Download Unit 1 Chapter 2 (Number systems)

Unit-1
Chapter 2
Number Systems
2.1 Sets of number
Important sets which we meet in elementary mathematics are sets of numbers. Of
particular importance, especially in analysis, is the set of real number which is denoted by
R. The set of real numbers and its properties is called the real number system.
2.1.1 List of numbers:
Take a straight line (which is extended to infinites on both sides) and fix a point 0 on
it. Choose a suitable segment as unit length. To the right side of 0 we mark off points
corresponding to positive integral multiple of unit length, to the left side, those
corresponding to negative integral multiple. This line is known as the line of numbers.
-∞
+∞
-2 -1
0
1
2
2.1.2 Real number R:
Every point on this line corresponds to a real number and every real number can be
represented by a point on this line.
2.1.3 Integer Z:
We denote the integers by Z and,
Z = {……-3, -2, -1, 0, 1, 2, 3……}
The integers can be also referred to as the whole numbers, & Z  R
2.1.4 Rational Number Q:
The rational numbers are those real numbers which can be expressed as the ratio of
two integers. We denote the set of rational numbers by Q. Accordingly
a
Q = {x│x = where a, b  Z, b ≠ 0} and Z  Q  R
b
The sum, product, difference and quotient of two rational numbers is again a rational
number.
2.1.5 Natural Number N:
The natural numbers are the nonnegative integers. So,
N = {0,1, 2, 3…….}
The natural numbers are closed only under the operations of additions and multiplication.
N  Z  Q  R.
2.1.6 Prime Numbers P:
The prime numbers are those natural numbers P, excluding 1, which are divisible by 1
and P itself.
e.g.: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31….
2.1.7 Irrational numbers:
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A real number which is not rational is called irrational. e.g. 2 , Π etc are irrational
numbers.
2.2 Properties of Real numbers:
If a, b, c are real numbers then,
1.
2.
3.
4.
5.
Only one of the relations a<b, a>b or a =b is true for a and b
If a<b then a  c  b  c
If a<b then a  c  b  c
If a  c  b  c and c  0 then ac  bc
If a<b and c<0 then bc  ac
1
6. If a>0 then  0
a
1 1
7. If a and b are both positive or both negative and a<b then 
b a
8. If a<b and b<c then a<c.
2.2.1 Absolute Value:
The absolute value of a real number x is denoted │x│ = {x if x  0 }
= {- x if x < 0}
i.e. if x is positive or zero then │x│equals to x and if x is negative then │x│equals – x.
Consequently the absolute value of any number is always non-negative. i.e. |x|  0 for
every x  R.
Example: │- 5│= 5; │5│= 5;
│3- 6│=│- 3│= 3; │6 - 3│=│3│= 3
The statement │x│< 5 means the distance of the real number x from 0 is 5 unit. And x
must lie between -5 and 5 on the real line.
i.e. │x│< 5; -5 < x < 5
-5
5
│x│≤ 5; -5 ≤ x ≤ 5
-5
2.2.2 Intervals
Consider the following sets of numbers:
A1 = {x│2 <x < 5} ------------- open interval
A2 = {x│2 ≤ x ≤ 5} ------------ closed interval
5
-2
0
2
5
-2
0
2
5
0
2
5
A3 = {x│2 < x ≤ 5} ------------ open-closed interval
2
A4 = {x│2 ≤ x < 5} ------------ closed-open interval
0
A1  A2 =
-2
A1  A2 =
0
2
2
5
5
-2
0
2
5
The four sets contain only the points which lie between 2 and 5 with possible
expectations of 2 and/or 5. We call those sets intervals, the numbers 2 and 5 being the
end points of each interval.
A parenthesis is used to designate an open end point i.e. an end point that is not in
the interval, and a bracket is used to designate a closed end point.
A1 = (2, 5)
A2 = [2, 5]
A3 = (2, 5]
A4 = [2, 5)
A1  A2 = [2, 5]
A1  A2 = (2, 5)
2.2.3 Properties of intervals:
a) Null set  and single points a = [a, a] are also considered to be intervals.
b) Intersection of two intervals is an interval.
c) The union of two non-disjoint intervals is an interval.
d) The difference of two non-comparable intervals is an interval.
A=
-∞
0
4
8
2 3 4
8
2
+∞
B=
0
Let A = [2, 4), B = (3, 8)
Then,
A  B = (3, 4)
A  B=
A  B = [2, 8)
A – B = [2, 3]
B – A = [4, 8)
2 3 4
0
A  B=
8
0
2
4
8
0
2 3 4
8
A-B=
B-A=
0
2
4
8
2.2.4 Infinite Intervals:
Set of the form
A = {x│x > 1}
3
B = {x│x ≥ 5}
C = {x│x < 2}
D = {x│x ≤ 4}
E = {x│x  R}
are called infinite intervals and are also denoted by
A = (1, ∞)
B = [5, ∞)
-∞
-∞
+∞
5
1
+∞
5
1
C = (-∞, 2)
D = (-∞, 4]
E = (-∞,∞)
+∞
2
-∞
+∞
4
2
1
2.3 Supplementary Problems
-∞
4
5
+∞
2.3.1. Find the distance d between each pair of real numbers ,
(a) 2 and -5 , (b)-6 and 3, (c) 2 and 8, (d) 3 and -3, (f) -7 and -9,
2.3.2Find all integers n such that : (a) 3  2n  4  10 (b) 1  6  3n  13
2..3.3Rewrite each interval in set –builder form:
(a) A=  1,6 (b) (2,5)
(c)  3,0
(d) 1,4
2.3.4Find the interval satisfying each inequality ,i.e rewrite the inequality in terms of x
alone;
(a) 1  x  2  4, (b)  3  x  4  7, (c)  6  3x  12, (d)  4  2x  6,
2.3.5. Find the interval satisfying each inequality :
(a) 3  2 x  5  7, (b)  8  4  3x  7,
2.3.6. Rewrite without the absolute value sign:
(a)
x  2 (b) x  3  5 (c) 2x  5  5
2.3.7. write each open interval in the form | x - a | r ,
(a) 3  x  9, (b)  5  x  1,
2.3.8.Rewrite each set using an infinite interval notation.:
(a) x  4, (b) x  5, (c) x  2 (d) x  3
2.3.9. Let A=  4,2 , B= (1,6) , C= (,1] , Find and write in interval notation:
(a) A  B, (b) A  B, (c) A  B, (c) A \ B, (d) B \ A, (e) A  C, (f) A  C, (g) A \ C, (h)
C \ A, (i) B  C, (j) B  C, (k) B \ C, (l) C \ B .
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