# Rational and Real Numbers

• The Rational Numbers are a field

• Rational Numbers are an integral domain,

since all fields are integral domains

• What other properties do the Rational

Numbers have that characterize them?

## Rational Order

Is (Q,+,•) an ordered integral domain?

Recall the definition of ordered.

Ordered Integral Domain: Contains a

subset D^{+ }with the following properties.

1. If a, b ∈ D^{+}, then a + b ∈ D^{+ }(closure)

2. If a , b ∈ D^{+}, then a • b ∈ D^{+ }(closure)

3. For each a ∈ Integral Domain D exactly

one of these holds

a = 0, a ∈ D^{+}, -a ∈ D^{+ }(Trichotomy)

• How can we define the positive set of

rational numbers?

• Verify closure of addition for the positive

set

• Suppose

• Verify closure of multiplication for the

positive set

• Suppose

• Verify the Trichotomy Law

• If a/b is a Rational Number then either a/b

is positive, zero , or negative .

## Dense Property

• Between any two rational numbers r and s

there is another rational number.

• Determine a rule for finding a rational

number between r and s. Verify it.

## Rational Holes

• Can any physical length be represented by

a rational number?

• Is the number line complete – does it still

have gaps?

## Pythagorean Society

• Believed all physical distances could be

represented as ratio of integers – our

rational numbers.

• 500 B.C discovered the following :

• h^{2} = 1^{2} + 1^{2}, h^{2} = 2, h = ? (not rational)

## Spiral Archimedes

## Rational Incompleteness

• Rational Numbers are sufficient for simple

applications to physical problems

• Theoretically the Rational Numbers are

inadequate

• Are these equations solvable over Q:

4x^2 = 25

x^2 = 13

Rational Incompleteness

• Where does reside on the

number line?

• Are the Rational Numbers sufficient to

complete the number line?

## Existence of Irrational Numbers

• Prove is an
irrational number.

Proof:

## Eudoxus of Cnidus

**Born: 408 BC in Cnidus (on
Resadiye peninsula), Asia Minor
(now Turkey)
Died: 355 BC in Cnidus, Asia
Minor (now Turkey)**

• Created the first known definition of the

real numbers .

• A number of authors have discussed

the ideas of real numbers in the work of

Eudoxus and compared his ideas with

those of Dedekind, in particular the

definition involving 'Dedekind cuts' given

in 1872.

## Julius Wihelm Richard

Dedekind

**Born: 6 Oct
1831 in
Braunschweig,
(now
Germany)
Died: 12 Feb
1916 in
Braunschweig**

• His idea was that every real number r

divides the rational numbers into two

subsets, namely those greater than r

and those less than r.

• Dedekind’s brilliant idea was to

represent the real numbers by such

divisions of the rationals.

• Among other things, he provides a

definition independent of the concept of

number for the infiniteness or finiteness

of a set by using the concept of

mapping.

• Presented a logical theory of number and

of complete induction, presented his

principal conception of the essence of

arithmetic , and dealt with the role of the

complete system of real numbers in

geometry in the problem of the continuity

of space.

## George Ferdinand Ludwig

Philipp Cantor

**Born: 3 March 1845 in
St Petersburg, Russia
Died: 6 Jan 1918 in Halle,
Germany**

• Dedekind published his definition of the

real numbers by "Dedekind cuts" also in

1872 and in this paper Dedekind refers

to Cantor's 1872 paper which Cantor

had sent him.

• However his attempts to decide whether

the real numbers were countable

proved harder.

• He had proved that the real numbers

were not countable by December 1873

and published this in a paper in 1874.

## What are the Real Numbers?

• Some common definitions

– Extension of the rational numbers to include the

irrational numbers

– Converging sequence of rational numbers, the limit of

which is a real number

– A point on the number line

• Microscope analogy: If you magnify the number

line at a very high power ,

– Would the Real Numbers look the same?

– Would the Rational Numbers look the same or be a

row of dots separated by spaces?

## Real Number Properties

• Real Numbers are an ordered field

• Theorem: Every ordered field contains, as

a subset, an isomorphic copy of the

rational numbers

– Thus the Rational Numbers are a subset of

every ordered field

– The Rational Numbers are subset of the Real

Numbers

## Upper Bound

• Upper Bound: Let S ⊆ ordered Field F. An

upper bound b ∈ F for S has the property

that x ≤ b for all x ∈ S.

• Least Upper Bound (l.u.b.) is the smallest

possible upper bound.

## Example

• Consider the following two sets.

• S = { x | x ∈Q, x < 9 / 2 }

• T = { x | x ∈Q, x^2 < 2 }

• Does an upper bound for S and T exist in

Q?

• Does a l.u.b. for S and T exist in Q?

## Dedekind Completeness

Property

• Let R be an ordered field. Any nonempty

set S ⊆ R which has an upper bound must

have a least upper bound.

• Are the Rational Number complete?

• Are the Real Numbers complete?

## Extension of Rational Numbers

into Real Numbers

• Theorem: There exists a Dedekind

complete ordered field.

• Verifying requires constructing it.

– Extension using decimal expansion

– Let R be the set of all infinite decimal

expansions and adopt the convention that

0.9999… = 1.0000…

– Can prove completeness holds, but very

difficult

• Theorem: There exists a Dedekind

complete ordered field.

– Extension using Dedekind cuts which are

pairs of nonempty subsets of Q such that for

any c ∈ Q: A = { r | r < c} and B = { r | r > c}

– Think of the cuts as representing the real

numbers

– The set of all cuts is a complete ordered field

## Characterization of the Reals

• Any other Dedekind complete ordered field

is an isomorphic copy of the Real

Numbers.

– R is an extension of Q

– R is an ordered field where Q^{+} ⊂ R^{+}

## Density of Real Numbers

• If a, b ∈ R with a < b, there exists a

rational number m/n such that

• If a, b ∈ R with a < b, there exists an

irrational number such that

Prev | Next |