#### Topics

1) Explain counting numbers?
The numbers 1, 2, 3, 4, 5, 6,.... are called counting numbers. These are also known as natural numbers or positive numbers.

2) Explain rational number?
A rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero.

3) Explain irrational Number?
An irrational Number is any real number that cannot be expressed as a ratio p/q, where p and q are integers, with q non-zero, and is therefore not a rational number.

4) Explain real number?
A real number is any positive or negative number. This includes all integers and all rational and irrational numbers.

5) How can we mark the position of π on a number line?
We know that the perimeter of a circle with diameter 1 units. Now make the perimeter of a  circle having diameter 1 unit is π units. Now make a circle having diameter 1 using a thin wire or draw a circle of diameter. 1 on a sheet of paper and take its perimeter using a thread. Now place one end of the wire or the thread on 0 and spread it on the line and the point where its other end touches the line is the position of π.

6) What is the length of AB in the following figure?

In this right angled triangle, AB2 = BC2 - AC2
= 52 - 32
= 25 - 9 = 16
AB = √16 cm = 4 cm

7) How many numbers are there between 0 and 1?
There is no counting number between 0 and 1. But there are fractions such as 0.1, 0.2, 0.3---- and 0.9 between 0 and 1.  Again between 0 and 0.1 there are 0.01, 0.02, 0.03,.... and 0.09. Also we can find out 0.001, 0.002, 0.003..... and 0.009 between 0 and 0.01. If we proceed like this we get an innumerable (infinite) number of both rational and irrational numbers between 0 and 1. As there are innumerable numbers between 0 and 1 there are infinite number of points on the line between 0 and 1.

8) Explain absolute value of a number?
Distance between zero and a number on the number line is the value of the number disregarding its sign. This value is called the absolute value of the number.

9) Find two numbers x and y for which |x + y| = |x| + |y|.
Let x = 3 and y = 7
|3 + 7| = |10| = 10
|3| + |7| = 3 + 7 = 10

10) Verify whether the following are true:
a) |7 + 5| = |7| + |5|
b) |22 + 8| = |22| + |8|

a) L.H.S  = |7 + 5| = |12| = 12
R.H.S  = |7| + |5| = 7 + 5 = 12
ie, |7 + 5| = |7| + |5| is true.
b) L.H.S  =  |22 + 8| = |30| = 30
R.H.S  =  |22| + |8| = 22 + 8 = 30
ie, |22 + 8| = |22| + |8| is true.

11) Check whether the following are true:
a) |-12 - (-4)| = |-12| - |-4|
b) |-6 + (+4)| ≠ |-6| + |4|

a) L.H.S = |-12 - (-4)| = |-12 + 4| = |-8| = 8
R.H.S = |-12| - |-4| = = 12 - 4 = 8
ie, |-12 - (-4)| = |-12| - |-4| is true.
b) L.H.S = |-6 + (+4)| = |-6 + 4| = |-2|  = 2
R.H.S  = |-6| + |4| = 6 + 4 = 10
ie, |-6 (+4)| ≠ |-6| + |4| is true.

12. Find the value of  |38 - (8)| ?
|38 - (8)| = |38 - 8| = |30| = 30

13) Write the absolute value for the following.
a) |-327|                b) |486|

a) |-327| = 327
b) |486| = 486

14) Find the distance between 4 and 9 .
Distance between 4 and 9 = |4 - 9| = |-5| = 5.

15) Find the value of x for the following: |x + 2| = 3.
We can write |x + 2| as |x - (-2)|
|x - (-2)| = 3
That is the distance between x and -2 is 3. x can be 3 unit to the right or left of -2 , 3 unit to the right of -2 is +1 and 3 unit left of -2 is -5
∴ x = 1 and -5

16) Find the value of x for the following : |x - 3| = |x -5|
Distance from 3 to x and distance from 5 to x are equal. That is the position of x is in the middle of 3 and 5 .

17) Mark given numbers on the number line and check whether the distance between them is equal to the number got by subtracting the smaller from the larger. 2, -6.

Larger = 2, Smaller = -6
Distance = 2 - -6 = 8
lsmaller - larger l
=| -6 - 2 | =|-8| = 8

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