about all types of numbers such as rational numbers, real numbers,irrational numbers,integers,whole numbers, natural numbers etc

1. number system

2. Real
Number:A
real number
may be
either
rational or
irrational;
either
algebraic or
transcenden
tal; and
either
positive,
negative, or
zero

5. RATIONAL NUMBER
All terminating decimal are rational number
All non-terminating but repeating decimals are rational
number
All integer that can be expressed in the form of
p/q(where p and q are integer and q ≠ 0 is called
rational number.

6. Rational Number on Number line

7. Rational Number by successive
magnification(ex: 2.65)

8. REPRESENTATION OF 5.378

9. Irrational number on Number line

10. Geometrical representation of Real
Number

11. To proof square root3 is an irrational
number.
Solution:
To prove that this statement is true, let us assume that square root 3 is rational so
that we may write
Square root 3 = a/b
Here a and b = any two integers. We must then show that no two such integers can
be found.
Squaring both side
3 = a²/b²
3b² = a²
If b is odd then b² is odd. Similarly, if b is even, then b², a², and a are even. Since any
choice of even values of a and b leads to a ratio a/b that can be reduced by canceling
a common factor of 2.
Suppose a² is odd than then b is odd that is a=2m+1 and b=2n+1
Putting the value of a and b in the above equation
3(4n² + 4n + 1) = 4m² + 4m + 1
6n² + 6n + 1 = 2(m² + m)
The LHS of the above expression is odd and the RHS is even. That is a contradiction.
That is square root 3 is irrational.

12. Prove
square
root 2 is
irrational