v
CHAPTER
1 Number Systems
Animation 1.1: Complex Plane
Source & Credit: elearn.punjab
,1. Quadratic Equations eLearn.Punjab 1. Quadratic E
1. Number Systems eLearn.Punjab 1. Number Syst
1.1 Introduction 1.2 Ratio
In the very beginning, human life was simple. An early ancient herdsman compared
sheep (or cattle) of his herd with a pile of stones when the herd left for grazing and again on We kno
its return for missing animals. In the earliest systems probably the vertical strokes or bars
such as I, II, III, llll etc.. were used for the numbers 1, 2, 3, 4 etc. The symbol “lllll” was used by qUZ / q ≠ 0.
many people including the ancient Egyptians for the number of ingers of one hand.
Around 5000 B.C, the Egyptians had a number system based on 10. The symbol p
form whe
q
for 10 and for 100 were used by them. A symbol was repeated as many times as it was
needed. For example, the numbers 13 and 324 were symbolized as and
respectively. The symbol was interpreted as 100 + 100 +100+10+10+1+1+1 Irratio
+1. Diferent people invented their own symbols for numbers. But these systems of notations
proved to be inadequate with advancement of societies and were discarded. Ultimately the p, qUZ and q
set {1, 2, 3, 4, ...} with base 10 was adopted as the counting set (also called the set of natural
numbers). The solution of the equation x + 2 = 2 was not possible in the set of natural
numbers, So the natural number system was extended to the set of whole numbers. No 1.2.1 Decim
number in the set of whole numbers W could satisfy the equation x + 4 = 2 or x + a = b , if
a > b, and a, b, UW. The negative integers -1, -2, -3, ... were introduced to form the set of 1) Terminat
integers Z = {0, ±1, ±2 ,...). part, is called
Again the equation of the type 2x = 3 or bx = a where a,b,UZ of terminati
and b ≠ 0 had no solution in the set Z, so the numbers of the form Sinc
terminating
a
where a,b,UZ and b ≠ 0, were invented to remove such diiculties. The set
b
2) Recurring
a
Q = { I a,b,UZ / b ≠ 0} was named as the set of rational numbers. Still the solution of equations periodic dec
,1. Quadratic Equations eLearn.Punjab 1. Quadratic E
1. Number Systems eLearn.Punjab 1. Number Syst
Example 1: Squari
.25 ( =
25
i) ) is a rational number. 2
100
.333...( = ) is a recurring decimal, it is a rational number.
1
ii) The R.H
3
Now a
iii) 2.3(= 2.333...) is a rational number. square. The
0.142857142857... ( =
so that equa
1
iv) ) is a rational number. 4
7
i.e., 2
v) 0.01001000100001 ... is a non-terminating, non-periodic decimal, so it is an In the las
irrational number. that equatio
vi) 214.121122111222 1111 2222 ... is also an irrational number. 2
vii) 1.4142135 ... is an irrational number.
viii) 7.3205080 ... is an irrational number. From equati
3.141592654... is an important irrational number called it p(Pi) which
ix) 1.709975947 ... is an irrational number. p
x) and from eq
denotes the constant ratio of the circumference of any circle to the length q
of its diameter i.e.,
∴
p
p=
circumference of any circle q
length of its diameter.
An approximate value of p is ,a better approximation is
22 355
and a still better This co
7 113
approximation is 3.14159. The value of p correct to 5 lac decimal places has been
Example 3:
determined with the help of computer.
, 1. Quadratic Equations eLearn.Punjab 1. Quadratic E
1. Number Systems eLearn.Punjab 1. Number Syst
The R.H.S. of this equation has a factor 3. Its L.H.S. must have the same factor. same Univer
Now a prime number can be a factor of a square only if it occurs at least twice in _ usu
the square. Therefore, p2 should be of the form 9p‘2 so that equation (1) takes the form: addition (+)
9p‘2 = 3q2 (2) for real num
’2 2
i.e., 3p = q (3)
1. Additi
In the last equation, 3 is a factor of the L.H.S. Therefore, q2 i) C
should be of the form 9q’2 so that equation (3) takes the form [
3p‘2 = 9q2 i.e., p’2 = 3q’2 (4) ii) A
From equations (1) and (2), [
P = 3P’ iii) A
and from equations (3) and (4) [
q = 3q’ (\
p 3 p′
0
∴ =
q 3q′
iv) A
[
p
This contradicts the hypothesis that is in its lowest form. a
q
Hence 3 is irrational. v) C
[
Note: Using the same method we can prove the irrationality of 2. Multip
5, 7,...., n where n is any prime number. vi) C
[
vii) A
1.3 Properties of Real Numbers
[
viii) M
We are already familiar with the set of real numbers and most of their properties. We
CHAPTER
1 Number Systems
Animation 1.1: Complex Plane
Source & Credit: elearn.punjab
,1. Quadratic Equations eLearn.Punjab 1. Quadratic E
1. Number Systems eLearn.Punjab 1. Number Syst
1.1 Introduction 1.2 Ratio
In the very beginning, human life was simple. An early ancient herdsman compared
sheep (or cattle) of his herd with a pile of stones when the herd left for grazing and again on We kno
its return for missing animals. In the earliest systems probably the vertical strokes or bars
such as I, II, III, llll etc.. were used for the numbers 1, 2, 3, 4 etc. The symbol “lllll” was used by qUZ / q ≠ 0.
