all these examples, and . So, 0 is rational.
2. Find six rational numbers between 3 and 4.
Answer:
Some rational numbers between 3 and 4 are:
Explanation:
To find rational numbers between 3 and 4, write 3 and 4 with the same denominator.
Let’s choose denominator = 7 (arbitrary choice for convenience):
Now pick numbers between and :
These are six rational numbers between 3 and 4.
3. Find five rational numbers between and .
Answer:
Explanation:
We can write both and with a common denominator:
Now, choose rational numbers between and :
These are the five rational numbers between and .
4. State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number.
Answer: True
Explanation:
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Natural numbers: 1, 2, 3, 4, ...
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Whole numbers: 0, 1, 2, 3, 4, ...
So, every natural number is included in the whole numbers.
(ii) Every integer is a whole number.
Answer: False
Explanation:
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Integers include negative numbers: ..., –3, –2, –1, 0, 1, 2, 3, ...
-
Whole numbers are only: 0, 1, 2, 3, ...
So, negative integers like –2, –1 are not whole numbers.
(iii) Every rational number is a whole number.
Answer: False
Explanation:
Rational numbers include fractions like , etc.
These are not whole numbers.
Only specific rational numbers (like 0, 1, 2, etc.) are whole numbers.
Here are the detailed solutions with explanations for Exercise 1.2 from the Real Numbers chapter:
✅ Exercise 1.2 – Solutions with Explanations
1. State whether the following statements are true or false. Justify your answers.
(i) Every irrational number is a real number.
Answer: ✅ True
Explanation:
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Real numbers include all rational and irrational numbers.
-
So, by definition, irrational numbers are a subset of real numbers.
Examples of irrational numbers:
All of them are real.
(ii) Every point on the number line is of the form , where is a natural number.
Answer: ❌ False
Explanation:
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Points like , , or are on the number line, but they are not of the form .
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Also, only gives positive irrational or rational values, while the number line contains:
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Rational numbers
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Negative numbers
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Irrationals like , which is not a square root of any natural number.
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Hence, not every point is of the form .
(iii) Every real number is an irrational number.
Answer: ❌ False
Explanation:
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Real numbers = Rational + Irrational numbers
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So, real numbers include rational numbers too, like:
-
Therefore, not all real numbers are irrational.
2. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.
Answer: ❌ No, not all square roots of positive integers are irrational.
Example:
Explanation:
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If a number is a perfect square (like 1, 4, 9, 16...), then its square root is a rational number.
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Otherwise, it’s irrational.
So, is irrational, but is rational.
3. Show how can be represented on the number line.
Answer:
You can represent using the Pythagoras Theorem and geometric construction.
Steps (explained):
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Draw a line segment AB = 2 units.
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At point B, draw a perpendicular line segment BC = 1 unit.
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Join AC. Using the Pythagoras Theorem:
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Using a compass, place the needle on point A and mark an arc of radius on the number line.
✅ This point on the number line represents .
You can use this method for any square root value like , etc.
This activity describes a Square Root Spiral, also known as Theodorus Spiral or Spiral of Square Roots. Here's how you can construct it and understand the math behind it:
✅ Constructing the Square Root Spiral – Step-by-Step Explanation
🧭 Objective:
Create a geometric figure where the length of each new segment represents the square root of a natural number. Visually, you will see emerge.
✍️ Materials Needed:
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A large sheet of paper
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A ruler
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A protractor or compass
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Pencil
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Eraser
🧱 Construction Steps:
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Start Point: Mark a point O on the paper.
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First Segment (Unit length):
Draw a straight horizontal line from O to P₁ such that: -
Second Segment :
From point P₁, draw a perpendicular line upward. Mark a point P₂ such that:Then, join O to P₂. Using the Pythagoras Theorem:
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Third Segment :
From P₂, draw a line segment P₂P₃ = 1 unit, perpendicular to OP₂.
