Still it is unknown, if ( π + e) is rational or not? |

**Important Points about Rational and Irrational Numbers**

**1.**The sum or difference of a rational number and an irrational number is always an irrational number.

If a is a rational number and √b is an irrational number then,

(i) a + √b is irrational

(ii) a - √b is irrational

(iii) a√b is irrational

(iv) a

**÷**√b is irrational
(v) √b

**÷**a is irrational**2.**The product or quotient of non-zero rational number and an irrational number is also an irrational number.

**3.**Sum, difference, product or quotient of two irrational numbers need not be irrational. The result may be rational or irrational.

e.g. √m and √n are irrational. √m x √n = √(mxn) may be rational or irrational.

**Exercise 1.5**

**Q1**: Classify the following numbers as rational or irrational:

(i) 2 − √5

(ii) (3 + √(23)) − √(23)

(iii) 2√7/7√7

(iv) 1/√2

(v) 2π

Answer:

(V) 2π = 2 x 3.141528... = 6.283056...

Since it is non-terminating, non-recurring, it is an irrational number.

Since it is non-terminating, non-recurring, it is an irrational number.

**Q2**: Simplify each of the following expressions:

(i) (3 + √3) (2 + √2)

(ii) (3 + √3) (3 − √3)

(iii) (√5 + √2)

^{2}
(iv) (√5 - √2)(√5 + √2)

Answer:

(See the formula listed above in point 5)

(i) (3 + √3) (2 + √2) = 3x2 + 3√2 + 2√3 + √3x√2 = 6 + 3√2 + 2√3 + √6

(ii) (3 + √3) (3 − √3) = (3)

^{2}- (√3)^{2}= 9 - 3 = 6
(iii) (√5 + √2)

^{2}= (√5)^{2}+ (√2)^{2}+ 2(√5)(√2) = 5 + 2 + 2√10 = 7 + 2√10^{2}- (√2)

^{2}= 5 - 2 = 3

**Q3. Recall,**π

**is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is, p = c/d. This seems to contradict the fact that p is irrational. How will you resolve this contradiction?**

Answer: It does not contradict. When we measure some value with a scale or device, we often approximate the values. i.e. π is almost equal to 22/7 (i.e. π ≈ 22/7).

In fact π = 3.141528… while 22/7 = 3.142857…

**Q4. Represent √(9.3) on the number line.**

Answer:

- Draw a line AB = 9.3 units on number line.
- Extend B to C such that BC = 1 unit.
- Find the mid-point AC say point D and draw a circle on OC.
- Draw a perpendicular to line AC passing through point B. Let it intersect the circle at F.
- Taking B as centre and BF as radius, draw an arc intersecting number line at G. BF is √(9.3)

**Q5: Rationalise the denominators of the following**

**Q6: Solve √8 x √18**

Answer: √8 x √18 = √(8 x 18) = √(144) = √(12)

^{2}= 12.

interesting..will it be available for all chapters. will be of great use to me to teach my son who is in the 9th standard.

ReplyDeleteThanks for appreciation. We are a small team. We do our best to provide as many topics as possible.

Deletewe all are waiting for you to publih all the other chapters tooo.recurit some more members and u r truly awesome

Deletei am understanding every thing thank you for the presentation

ReplyDeletei can easily understand you presentation

ReplyDeletenice i like it cant we copy it

ReplyDeletenice i like it cant we copy it

ReplyDeletewe can just right click the image part and save it or copy

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