REAL NUMBER & property of real numbers

Real Number : 

The set of rational and irrational numbers are known as Real number .

1. It is represented by R .

R = Q U Q^c 

where 

Q represent rational number & Q^c irrational number .

2. Every real  number can be represented at a point on  number line or real number line .

Property of real numbers : 

(a) . Rational number + Rational number = Rational number .

3/4 + 1/2 = 5/4 .

(b) . Rational - Rational = Rational number 

3/4 - 1/2 = 1/4 .

(C) . Rational × Rational = Rational number 

 3/4 × 1/2  = 3/8 . 

(d) . Rational / Rational = Rational 

3/4 ÷ 1/2 = 3/2 .

(e) . Irrational + Irrational = Irrational or rational number .

(2+√3) + (√3 - 2 ) = 2√3 (irrational number)

(4+√5) + (4 - √5) = 8(Rational number)

(f) . irrational - irrational = irrational or rational number .

(√7+2) - (√7-2) = 4 (Rational number)

(√7+2) - (2 - √7) = 2√7(irrational number)

(g) . Irrational × Irrational = Irrational or Rational number . 

(2+√3) × (2-√3) = 1 . (Rational number ) 

(2+√3) × (2+√3) = 4+ 4√3+3 = 7+ 4√3 (Irrational number) . 

(h) . Irrational ÷ Irrational = x÷y 

(Where y  not equal to zero ) = Rational or Irrational number .

(2+√3) ÷ (2+√3) = 1 ( Rational number)

(2+√3) ÷ (√3) = - (2√3 + 3)/3 = 1+( 2/√3) .

(I) . Rational + Irrational = Irrational 

3 + √7 .

(J) . Rational - Irrational = Irrational number .

3 - √7 .

(K) . Irrational - Rational = Irrational number .

√7 - 3 .

(l) . Irrational × Rational = Irrational number . 

7×√3 = 7√3 .

(m) . Rational ÷ Irrational  = Irrational .

4/√5 .

(n) . Irrational ÷ Rational = Irrational number . 

{√3+2} ÷ 2 = 1 + (√3/2) .