Now join O to P₃: -
Continue the Process:
Each time, you draw a line segment of 1 unit length, perpendicular to the previous line from O to Pₙ, and then connect the new point to O. -
You get a Spiral:
This forms a spiral where:
📐 Mathematical Concept Behind the Spiral
Each new triangle you build is a right-angled triangle, and by applying the Pythagoras Theorem:
So,
🌀 Why is this called a Square Root Spiral?
Because as you go outward, each new line from O to a point Pₙ has a length equal to . Visually, you are plotting the square roots on the paper!
🖼️ Optional: Want a visual illustration?
I can generate an image of the spiral to show you how it looks. Would you like me to draw that for you?
Here are the fully explained solutions for Exercise 1.3 from the Real Numbers chapter:
✅ Exercise 1.3 – Solutions with Explanations
1. Write the following in decimal form and say what kind of decimal expansion each has:
(i)
Decimal form:
Type: Terminating decimal.
Explanation:
The division ends after two decimal places. No repeating digits.
(ii)
Decimal form:
Type: Non-terminating, repeating (recurring) decimal.
Explanation:
Digits "09" keep repeating forever: 0.09090909...
(iii)
Decimal form:
Type: Terminating decimal.
Explanation:
It's a whole number (rational), so its decimal form ends.
(iv)
Decimal form:
Type: Non-terminating, repeating decimal.
Explanation:
It has a repeating block of 6 digits: 230769
(v)
Decimal form:
Type: Non-terminating, repeating decimal.
Explanation:
Repeats the digits 18: 0.181818...
(vi)
Decimal form:
Type: Terminating decimal.
Explanation:
Division ends after 4 decimal places. No repetition.
2. You know that . Can you predict the decimal expansions of ?
Answer & Explanation:
The decimal expansion of
This repeating block of 6 digits cycles through all other numerators (2 to 6):
| Fraction | Decimal Form |
|---|---|
Explanation:
Each is just a cyclic permutation of the repeating block 142857.
3. Express the following in the form , where and are integers and :
(i)
Let
Multiply by 10:
Subtract:
So,
✅ Answer:
(ii)
Let
Multiply by 100:
Subtract:
So,
✅ Answer:
(iii)
Let
The repeating block has 3 digits. Multiply by 1000:
Subtract:
So,
✅ Answer:
4. Express in the form . Are you surprised by your answer?
Let
Multiply by 10:
Subtract:
So,
✅ Answer:
Explanation:
This shows that 0.999... and 1 are exactly the same real number.
5. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of ? Perform the division to check your answer.
Answer: At most 16 digits.
Explanation:
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The length of the repeating block in the decimal expansion of , where is a prime number, is less than or equal to .
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Here, , so maximum length = 16.
If you divide 1 by 17, you will get:
✅ This block has 16 digits. So, prediction is correct.
6. Look at several examples of rational numbers in the form with terminating decimals. Can you guess what property must satisfy?
Answer:
must have only 2 and/or 5 as its prime factors.
Explanation:
A rational number (in lowest terms) has a terminating decimal if and only if the prime factorization of q contains only 2s or 5s or both.
Examples:
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→ denominator =
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→ denominator =
But:
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→ non-terminating since 7 is not 2 or 5.
7. Write three numbers whose decimal expansions are non-terminating non-recurring.
Answer:
Examples:
Explanation:
These numbers:
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Never end (non-terminating)
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Have no repeating pattern (non-recurring)
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Hence, they are irrational numbers.
8. Find three different irrational numbers between the rational numbers and .
Answer:
First, approximate the given values:
Now pick irrational numbers between them:
-
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(Too large! Try → Still big)
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So, valid answers:
✅ All are non-terminating, non-repeating decimals between 0.714 and 0.818.
