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)

Rational Order How can we define the positive set of rational numbers?

Rational Order Verify closure of addition for the positive set Suppose show

Rational Order Verify closure of multiplication for the positive set Suppose show

Rational Order 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 = , h 2 = 2, h = ? (not rational) 1 1 h

Spiral Archimedes  2  3  4  5   7

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 … = … –Can prove completeness holds, but very difficult

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 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

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