Rational and Real Numbers

1
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?

2
Rational Order

• Is (Q,,) an ordered integral domain? Recall the
  definition of ordered.
• Ordered Integral Domain Contains a subset D
  with the following properties.
• If a, b ? D ,then a b ? D (closure)
• If a , b ? D , then a b ? D (closure)
• For each a ? Integral Domain D exactly one of
  these holds
• a 0, a ? D, -a ? D (Trichotomy)

3
Rational Order

• How can we define the positive set of rational
  numbers?

4
Rational Order

• Verify closure of addition for the positive set
• Suppose show

5
Rational Order

• Verify closure of multiplication for the positive
  set
• Suppose show

6
Rational Order

• Verify the Trichotomy Law
• If a/b is a Rational Number then either a/b is
  positive, zero, or negative.

7
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.

8
Rational Holes

• Can any physical length be represented by a
  rational number?
• Is the number line complete does it still have
  gaps?

9
Pythagorean Society

• Believed all physical distances could be
  represented as ratio of integers our rational
  numbers.
• 500 B.C discovered the following
• h2 12 12, h2 2, h ? (not rational)

1
h
1
10
Spiral Archimedes
1
1
1

• ?3

• ?4

1

• ?2

• ?5

• ?6

1
1
1

• ?7

11
Rational Incompleteness

• Rational Numbers are sufficient for simple
  applications to physical problems
• Theoretically the Rational Numbers are inadequate
• Are these equations solvable over Q
• 4x2 25 x2 13

12
Rational Incompleteness

• Where does reside on the number
  line?
• Are the Rational Numbers sufficient to complete
  the number line?

3
4
3.5
13
Existence of Irrational Numbers

• Prove is an irrational number.
• Proof

14
Eudoxus of Cnidus

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

15

• 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.

16
Julius Wihelm Richard Dedekind

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

17

• 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.
• Dedekinds brilliant idea was to represent the
  real numbers by such divisions of the rationals.

18

• 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.

19

• 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.

20
George Ferdinand Ludwig Philipp Cantor

• Born 3 March 1845 in
• St Petersburg, RussiaDied 6 Jan 1918 in Halle,
  Germany

21

• 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.

22
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?

23
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

24
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.

25
Example

• Consider the following two sets.
• S x x ?Q, x lt 9 / 2
• T x x ?Q, x2 lt 2
• Does an upper bound for S and T exist in Q?
• Does a l.u.b. for S and T exist in Q?

26
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?

27
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

28
Extension of Rational Numbers into Real Numbers

• 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 lt c and B r r gt c
• Think of the cuts as representing the real
  numbers
• The set of all cuts is a complete ordered field

29
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

30
Density of Real Numbers

• If a, b ? R with a lt b, there exists a rational
  number m/n such that
• If a, b ? R with a lt b, there exists an
  irrational number such that

31

• Thank You !!