9. Classify the following numbers as rational or irrational:
(i) → Irrational
(ii) → Rational
(iii) → Rational (terminates)
(iv) → Rational (repeats)
(v) → Irrational (non-terminating, non-repeating)
Here are the step-by-step solutions with explanations for Exercise 1.4 from the Real Numbers chapter:
✅ Exercise 1.4 – Solutions with Explanations
1. Classify the following numbers as rational or irrational:
(i)
Answer: ✅ Irrational
Explanation:
Both and are irrational.
The difference of two irrational numbers can be rational but usually is not unless they're designed to cancel each other out.
Since and ,
This is not a terminating or repeating decimal, hence irrational.
(ii)
Answer: ✅ Rational
Explanation:
Use identity:
So,
✅ Result is rational
(iii)
Answer: ✅ Irrational
Explanation:
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Both numerator and denominator are irrational.
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Their ratio is generally irrational unless the square roots simplify (which they don't here).
So:
(iv)
Answer: ✅ Irrational
Explanation:
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is irrational.
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Dividing 1 by an irrational number gives another irrational number.
Even after rationalising the denominator:
(v)
Answer: ✅ Irrational
Explanation:
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is irrational.
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Multiplying an irrational number by a non-zero rational number (like 2) results in an irrational number.
So, is irrational.
2. Simplify each of the following expressions:
(i)
Answer:
Explanation:
Just like adding like terms:
(ii)
Answer:
Explanation:
, and anything multiplied by 0 is 0.
(iii)
Answer:
Explanation:
Use identity:
Here,
(iv)
Answer:
Explanation:
Use identity:
3. π is defined as the ratio of the circumference (c) of a circle to its diameter (d). That is, . This seems to contradict the fact that π is irrational. How will you resolve this contradiction?
Answer:
There is no contradiction.
Explanation:
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is not saying that π is a ratio of two integers.
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Instead, it defines a relationship between the length of the circumference and the diameter.
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Both c and d are real measurements, not necessarily integers or rational numbers.
So:
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Although , it does not mean π is rational.
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In fact, even if c and d are irrational, their ratio can be irrational — which π is.
✅ No contradiction — π is still irrational.
4. Represent on the number line.
Answer:
You can use the semi-circle construction method:
Steps:
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Mark a segment AB = 9.3 units on a number line.
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Extend 1 unit further from B to a point C so that AC = 10.3 units.
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Find the midpoint O of AC.
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Draw a semi-circle with diameter AC.
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At point B (which is at 9.3 units from A), draw a perpendicular line that intersects the semi-circle at point D.
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The length BD = √9.3
Then, place BD along the number line using a compass from zero. That gives you the location of .
📌 This method is based on the Pythagorean Theorem, just like in the book’s Fig. 1.11–1.13.
Would you like me to generate an illustration of this geometric construction for you?
5. Rationalise the denominators of the following:
(i)
Multiply numerator and denominator by :
(ii)
Multiply by conjugate:
(iii)
Multiply by conjugate:
(iv)
Multiply by conjugate:
Here are the fully explained solutions for Exercise 1.5 from the Real Numbers chapter, covering laws of exponents for real numbers.
✅ Exercise 1.5 – Solutions with Explanations
1. Find:
(i)
Answer:
Explanation:
The exponent means "square root".
Since , we get:
✅
(ii)
Answer:
Explanation:
The exponent means "fifth root".
Since , we have:
✅
(iii)
Answer:
Explanation:
The cube root of 125 is 5, because .
✅
2. Find:
(i)
Answer:
First, express 9 as :
Or evaluate directly:
✅ Final answer:
(ii)
Answer:
Explanation:
-
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Square it:
✅ Answer:
(iii)
Answer:
Now,
✅ Final Answer:
(iv)
Answer:
Explanation:
Negative exponent means reciprocal.
✅ Answer:
3. Simplify:
(i)
Use Law:
✅ Answer:
(ii)
Answer:
This is the cube root of .
There’s no simplification; leave it as:
✅ Final Answer:
Or approximate:
(iii)
Use Law:
✅ Answer:
(iv)
Use Law:
✅ Answer:
🔁 Summary of Exponent Laws Used
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