number system

Real Number:A real number may be either rational or irrational; either algebraic or transcendental; and either positive, negative, or zero

RATIONAL NUMBER

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

Rational Number on Number line

Rational Number by successive magnification(ex: 2.65)

REPRESENTATION OF 5.378

Irrational number on Number line

Geometrical representation of Real Number

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 writeSquare 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 side3 = a²/b² 3b² = a²If 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+1Putting the value of a and b in the above equation 3(4n² + 4n + 1) = 4m² + 4m + 16n² + 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.

Prove square root 2 is irrational