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NCERT Solutions for Class 9 Maths Chapter 1 - Number System (Exercise 1.2)

Q1. State whether the following statements are true or false. Justify your answers.
(i) Every irrational number is a real number.
Ans: True

Reason: Irrational Numbers – A number is said to be irrational, if it cannot be written in the p/q, where p and q are integers and q ≠ 0.
i.e., Irrational numbers = π, e, √3, 5+√2, 6.23146…. , 0.101001001000….
Real numbers – The collection of both rational and irrational numbers are known as real numbers.
i.e., Real numbers = √2, √5, 56 , 0.102…
Every irrational number is a real number, however, every real number is not an irrational number.

(ii) Every point on the number line is of the form √m where m is a natural number.
Ans: False

Reason: The statement is false because as per the rule, a negative number cannot be expressed as square roots.
E.g., √9 =3 is a natural number.
But √2 = 1.414 is not a natural number.
Similarly, we know that negative numbers exist on the number line, but their square roots are not real numbers; they are complex.
E.g., √-7 = 7i, where i = √-1
The statement that every point on the number line is of the form √m, where m is a natural number is false.

(iii) Every real number is an irrational number.
Ans: False

Reason: The statement is false, the real numbers include both irrational and rational numbers. Therefore, every real number cannot be an irrational number.
Real numbers – The collection of both rational and irrational numbers are known as real numbers.
i.e., Real numbers = √2, √5, , 0.102…
Irrational Numbers – A number is said to be irrational if it cannot be written in the p/q, where p and q are integers and q ≠ 0.
i.e., Irrational numbers = π, e, √3, 5+√2, 6.23146…. , 0.101001001000….
Every irrational number is a real number, however, every real number is not irrational.


Q2. 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.
Ans: No, the square roots of all positive integers are not irrational.
For example,
√4 = 2 is rational.
√9 = 3 is rational.
Hence, the square roots of positive integers 4 and 9 are not irrational. ( 2 and 3, respectively).


Q3. Show how √5 can be represented on the number line.
Ans:
Step 1: Let line AB be of 2 unit on a number line.
Step 2: At B, draw a perpendicular line BC of length 1 unit.
Step 3: Join CA
Step 4: Now, ABC is a right angled triangle. Applying Pythagoras theorem,
AB+ BC2 = CA2
2+ 12 = CA2 = 5
⇒ CA = √5 . Thus, CA is a line of length √5 unit.
Step 5: Taking CA as a radius and A as a center draw an arc touching the number line. The point at which number line get intersected by arc is at √5 distance from 0 because it is a radius of the circle whose center was A.
Thus, √5 is represented on the number line as shown in the figure.

√5 on number line√5 on number line
Q4. Classroom activity (Constructing the ‘square root spiral’) : Take a large sheet of paper and construct the ‘square root spiral’ in the following fashion. Start with a point O and draw a line segment OP1 of unit length. Draw a line segment P1 P2 perpendicular to OP1 of unit length (see Fig. 1.9). Now draw a line segment P2 P3 perpendicular to OP2 . Then draw a line segment P3 P4 perpendicular to OP3.
NCERT Solutions for Class 9 Maths Chapter 1 - Number System (Exercise 1.2)Continuing in this manner, you can get the line segment Pn–1Pn by drawing a line segment of unit length perpendicular to OPn–1. In this manner, you will have created the points P2 , P3 ,...., Pn ,... ., and joined them to create a beautiful spiral depicting √2, √3, √4, ....
Ans: 
NCERT Solutions for Class 9 Maths Chapter 1 - Number System (Exercise 1.2)Step 1: Mark a point O on the paper. Here, O will be the center of the square root spiral.
Step 2: From O, draw a straight line, OA, of 1cm horizontally.
Step 3: From A, draw a perpendicular line, AB, of 1 cm.
Step 4: Join OB. Here, OB will be of √2
Step 5: Now, from B, draw a perpendicular line of 1 cm and mark the end point C.
Step 6: Join OC. Here, OC will be of √3
Step 7: Repeat the steps to draw √4, √5, √6….

The document NCERT Solutions for Class 9 Maths Chapter 1 - Number System (Exercise 1.2) is a part of the Class 9 Course Mathematics (Maths) Class 9.
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FAQs on NCERT Solutions for Class 9 Maths Chapter 1 - Number System (Exercise 1.2)

1. What are the different types of numbers in the number system?
Ans.The number system consists of various types of numbers, including natural numbers (1, 2, 3, ...), whole numbers (0, 1, 2, 3, ...), integers (..., -2, -1, 0, 1, 2, ...), rational numbers (numbers that can be expressed as a fraction, like 1/2, 3/4), and irrational numbers (numbers that cannot be expressed as a fraction, like √2, π).
2. How do you convert a decimal number to a fraction?
Ans.To convert a decimal number to a fraction, first count the number of decimal places. For example, for 0.75, there are two decimal places. This means you can write it as 75/100. Then simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD). In this case, 75/100 simplifies to 3/4.
3. What is the significance of the number line in the number system?
Ans.The number line is a visual representation of the number system that helps to illustrate the relationships between different types of numbers. It allows us to see where integers, fractions, and irrational numbers lie in relation to each other, making it easier to understand concepts like positive and negative values, as well as the density of rational and irrational numbers.
4. How to determine if a number is rational or irrational?
Ans.A number is considered rational if it can be expressed as the quotient of two integers, where the denominator is not zero. For example, 1/3 and -2 are rational numbers. On the other hand, a number is irrational if it cannot be expressed as a fraction, such as √3 or π. To determine if a number is rational, check if it can be written in the form a/b, where a and b are integers and b ≠ 0.
5. What are the properties of rational numbers?
Ans.Rational numbers possess several key properties: they can be added, subtracted, multiplied, and divided (except by zero); they can be expressed as fractions; they can be ordered on a number line; and they include both positive and negative values. Additionally, the sum or product of two rational numbers is also a rational number, demonstrating their closure property.
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