1.1. REAL NUMBERS

A number which is either rational or irrational is called a real number. Thus the union of the sets of rationals and irrationals is the set of reals, denoted by R.

Since every point on the number line represents either a rational number or an irrational number, every point on the number line represents a real number. Thus there is a one-one correspondence between the real numbers and the points on the number line.

1.2. BASIC PROPERTIES OF REAL NUMBERS

(All these properties hold good for rational numbers also)

1. Closure Laws

   ? a, b ? R, a + b, a b, a. b, ( b ? 0) are real numbers. Thus R is closed under the four fundamental operations (excluding division by zero).

2. Commutative Laws

   ? a, b ? R, a + b = b + a, a . b = b . a.

3. Associative Laws

   ? a, b, c ? R, a + ( b + c) = ( a + b) + c, a.( b.c) = ( a.b). c.

4. Additive Identity

   ? a ? R, ? 0 ? R such that a + 0 = 0 + a = a 0 is called the additive identity in R.

5. Additive Inverse

   ? a ? R, ? b ?