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**NCERT Chapter Solutions**

Is Pi rational or irrational? (src:openclipart.org) |

**Q1: State whether the following statements are true or false. Justify your answers.**

Answer:

(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)

Construction:

- 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?**

Answer: Infinite

**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.

⟵Go to CH1-Number Systems - Ex 1.1 Goto CH1 - Number Systems Ex 1.3→

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ReplyDeletesq root of (3+2*sq root of2)