Exercise 9.1
chapter 9.1.1
- As we have to find 6 rational numbers,We multiply the numbers by 7/7.
- For 3 = 3 x 7/7 =
. - For 4 = 4 x 7/7 =
. - 6 rational numbers between 3 and 4 are from 21/7 to 28/7.
- They are :
, , , , , .
3. Find five rational numbers between 3/5 and 4/5.
- As we have to find 5 rational numbers,We multiply the numbers by 6/6.
- For 3/5 = 3/5 x 6/6 =
. - For 4/5 = 4/5 x 6/6 =
. - 5 rational numbers between 3/5 and 4/5 are from 18/30 to 24/30.
- They are :
, , , , .
(i) Every natural number is a whole number :
Natural numbers are the set of positive integers starting from 1,... Whole numbers include all natural numbers and the number 0,....
(ii) Every integer is a whole number :
Integers include all positive and negative whole numbers, as well as zero.Whole numbers are only non-negative integers.
(iii) Every rational number is a whole number :
Rational numbers are numbers that can be expressed as a fraction p/q where p and q are integers and q ≠ 0.
Exercise 9.1.2
1. State whether the following statements are true or false. Justify your answers.(i) Every irrational number is a real number :
Since irrational numbers fall within the category of real numbers, every irrational number is indeed a real number.
(ii) Every point on the number line is of the form where m is a natural number :
The number line includes all real numbers, which means it contains not just natural numbers but also integers,rational numbers,and irrational numbers.
(iii) Every real number is an irrational number :
Since there are real numbers not every real number is irrational.
:
Example : Take positive integer 4 : √4 =
2 is a
Exercise 9.1.3
1. Write the following in decimal form and say what kind of decimal expansion each has.(i).36/100 :
(ii).1/11 :
(iii).
(iv). 3/13 :
(v).2/11 :
(vi).329/400 :
(i)0.6 :
(ii)0.47 :
(iii)0.001 :
= 0.9999999 =
Hence, we can say that 0.99999 = 1 which is algebraically proven.
= 1/17 =
The maximum number of digits in the quotient are
= 2/5 :
=27/16 :
We observe that the denominators of the above rational numbers are in the form of 2a × 5b, where a and b are whole numbers.
Hence if q is in the form 2a × 5b then p/q is a terminating decimal.
All irrational numbers are non-terminating and non-repeating.
Example :
(i) √23 :
(ii)√225 :
(iii) 0.379 :
(iv) 7.478478... :
(v) 1.101001000100001... :
Exercise 9.1.4
1. Classify the following numbers as rational or irrational.(i)2-√5 :
(ii)1/√2 :
(iii)(3+√23)-√23 :
(iv)2π :
(v)2√7/7√7 :
2.Simplify each of the following expressions:
(i) (3 + √3)(2 + √2)
- We use Distributive Propertry.
- Apply (a + b) (c + d) = ac + ad + bc + bd
- (3 + √3)(2 + √2) = 3 ×
+ 3√2 + √3 × + √3 × √2 -
+ √2 + √3 + √6
(ii) (3 + √3)(3 - √3)
- We use identity Propertry.
- Apply (a + b) (a - b) = a² - b²
- (3 + √3)(3 - √3) =
² - (√3)² - =
- - =
(iii) (√5 + √2)²
- We use identity Propertry.
- Apply (a + b) ² = a² + 2ab + b²
- (√5 + √2)² = (√5)² + (2×√5×√2) + (√2)²
- =
+ 2√10 + - =
+ 2√10
(iv) (√5 - √2)(√5 + √2)
- We use identity Propertry.
- Apply (a + b) ² = a² + 2ab + b²
- (√5 - √2)( √5 + √2) = (√
)² - (√2)² - =
- - =
.
π is defined as the ratio of the circumference of a circle to its diameter, that is, π = c/d. Hence, we see that π is a
But, we know that π is an irrational number.In fact, the value of π is calculated as the non-terminating, non-recurring decimal number as π = 3.14159...and hence π is not exactly equal to 22/7. In conclusion, π is an
4. Represent √9.3 on the number line.
Exercise 9.1.5
1. Find :(i)
(ii)
(iii)
2.Find :
(i)
(ii)
(iii)
(iv)
3.Simplify :
(i)
(iii)
(iv)
Exercise 9.3.1
1. How will you describe the position of a table lamp on your study table to another person?Table Size : length=80cm, width=40cm.
If the Lamp is at center of the table.
Position of lamp =
All the other streets of the city run parallel to these roads and are 200 m apart. There are 5 streets in each direction. Using 1cm = 200 m, draw a model of the city on your notebook. Represent the roads/streets by single lines.There are many cross- streets in your model. A particular cross-street is made by two streets, one running in the North - South direction and another in the East - West direction. Each cross street is referred to in the following manner : If the 2nd street running in the North - South direction and 5th in the East - West direction meet at some crossing, then we will call this cross-street (2, 5). Using this convention, find: (i)how many cross - streets can be referred to as (4, 3). (ii)how many cross - streets can be referred to as (3, 4).