2. -
3
-
1
-
2
-1/2 0 1/2 1 2 3
A number line consist of all
types :
• Natural numbers
• Whole numbers
• Integers
• Rational numbers
• Irrational numbers
3.
4. Counting numbers are called Natural numbers. 1 is the first
natural number and there is no last natural number. The
collection of all natural numbers is denoted by N.
1 2 3 4 5 6 7 oo
5. 0 is the first whole number and there is no last whole number.
The collection of all whole numbers is denoted by W.
0 1 2 3 4 5 6 oo
6. When we include negative numbers like -1,-2,-3,-4 etc in the
collection of whole numbers then we will get the collection of
integers. It is denoted by Z.
-3 -2 -1 0 1 2 3 oo
-00
7. 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 is not equal to 0. The collection of rational numbers is
denoted by Q.
-6/2 -4/2 -2/2 0 2/2 4/2 6/2 00
-00 -1/2 1/2
8. A number r is called a irrational number. If it cannot be written in the
form p/q, where p and q are integers and q is not equal to 0.
√15
√13 √14 √3
√11
√5
√2
√12
√17
√19
√18
√20
√10 √6 √7
9.
10. A √2
P
O B 2 3 4 5 6
1. Firstly draw a number line.
2. Then take OB as the base.
3. Then make a line AB with measurement of 1 cm.
4. Then join A and O.
5. Finally use a compass with centre O and radius OB, draw an arc
intersecting the number line at the point P.
11.
12. Now we will look at the decimal expansion of rational and
irrational numbers to distinguish between them.
Try the decimal expansion of the following numbers
10/3
7/8
√5
10/3 = 3.3333…
7/8 = 0.875
√5 = 2.23606797749979
We have noticed that :-
1. The remainders either became 0, or start repeating themselves.
2. The number of entries in the repeating string of remainders is
less than the divisor.
3. If the remainders repeat, then we get a repeating block of
digits in the quotient.
13. NOTE :-
In case (i) the remainder never becomes zero and
repeats after a certain stage forcing the decimal
expansion to go for ever. These type of decimal
expansions are known as non-terminating repeating
decimal expansion.
In case (ii) the remainder becomes zero after a certain
stage. This type of decimal expansion is commonly
known as terminating decimal expansion.
In case (iii) the reminder never becomes zero and
never repeats. These type of decimal expansion are
called non-terminating non-repeating decimal
expansion.
14.
15. In order to convert a rational number having finite
number of digits after the decimal point, we follow the
following steps:-
STEP 1- Obtain the rational number.
STEP 2- Determine the number of digits in its decimal
part.
STEP 3- Remove decimal point from the numerator.
Write 1 in the denominator and put as many zeros on
the right side of 1 as the number of digits in the
decimal part of the given rational number.
STEP 4- Find a common divisor of the numerator and
denominator and express the rational number to
lowest terms by dividing its numerator and
denominator by the common divisor.
16. Example – Express the following decimal in the
form p/q.
a) 0.15 b) 0.675
Solution:-
a) Let x= 0.15 (i)
Multiplying (i) by 100
100x = 15(ii)
0.15 = 15/100 = 3/20
b) Let x=0.675 (i)
Multiplying (i) by 1000
1000x = 675 (ii)
0.675 = 675/1000 = 27/40
17. In order to convert a pure recurring decimal to the form
p/q, we follow the following steps:-
STEP 1- Obtain the repeating decimal and put it equal to x.
STEP 2- Write the number in decimal form by removing bar
from the top of repeating digit and listing repeating digit at
least twice.
STEP 3- Determine the number of digits having bar on their
heads.
STEP 4- If the repeating decimal has one place repetition,
multiply by ten; a two place repetition, multiply by hundred;
a three place repetition, multiply by thousand and so on.
STEP 5- Subtract the number in step two from the number
obtained in step four.
18. STEP6- Divide both sides of the equation by the coefficient of x.
STEP 7-Write the rational number in its simplest form.
Example – Express the following decimal in the form p/q.
a) 0.111… b) 0.66666…..
Solution:-
a) Let x= 0.111… (i)
Multiplying (i) by 10
10x = 1.111…(ii)
Subtracting (i) by (ii)
9x = 1
0.111… = 1/9
b) Let x=0.666…(i)
Multiplying (i) by 10
10x = 6.666… (ii)
Subtracting (i) by (ii)
9x = 6
0.666… = 2/3
19. In order to convert a mixed recurring decimal to the form
p/q, we follow the following steps:-
STEP 1- Obtain the mixed recurring decimal and put it
equal to x.
STEP 2- Determine the number of digits after the decimal
point which do not bar on them. Let there be n digits
without bar just after the decimal point.
STEP 3- Multiply both sides of x by 10n so that only
the repeating decimal is on the right side of the
decimal point.
STEP 4- Use the method of converting pure recurring
decimal to the form p/q and obtain the value of x.
20. Example – Express the following decimal in the form p/q.
a) 0.3222… b) 0.12333……
Solution:-
a) Let x= 0.3222… (i)
Multiplying (i) by 10
10x = 3.222…(ii)
Multiplying (ii) by 10
100x = 32.222…(iii)
Subtracting (ii) by (iii)
90x = 29
0.3222… = 29/90.
b) Let x=0.12333… (i)
Multiplying (i) by 100
100x =12.333… (ii)
Multiplying (ii) by 10
1000x = 123.333… (iii)
Subtracting (ii) by (iii)
900x = 111
0.12333… = 111/900
21.
22. In order to represent a irrational number geometrically, we
follow the following steps:-
STEP 1- Obtain the positive real number x.
STEP 2- Draw a line and mark a point A on it.
STEP 3- Mark a point B on the line such that AB=x units.
STEP 4- From point b mark a distance of 1 unit and mark a
new point as C.
STEP 5 – Find the mid-point of AC and mark the point as
O.
STEP 6 – Draw a circle with centre O and radius OC.
STEP 7 – Draw a line perpendicular to AC passing through
B and intersecting the semi-circle at D. Length BD is equal
to √x.
