1.6 Real Numbers:

 

We have studied properties of  Natural numbers, Whole Numbers, Integers, fractions, irrational number such as and . We have also studied that every non zero number has a negative number associated with it such that their sum is zero.

 

The combined set of rational numbers and irrational numbers is called ‘real number’ and is denoted  by R. Note that a number can be either a rational number or an irrational number and it can not be both.

Therefore

If Q is set of rational numbers and I_r is set of irrational numbers then we say Q I_r = (null set)

The relationship between various types of numbers can be represented in a tree structure as follows:

If

N is Set of Natural numbers, W is set of Whole numbers, Z is set of Integers ,Q is set of Rational numbers, R is set of Real numbers and I_r  is set of Irrational numbers

then

 

 

The above relationship can also be expressed using a Diagram (called Venn diagram) as follows:

 

Notice  that

NW  Z  Q R and I_rR and QI_r = R

 

If a, b, c  R where R is the set of real numbers then

 

No	Relationships	Name of  the property
1	a=a	Reflexive property
2	If a=b then b=a	Symmetric property
3	If a=b and b=c then a=c	Transitive property
4	If a=b then a+c =b+c, ac=bc
5	If ac=bc and c  0then a=b
6	a+b    R	Closure property of addition
7	a-b    R	Closure property of subtraction
8	a*b   R	Closure property of multiplication
9	a/b   R provided b0	Closure property of division
10	a+b = b+a	Commutative property of addition
11	a*b = b*a	Commutative property of multiplication
12	(a+b)+c = a+(b+c)	Associative property of addition
13	a*(b*c) = (a*b)*c	Associative property of  multiplication
14	a*(b+c) = a*b + a*c, (b+c)*a = b*a+c*a	Distributive law
15	a+0 =0+a =a	0 is additive  identity
16	a*1= 1*a=a	1 is multiplicative  identity
17	a+ (-a) = 0	-a, the additive inverse exists  for every a
18	a*1/a =1 provided a0	1/a,  the multiplicative inverse exists for every a

 

If a, b and c are real numbers then their order relations are:

1	Either a=b or a<b or a>b
2	If a <b	Then b>a
3	If a<b  and b <c	Then a<c
4	If a<b  and for any value of c	Then a+c < b+c
5	If a<b	Then ac< bc if c>0
		Then ac > bc  if c<0

 

1.6 Problem 1: Solve (x-3)/x²+4 >= 5/x²+4

 

Solution:

 

Solve means finding value of x

Multiplying both sides of the given statement by x²+4 we get

(x-3)>= 5( x²+4 >0)

x >=5+3 (Add 3 to both sides)

x >=8

Verification: By substituting value of x =8,9 notice that the the given statement is satisfied.

 

 

1.6 Summary of learning

 

No	Points studied
1 2	Real numbers and their properties The relationship between real numbers and other types of numbers.