many people including the ancient Egyptians for the number of ingers of one hand.
Around 5000 B.C, the Egyptians had a number system based on 10. The symbol p
form whe
q
for 10 and for 100 were used by them. A symbol was repeated as many times as it was
needed. For example, the numbers 13 and 324 were symbolized as and
respectively. The symbol was interpreted as 100 + 100 +100+10+10+1+1+1 Irratio
+1. Diferent people invented their own symbols for numbers. But these systems of notations
proved to be inadequate with advancement of societies and were discarded. Ultimately the p, qUZ and q
set {1, 2, 3, 4, ...} with base 10 was adopted as the counting set (also called the set of natural
numbers). The solution of the equation x + 2 = 2 was not possible in the set of natural
numbers, So the natural number system was extended to the set of whole numbers. No 1.2.1 Decim
number in the set of whole numbers W could satisfy the equation x + 4 = 2 or x + a = b , if
a > b, and a, b, UW. The negative integers -1, -2, -3, ... were introduced to form the set of 1) Terminat
integers Z = {0, ±1, ±2 ,...). part, is called
Again the equation of the type 2x = 3 or bx = a where a,b,UZ of terminati
and b ≠ 0 had no solution in the set Z, so the numbers of the form Sinc
terminating
a
where a,b,UZ and b ≠ 0, were invented to remove such diiculties. The set
b
2) Recurring
a
Q = { I a,b,UZ / b ≠ 0} was named as the set of rational numbers. Still the solution of equations periodic dec
,1. Quadratic Equations eLearn.Punjab 1. Quadratic E
1. Number Systems eLearn.Punjab 1. Number Syst
Example 1: Squari
.25 ( =
25
i) ) is a rational number. 2
100
.333...( = ) is a recurring decimal, it is a rational number.
1
ii) The R.H
3
Now a
iii) 2.3(= 2.333...) is a rational number. square. The
0.142857142857... ( =
so that equa
1
iv) ) is a rational number. 4
7
i.e., 2
v) 0.01001000100001 ... is a non-terminating, non-periodic decimal, so it is an In the las
irrational number. that equatio
vi) 214.121122111222 1111 2222 ... is also an irrational number. 2
vii) 1.4142135 ... is an irrational number.
viii) 7.3205080 ... is an irrational number. From equati
3.141592654... is an important irrational number called it p(Pi) which
ix) 1.709975947 ... is an irrational number. p
x) and from eq
denotes the constant ratio of the circumference of any circle to the length q
of its diameter i.e.,
∴
p
p=
circumference of any circle q
length of its diameter.
An approximate value of p is ,a better approximation is
22 355
and a still better This co
7 113
approximation is 3.14159. The value of p correct to 5 lac decimal places has been
Example 3:
determined with the help of computer.
, 1. Quadratic Equations eLearn.Punjab 1. Quadratic E
1. Number Systems eLearn.Punjab 1. Number Syst
The R.H.S. of this equation has a factor 3. Its L.H.S. must have the same factor. same Univer
Now a prime number can be a factor of a square only if it occurs at least twice in _ usu
the square. Therefore, p2 should be of the form 9p‘2 so that equation (1) takes the form: addition (+)
9p‘2 = 3q2 (2) for real num
’2 2
i.e., 3p = q (3)
1. Additi
In the last equation, 3 is a factor of the L.H.S. Therefore, q2 i) C
should be of the form 9q’2 so that equation (3) takes the form [
3p‘2 = 9q2 i.e., p’2 = 3q’2 (4) ii) A
From equations (1) and (2), [
P = 3P’ iii) A
and from equations (3) and (4) [
q = 3q’ (\
p 3 p′
0
∴ =
q 3q′
iv) A
[
p
This contradicts the hypothesis that is in its lowest form. a
q
Hence 3 is irrational. v) C
[
Note: Using the same method we can prove the irrationality of 2. Multip
5, 7,...., n where n is any prime number. vi) C
[
vii) A
1.3 Properties of Real Numbers
[
viii) M
We are already familiar with the set of real numbers and most of their properties. We
