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Operation Of Real Numbers
Properties of addition on R :
Closure Property : The sum of two real numbers is always a real number. We say that R is closed for addition. ( i.e. a R, b
R
a + b for all a, b
R. )
Associative Property : (a + b) + c = a + (b + c) : a, b
R.
Commutative Law : a + b = b + a : a, b
R.
Additive Identity : Zero is a real number such that 0 + a = a + 0 = a : a
R.
0 is called the additive identity in R.
Additive Inverse : For each a R, there exists – a
R such that
a + (– a) = (– a) + a = 0.
The real number – a is called the additive inverse of a.
Properties of Multiplication on R :
R is closed for multiplication
i.e. a R, b
R
a
b for all a, b
R.
(ab) c = a (bc) for all a, b R.
ab = ba for all a, b R.
a (b + c) = ab + ac
1 R such that 1 · a = a · 1 = a for all a
R.
1 is called the multiplicative identity in R.
For each non-zero real number a R there exists real number
such that
a · =
· a = 1
The number is called the multiplicative inverse or reciprocal of ‘a’.
Note that 0 has no reciprocal.
Properties of Subtraction & Division on R :
▪ R is closed for subtraction.
▪ Subtraction on R does not satisfy the commutative and associative laws.
▪ R is not closed for division, since
2 R, 0
R but
R.
Some More Properties of Real Numbers :
Factors & Multiples : For real numbers ‘a’ and ‘b’, we say that ‘a’ is greater than ‘b’, if b = ac for some real number ‘c’ and we write, a | b.
If a | b then ‘b’ is called a multiple of ‘a’.
a | b, b | c a | c (Transitivity)
a | a for all a R (Reflexivity)
a | b and a | c a | (b + c) and a | (b – c).
Ordering : For real numbers ‘a’ and ‘b’, we say that ‘a’ is greater than ‘b’, if
a = b + c for some c R, and write, a > b.
Clearly, a > b b < a.
Trichotomy Law : For any two real numbers ‘a’ and ‘b’ one and only one of the following holds :
(i) a > b (ii) a < b (iii) a = b
If a and b are positive real numbers, then
(i) a > b
<
(ii) a > b
– a < – b
(iii) a > b a + c > b + c (iv) a > b
ac > bc, when c > 0.
ab = 0 a = 0 or b = 0 for all, a,b
R.
Absolute value of a Real Number : the numerical value of a real number ‘x’ is called its absolute value, to be denoted by | x |.
Thus, | 2 | = 2 and | – 2 | = 2.
If ‘x’ is any real number, then we define: x when x 0
e.g. Find the value of | 2x – 5 | + | 3 – 5x |, when x = 2.
Soln. | 2x – 5 | + | 3 – 5x | = | 2 2 – 5 | + | 3 – 5 2 |, when x = 2
= | – 1 | + | – 7 |
= (1 + 7)
= 8
e.g. Find all real values of x for which | x | < 2.
Soln. Note that | x | < 2 means x < 2 and – x < 2.
| x | < 2 x < 2 and – x < 2
x < 2 and x > – 2
– 2 < x < 2.
If | x | < a – a < x < a
