NUMBER SYSTEMS AND ANALYSIS

Real Numbers

Contents

Representation of Real Numbers
Laws of Algebra
Laws of Order
Roots of Polynomials
Absolute Values
Sequences
Completeness
Bounded Sequences
Series


Representation of Real Numbers

The real number system consists of whole numbers, fractions, and irrational numbers. These numbers may be positive, zero, or negative. Real numbers may be represented by infinite decimals, for example:
11 = 11.00000000...
2/11 = 0.18181818...
sqrt 2 = 1.41421356...
pi = 3.14159265...
In practical calculations we use finite decimals as approximations to real numbers.

A real number may be represented by a point on the real line (see Fig. 1).

Fig. 1.

Fig. 1. The real number x on the real line.

Laws of Algebra

The real numbers are a commutative group under addition:
Associative law of addition:
x + (y + z) = (x + y) + z.
Commutative law of addition:
x + y = y + x.
Identity law of addition:
There is a unique zero 0 such that, for all x:
x + 0 = x.
Inverse law of addition:
Every real number x has a unique negative -x such that:
x + (-x) = 0.
In the real number system multiplication is distributive over addition:
x.(y + z) = x.y + x.z.
The non-zero real numbers are a commutative group under multiplication:
Associative law of multiplication:
x.(y.z) = (x.y).z.
Commutative law of multiplication:
x.y = y.x.
Identity law of multiplication:
There is a unique unit 1 such that, for all x:
x.1 = x.
Inverse law of multiplication:
Every non-zero real number x has a unique reciprocal 1/x such that:
x.(1/x) = 1.
These laws of algebra show that the real number system is a field. Important laws derived from the laws of a field are as follows:

Laws of Order

The real numbers are simply ordered. They obey the following laws of order:
Trichotomy law:
For two given real numbers, x and y, exactly one of the following relations is true:
x < y, x = y, x > y.
Transitive law:
If x < y and y < z, then x < z.
Order and addition:
If x < y, then x + z < y + z.
Order and multiplication:
If x < y and z > 0, then x.z < y.z.
Important laws derived from the laws of order are as follows:

Roots of Polynomials

It follows from the laws of order that there is no real number x such that x2 < 0. For example, there is no real number x such that x2 = -1. More generally, there is not always a real number x which satisfies a given polynomial equation:
xn + a1xn-1 + ... + an-1x + an = 0.

Absolute Values

The absolute value | x | of a real number x is defined as follows:
If x > 0, then | x | = x,
If x = 0, then | x | = 0,
If x < 0, then | x | = -x.
Important laws with absolute values are as follows: Note: The following relations are not always true:

Sequences

A sequence is an infinitely long list of numbers:
x1, x2, x3, ...
These numbers are called the terms of the sequence. The sequence may also be written:
xk, where k = 1, 2, 3, ...
When the difference between the terms of a sequence and a number u approaches zero as k increases we call u a limit of the sequence, and we write:
lim xk = u.
The exact definition of the limit is as follows:
The number u is a limit of the sequence xk if and only if, for all epsilon > 0, there exists n such that k > n implies | xk - u | < epsilon.
When a sequence has a limit u we also say the sequence converges to u.

Important properties of limits are as follows:

A limit (when it exists) is unique:
If lim xk = u, and lim xk = v, then u = v.
Limits and sums:
If lim xk = u and lim yk = v, then lim (xk + yk) = u + v.
Limits and products.
If lim xk = u and lim yk = v, then lim (xk.yk) = u.v.
A sequence xk which has a limit satisfies the Cauchy condition:
For all epsilon > 0, there exists n such that j > n and k > n imply | xk - xj | < epsilon. (In other words: if the terms of a sequence approach a limit, then the terms of the sequence approach each other.)

Completeness

A sequence which satisfies the Cauchy condition is called a Cauchy sequence. Therefore, every sequence which has a limit is a Cauchy sequence. In the real number system the converse is also true:
Every Cauchy sequence has a limit.
This property of the real number system is called completeness.

We may regard the infinite decimals which represent real numbers as special Cauchy sequences. For example, the infinite decimal

sqrt 2 = 1.41421356...
may be regarded as the Cauchy sequence
1, 1.4, 1.41, 1.414, 1.4142, 1.41421, 1.424213, 1.4142135, 1.41421356, ...
This represents the real number sqrt 2, because sqrt 2 is the limit of the sequence.

Bounded Sequences

A sequence xk is called bounded when there exist real numbers a and b such that, for all k, a < xk < b.

The following theorems show the relationship between bounded sequences and Cauchy sequences:

  1. Every Cauchy sequence is bounded.
  2. The Bolzano-Weierstrass Theorem: Every bounded sequence has a Cauchy subsequence.
Since a sequence of real numbers is a Cauchy sequence if and only if it has a limit, it follows from the above theorems that:
  1. If a sequence has a limit, then it is bounded.
  2. Every bounded sequence has a subsequence with a limit.

Series

A series is an infinite list of numbers added together:
x1 + x2 + x3 + ...
A series is summable when the sequence of partial sums
x1,   x1 + x2,   x1 + x2 + x3, ...
has a finite limit s. This limit is the sum of the series, and we write:
x1 + x2 + x3 + ... = s.
We also say the series converges to s. A series which is not summable is said to diverge.

If the series x1 + x2 + x3 + ... is summable, then the sequence x1, x2, x3, ... converges to zero. But the converse is false. For example, lim(1, 1/2, 1/3, 1/4, ...) = 0, but the series 1 + 1/2 + 1/3 + 1/4 + ... diverges.

A series x1 + x2 + x3 + ... is absolutely summable, or absolutely convergent, when the series

| x1 | + | x2 | + | x3 | + ...
is summable.

A series x1 + x2 + x3 + ... converges to a unique sum independent of the arrangement of the terms if and only if it is absolutely summable.

EXAMPLE

The following series are all rearrangements of the same series:

These series are not absolutely summable because 1 + 1/2 + 1/3 + 1/4 + ... diverges.


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By R. H. B. Exell, 1998. King Mongkut's University of Technology Thonburi.