Fundamentals of Mathematics(MATH 1510)

Instructor: Lili ShenEmail: shenlili@yorku.ca

Department of Mathematics and StatisticsYork University

September 11, 2015

About the course

Name: Fundamentals of Mathematics, MATH 1510,Section A.Time and location: Monday 12:30-13:30 CB 121,Wednesday and Friday 12:30-13:30 LSB 106.Please check Moodle regularly for course information,announcements and lecture notes:http://moodle.yorku.ca/moodle/course/view.php?id=55333Be sure to read Outline of the Course carefully.

Textbook

James Stewart, Lothar Redlin, Saleem Watson. Precalculus:Mathematics for Calculus, 7th Edition. Cengage Learning,2015.

Towards real numbers

In mathematics, a set is a collection of distinct objects. Wewrite a ∈ A to denote that a is an element of a set A.

An empty set is a set with no objects, denoted by ∅.

Starting from the empty set ∅, we will constructthe set N of natural numbers: 0,1,2,3,4, . . . ,the set Z of integers: . . . ,−4,−3,−2,−1,0,1,2,3,4, . . . ,the set Q rational numbers: all numbers that can beexpressed as the quotient

mn

of integers m,n, where n 6= 0,

and finally, the set R of real numbers.

Natural numbers

Natural numbers are defined recursively as:

Definition (Natural numbers)0 = ∅,n + 1 = n ∪ {n}.

Natural numbers

In details, natural numbers are constructed as follows:0 = ∅,1 = 0 ∪ {0} = {0} = {∅},2 = 1 ∪ {1} = {0,1} = {∅, {∅}},3 = 2 ∪ {2} = {0,1,2} = {∅, {∅}, {∅, {∅}}},. . . . . .

n + 1 = n ∪ {n} = {0,1,2, . . . ,n},. . . . . .

Integers and rational numbers

Four basic operations in arithmetic: + − × ÷

Natural numbers

Integers

Rational numbers

−

÷

Subtraction

Division

Integers and rational numbers

TheoremIntegers are closed with respect to addition, subtractionand multiplication.Rational numbers are closed with respect to addition,subtraction, multiplication and division.

In the terminologies of modern algebra, the set Z of integers isa ring, while the set Q of rational numbers is a field.

Rational numbers as decimals

The decimal representation of a rational number eitherterminates or becomes periodic after a finite number of digits:

Terminating decimals:

58= 0.625.

Repeating decimals (or recurring decimals):

87= 1.142857142857 · · · = 1.142857,

1355

= 0.2363636 · · · = 0.236.

Rational numbers as decimals

Conversely, every repeating or terminating decimal representsa rational number.

Interested students may visit here for the methods of convertingrepeating decimals to fractions (see Section 5):http://en.wikipedia.org/wiki/Repeating_decimal

Number line

Draw a horizontal line continuing indefinitely in each direction,fix a point O as the origin and a distance to O as the unit length:

−3 −2 −1 O 1 2 3

This line is known as the number line.

Number line and rational numbers

QuestionIs there a one-to-one correspondence between rationalnumbers and points of the numbers line? In other words, canevery point in the number line be represented by a rationalnumber?

Number line and rational numbers

The answer is negative.

For example, the length of the diagonal of a unit square,√

2, isnot a rational number.

O 1√

2

√2 is not a rational number

Theorem√

2 is not a rational number.

Proof.

Suppose that√

2 =mn

is an irreducible fraction, i.e., m,n areintegers with no common divisors greater than 1.Then m2 = 2n2 and consequently m2 is an even number, thusm is even. Since m, n have no common divisors greater than 1,n must be an odd number.But m being even means that m = 2p for some integer p, whichleads to m2 = 4p2 = 2n2, and it follows that n2 = 2p2 is even,thus n is even, a contradiction.

From rational numbers to real numbers

That is to say, although rational numbers are dense in thenumber line, there are numerous gaps between rationalnumbers.

In order to fill the whole number line, we need to construct andunderstand real numbers.

Cuts of rational numbers

Suppose A and B are non-empty sets of rational numbers. Ifthey satisfy

a < b for all a ∈ A and b ∈ B,A ∪ B = Q,

We say that A and B form a cut of rational numbers.

Note that the above conditions guarantee that every rationalnumber is either in A or B, but cannot be in both A and B.

