NCERT Class 9 Mathematics Textbook for Blind Students made Screen Readable by Professor T K Bansal.

In this chapter, we have studied the following points:

1. A number r is called a rational number, if it can be written in the form p ÷ q , where p and q are integers and q ≠ 0.

2. A number s is called a irrational number, if it cannot be written in the form p ÷ q , where p and q are integers and q ≠ 0.

3. The decimal expansion of a rational number is either terminating or non-terminating recurring. Moreover, a number whose decimal expansion is terminating or non-terminating recurring is a rational number.

4. The decimal expansion of an irrational number is non-terminating non-recurring. Moreover, a number whose decimal expansion is non-terminating non-recurring is an irrational number.

5. All the rational and irrational numbers, put together, make up the collection of real numbers.

6. There is a unique real number corresponding to every point on the Real Number Line. Also, corresponding to each real number, there is a unique point on the Real Number Line.

7. If r is a rational number and s is an irrational number, then r + s and r − s are irrational numbers, and r × s and r ÷ s are also irrational numbers, provided r ≠ 0.

8. For positive real numbers a and b, the following identities hold:

\[(i)\ \sqrt{(a\ ×\ b)}\ =\ √a\ ×\ √b\]

\[(ii)\ \sqrt{(a\ ÷\ b)}\ =\ √a\ ÷\ √b\]

\[(iii)\ (√a\ +\ √b)\ ×\ (√a\ −\ √b)\ =\ a\ −\ b\]

\[(iv)\ (a\ +\ √b)\ ×\ (a\ −\ √b)\ =\ a^2\ −\ b\]

\[(v)\ (√a\ +\ √b)^2\ =\ a\ +\ b\ +\ 2\sqrt{(ab)}\]

9. To rationalise the denominator of

\[\frac{1}{(√a\ +\ b)}\]

we multiply this by,

\[\frac{(√a\ −\ b)}{(√a\ −\ b)}\]

where a and b are integers.

10. Let, a > 0, be a real number and p and q be rational numbers. Then

\[(i)\ a^p\ ×\ a^q\ =\ a^{(p+q)}\]

\[(ii)\ (a^p)^q\ =\ a^{(pq)}\]

\[(iii)\ a^p\ ÷\ a^q\ =\ a^{(p−q)}\]

\[(iv)\ a^p\ ×\ b^p\ =\ (a×b)^p\]

Acknowledgements
1. We at blind to visionaries Trust, are thankful to Mr. Arun kumar for typing and setting this chapter with full devotion and dedication.
2. In case you suggest us any improvement or can point us any mistakes in this lesson your name will also be added to this list of contributors.

End of Chapter 1.