Representation of Real Numbers

Laws of Algebra

Laws of Order

Roots of Polynomials

Absolute Values

Sequences

Completeness

Bounded Sequences

Series

11 = 11.00000000...In practical calculations we use finite decimals as approximations to real numbers.

2/11 = 0.18181818...

sqrt 2 = 1.41421356...

pi= 3.14159265...

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

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

- 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.

x.(y + z) = x.y + x.z.The non-zero real numbers are a

- 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.

- For all x: x.0 = 0.
- If x.y = 0, then x = 0 or y = 0.
- If x.y is not 0, then x is not 0 and y is not 0.

- 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.

- If u < v and x < y, then u + x < v + y.
- If x > 0 and y > 0, then x.y > 0.
- If x < 0 and y > 0, then x.y < 0.
- If x < 0 and y < 0, then x.y > 0.

x^{n}+ a_{1}x^{n-1}+ ... + a_{n-1}x + a_{n}= 0.

If x > 0, then | x | = x,Important laws with absolute values are as follows:

If x = 0, then | x | = 0,

If x < 0, then | x | = -x.

- If | x | = 0, then x = 0.
- | x | = | -x |.
- | x | - | y |
__<__| x + y |__<__| x | + | y |. - | x | - | y |
__<__| | x | - | y | |__<__| x - y |__<__| x | + | y |. - | x.y | = | x |.| y |.
- | x + y |
^{2}__<__| x |^{2}+ 2.| x |.| y | + | y |^{2}. - | x + y |
^{2}+ | x - y |^{2}= 2.| x |^{2}+ 2.| y |^{2}.

- | x - y |
^{2}__<__| x |^{2}- 2.| x |.| y | + | y |^{2}. [*generally false*] - | x + y |
^{2}__<__| x |^{2}+ | y |^{2}. [*generally false*]

xThese numbers are called the_{1}, x_{2}, x_{3}, ...

xWhen the difference between the terms of a sequence and a number_{k}, where k = 1, 2, 3, ...

lim xThe exact definition of the limit is as follows:_{k}= u.

The number u is a limit of the sequence xWhen a sequence has a limit_{k}if and only if, for allepsilon> 0, there exists n such that k > n implies | x_{k}- u | <epsilon.

Important properties of limits are as follows:

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

Every Cauchy sequence has a limit.This property of the real number system is called

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.

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

- Every Cauchy sequence is bounded.
- The
**Bolzano-Weierstrass Theorem**: Every bounded sequence has a Cauchy subsequence.

- If a sequence has a limit, then it is bounded.
- Every bounded sequence has a subsequence with a limit.

xA series is_{1}+ x_{2}+ x_{3}+ ...

xhas a finite limit_{1}, x_{1}+ x_{2}, x_{1}+ x_{2}+ x_{3}, ...

xWe also say the series_{1}+ x_{2}+ x_{3}+ ... = s.

If the series x_{1} + x_{2} + x_{3} + ... is summable, then the sequence x_{1}, x_{2}, x_{3}, ... 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 x_{1} + x_{2} + x_{3} + ... is **absolutely summable**, or absolutely convergent, when the series

| xis summable._{1}| + | x_{2}| + | x_{3}| + ...

A series x_{1} + x_{2} + x_{3} + ... converges to a unique sum independent of the arrangement of the terms if and only if it is absolutely summable.

- 1 - 1/2 + 1/3 - 1/4 + 1/5 - 1/6 + 1/7 - 1/8 ... = 0.693147...
- 1 + 1/3 - 1/2 + 1/5 + 1/7 - 1/4 + ... = 1.03972...
- 1 - 1/2 + 1/3 + 1/5 - 1/4 + 1/7 + 1/9 + 1/11 - 1/6 + ... diverges.

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.
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