Cuts of rational numbers

Logically, there are four cases for cuts of rational numbers:(1) A has a greatest element, and B has no least element.(2) A has no greatest element, and B has a least element.(3) A has no greatest element, and B has no least element.(4) A has a greatest element, and B has a least element.

Cuts of rational numbers

In fact, the fourth case cannot happen. If A has a greatestelement a0 and B has a least element b0, the definition of cutsshows a0 < b0, therefore

a0 <a0 + b0

2< b0.

That is, the rational numbera0 + b0

2is neither in A nor in B,

contradicting to the hypothesis A ∪ B = Q.

Furthermore, we will combine the first two cases as they makeno difference in the construction of real numbers.

Dedekind cuts

DefinitionA Dedekind cut, written as A/B, is a partition of the rationalnumbers into two non-empty sets A and B, such that

a < b for all a ∈ A and b ∈ B,A ∪ B = Q,A contains no greatest element.

Dedekind cuts

Now we have two cases of Dedekind cuts:(1) A has no greatest element, and B has a least element b0,

e.g.

A = {a ∈ Q | a < 0}, B = {b ∈ Q | b ≥ 0}.

(2) A has no greatest element, and B has no least element,e.g.

A = {a ∈ Q | a2 < 2 or a < 0},B = {b ∈ Q | b2 > 2 and b > 0}.

Irrational numbers

For the first case, we say that the Dedekind cut A/B determinesa rational number b0.

For the second case, the Dedekind cut A/B does not determineany rational number; that is, there is a gap between A and B.Therefore, we must introduce a new number, i.e., an irrationalnumber, to fill this gap. In the above example, the irrationalnumber filling the gap is

√2.

Irrational numbers

DefinitionLet A/B be a Dedekind cut of rational numbers. If A has nogreatest element and B has no least element, we say that A/Bdetermines an irrational number c. c is greater than anyrational number in A and less than any rational number in B.

Irrational numbers

For example, the irrational number√

2 determined by theDedekind cut

A = {a ∈ Q | a2 < 2 or a < 0},B = {b ∈ Q | b2 > 2 and b > 0}

is greater than any rational number in A and less than anyrational number in B.

The decimal representation of an irrational number neitherterminates nor infinitely repeats but extends forever withoutregular repetition. For example,

√2 = 1.41421356 . . . ,π = 3.14159265 . . . .

Real numbers

DefinitionThe set R of real numbers consists of all the rational numbersand all the irrational numbers determined by Dedekind cuts ofrational numbers.

Cuts of real numbers

We may define cuts of real numbers in a similar way toDedekind cuts of rational numbers:

DefinitionA cut A/B of real numbers consists of two non-empty sets Aand B, such that

a < b for all a ∈ A and b ∈ B,A ∪ B = R,A contains no greatest element.

Cuts of real numbers

TheoremLet A/B be a cut of real numbers. Then B must contain a leastelement.

This theorem states that there is no gap in the set of realnumbers.Equivalently speaking, every point in the number line can berepresented by a real number. Therefore, the number line isusually called the real line.

0.9 = 1?

Question

Is the repeating decimal 0.9 = 0.999... equal to 1?

0.9 = 1?

An elementary solution.Let

a = 0.999...,

then10a = 9.999....

Thus9a = 10a− a = 9.999...− 0.999... = 9,

and thereforea = 1.

0.9 = 1?

A rigorous proof.

Let x = 0.9, form two Dedekind cuts of rational numbers:A1 = {a ∈ Q | a < x}, B1 = {b ∈ Q | b ≥ x};A2 = {a ∈ Q | a < 1}, B2 = {b ∈ Q | b ≥ 1}.

In order to prove x = 1, it suffices to show that A1/B1 andA2/B2 are the same Dedekind cuts; or equivalently, A1 = A2.

First, if a rational number a ∈ A1, then a < x , and it is clear thata < 1, thus a ∈ A2. This means A1 ⊆ A2.

0.9 = 1?

Second, if a rational number a ∈ A2, then a < 1. Supposea > 0 and a =

mn

, where m,n are positive integers. Thenm < n, and it follows that

1− mn≥ 1

n> 0

=⇒ There exists a positive integer k such that1n≥ 1

10k > 0

=⇒ a =mn≤ 1− 1

n≤ 1− 1

10k = 0.99 . . . 9︸ ︷︷ ︸k

< x .

That is, a < x , i.e., a ∈ A1, and consequently A2 ⊆ A1.

Therefore A1 = A2, completing the proof.