NCERT Chapter Solutions
|Is Pi rational or irrational?|
(i) Every irrational number is a real number. ✓(TRUE)
Explanation: Real numbers are the collection of rational and irrational numbers.
(ii) Every point on the number line is of the form √m, where m is a natural number. ✗ (FALSE)
Explanation: Since -ve numbers cannot be expressed as square roots.
(iii) Every real number is an irrational number. ✗ (FALSE)
Explanation: Real numbers contains both rational and irrational numbers.
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 can be rational or irrational. e.g. √9 = 3 which is a rational number.
Q3: Show how √5 can be represented on the number line.
We can write √5 in Pythagoras form:
= √(4 + 1) = √(22 + 1)
- Take a line segment AB = 2 units (consider 1 unit = 2 cm) on x-axis.
- Draw a perpendicular on B and construct a line BC = 1 unit length.
- Join AC which will be √5 (Pythagoras theorem). Take A as center, and AC as radius draw an which cuts the x-axis at point E.
- The line segment AC represents √5 units.
Q4. Construct the ‘square root spiral’ (Class room activity.)
Following video shows the square root spiral.
Q5: Who discovered √2 or disclosed its secret?
Answer: Hippacus of Croton.
It is assumed Pythagoreans, followers of the famous Greek mathematician Pythagoras, were the first to discover the numbers which cannot be written in the form of a fraction. These numbers are called irrational numbers.
Q6: Who showed that showed that "Corresponding to every real number, there is a point on the real number line, and corresponding to every point on the number line, there exists a unique real number"?
Answer: Two German mathematicians, Deddkind and Cantor.
Q6: Is π (pi) rational or irrational?
Answer: π is an irrational number. 22/7 is just an approximation.
Q7: How many rational numbers exist between any two rational numbers?
Q8: Are irrational numbers finite?
Answer: No. There are infinite set of irrational numbers.
Q9: The product of any two irrational numbers is :
(A) always an irrational number.
(B) always a rational number.
(C) always an integer.
(D) sometimes rational, sometimes irrational number.
Answer: (D) sometimes rational, sometimes irrational number
e.g. √2 and √3 are two irrational numbers, √2×√3 = √6 an irrational number.
√2 and √8 are two irrational numbers, √2×√8 = √(16) = 4 a rational number.
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