NCERT Solutions for Class 9 Maths Chapter 1 – Number Systems Ex 1.2
NCERT Solutions for Class 9 Maths Chapter 1 – Number Systems Ex 1.2 are given here in simple, step-by-step form. NCERT stands for National Council of Educational Research and Training, the body that makes your textbook. Exercise 1.1 was about rational numbers, which can be written as a fraction. Exercise 1.2 introduces irrational numbers. An irrational number is a number that cannot be written as a simple fraction, and its decimal form goes on forever without repeating. Numbers like √2 and √3 are irrational.
This exercise has 4 questions. The first three are true-or-false and short reasoning questions about rational and irrational numbers. The last one asks you to show √5 on the number line using a geometric construction. Every question below is solved in short steps, with the final answer in bold. Students can use this page for homework and revision, parents can use it to check the work at home, and teachers can use it in class. A free printable PDF of these solutions is also available for offline study.
Exercise 1.2 – Questions and Answers
Q1. State whether the following statements are true or false. Justify your answers.
(i) Every irrational number is a real number.
(ii) Every point on the number line is of the form √m, where m is a natural number.
(iii) Every real number is an irrational number.
Answer:
(i) True. The real numbers are made up of all rational and all irrational numbers. So every irrational number is also a real number.
(ii) False. A point on the number line can be a negative number, but √m of a natural number is never negative. Also many points, like 1/2, are not the square root of a natural number. So not every point is of the form √m.
(iii) False. Real numbers include both rational and irrational numbers. A number like 2 or 1/3 is real but rational, not irrational. So 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.
Answer:
No, the square roots of all positive integers are not irrational.
If the integer is a perfect square, its square root is a whole number, which is rational.
For example, √4 = 2 and √9 = 3, and both 2 and 3 are rational numbers.
Final answer: No. For example, √4 = 2 is rational.
Q3. Show how √5 can be represented on the number line.
Answer: Use a right-angled triangle and the Pythagoras property. We look for two numbers whose squares add up to 5, and 22 + 12 = 4 + 1 = 5 works.
Step 1: Draw a number line and mark point O at 0 and point A at 2. So OA = 2 units.
Step 2: At A, draw a line AB perpendicular to the number line, with AB = 1 unit.
Step 3: Join OB. By the Pythagoras property, OB2 = OA2 + AB2 = 22 + 12 = 5, so OB = √5.
Step 4: With O as the centre and radius OB, draw an arc that cuts the number line at point P.
Then OP = OB = √5, so point P represents √5 on the number line.
Final answer: P is the point that represents √5.
Q4. Classroom activity (constructing the ‘square root spiral’). Construct a square root spiral to show √2, √3, √4, and so on.
Answer: This is a drawing activity. Follow these steps.
Step 1: Take a point O and draw a line segment OP1 of length 1 unit.
Step 2: At P1, draw a line P1P2 of length 1 unit, perpendicular to OP1. Join OP2. Then OP2 = √(12 + 12) = √2.
Step 3: At P2, draw P2P3 of length 1 unit, perpendicular to OP2. Join OP3. Then OP3 = √(√22 + 12) = √3.
Step 4: Keep repeating. Each new perpendicular of length 1 unit gives the next square root: OP4 = √4, OP5 = √5, and so on.
The line segments OP1, OP2, OP3, ... make a spiral shape, called the square root spiral.
This construction shows √2, √3, √4 and further roots one after another.