Page 55 - DMTH401_REAL ANALYSIS

Unit 3: Matric Spaces

x – y > 0. You will recall from Section 2.2 that for the validity of the properties of order relations Notes
or the inequalities. Such as the one concerning the multiplication of inequalities, it is essential to
specify that some of the numbers involved should be positive. For example, it is necessary that
z > 0 so that x > y implies xz > yz. Again, the fractional power of a number will not be real if the
1/2
number is negative, for instance x when x = –4. Many of the fundamental inequalities, which
you may come across in higher Mathematics, will involve such fractional powers of numbers. In
this context, the concept of the absolute value or the modulus of a real member is important to
which you are already familiar. Nevertheless, in this section, we recall the notion of the modulus
of a real number and its related properties which we need for our subsequent discussion.
Defination: Modulus of Real Number

Let x be any real number. The absolute value or the modulus of x denoted by |x| is defined as
follows:
|x| = x if x > 0

= –x if x < 0
= 0 if x = 0.
You can easily see that


"
x = x , x R .
Not that |–x| is different from –|x|.
3.2.1 Properties of the Modulus of Real Number

Since the modulus of a real number is essentially a non-negative real number, therefore the
operations of usual addition, subtraction, multiplication and division can be performed on
these numbers. The properties of the modulus are mostly related to these operations.

Property 1: For any real number x, |x| = Maximum of (x, – x),
Proof: Since x is any real number, therefore either x  0 or x < 0. If x  0, then by definition,
we have

|x| = X.
Also, x > 0 implies that – x  0. Therefore, maximum of (x, – x} = x = |x|
Again x < 0, implies that –x > 0. Therefore again maximum of {x, –x} = –x = |x|.
Thus,

Maximum (x, – x} = |x|
4
Now consider the numbers |5| , |–4.5|, . It is easy to see that
2
5
2
2
|5| = |5| = 5.5 = 5 = |–5| 2
|–4.5| = |–20| = 20 Also |–4|. |5| = 4.5 = 20
i.e. |–4.5| = |–4|. |5|

and
4 4 4 4
= and = i.e.
5 5 5 5

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