2008 W -C L Calculus(I)

Calculus (I)

W^EN-C^HING L^IEN

Department of Mathematics National Cheng Kung University

2008

HW 1:

1.5: 30,42,46,54 1.6: 84

1.7: 48, 53,54,55 S 1: Prove that √

x is not a rational function.

Ch1-1: Introduction

1. The real number system

(1) Natural numbers N= {1,2,3, . . .} (2) Integers

Z= {0,±1,±2, . . .} (3) Rational numbers

Q= {x :x = ^p_q, where p,q are integers and q 6=0} (4) Irrarional numbers

= {real numbers that are not rational numbers} ex:√

2, π, . . .

Ch1-1: Introduction

1. The real number system

(1) Natural numbers N= {1,2,3, . . .} (2) Integers

Z= {0,±1,±2, . . .} (3) Rational numbers

Q= {x :x = ^p_q, where p,q are integers and q 6=0} (4) Irrarional numbers

= {real numbers that are not rational numbers} ex:√

2, π, . . .

Ch1-1: Introduction

1. The real number system

(1) Natural numbers N= {1,2,3, . . .} (2) Integers

Z= {0,±1,±2, . . .} (3) Rational numbers

Q= {x :x = ^p_q, where p,q are integers and q 6=0} (4) Irrarional numbers

= {real numbers that are not rational numbers} ex:√

2, π, . . .

Ch1-1: Introduction

1. The real number system

(1) Natural numbers N= {1,2,3, . . .} (2) Integers

Z= {0,±1,±2, . . .} (3) Rational numbers

Q= {x :x = ^p_q, where p,q are integers and q 6=0} (4) Irrarional numbers

= {real numbers that are not rational numbers} ex:√

2, π, . . .

Ch1-1: Introduction

1. The real number system

(1) Natural numbers N= {1,2,3, . . .} (2) Integers

Z= {0,±1,±2, . . .} (3) Rational numbers

Q= {x :x = ^p_q, where p,q are integers and q 6=0} (4) Irrarional numbers

= {real numbers that are not rational numbers} ex:√

2, π, . . .

Ch1-1: Introduction

1. The real number system

(1) Natural numbers N= {1,2,3, . . .} (2) Integers

Z= {0,±1,±2, . . .} (3) Rational numbers

Q= {x :x = ^p_q, where p,q are integers and q 6=0} (4) Irrarional numbers

= {real numbers that are not rational numbers} ex:√

2, π, . . .

2. Intervals

(a,b], [a,b), (a,b), [a,b], (−∞,b], (a,∞) open , closed

R= (−∞,∞) =the set of all real numbers

3 Boundedness:

LetS be a set of real numbers.

(1)S is bounded above if∃M s.t. x ≤M ,∀x ∈ S. M is called an upper bound of S.

(2)S is bounded below if∃M s.t. x ≥M ,∀x ∈ S. M is called a lower bound ofS.

(3)S is bounded if bounded above and below.

2. Intervals

(a,b], [a,b), (a,b), [a,b], (−∞,b], (a,∞) open , closed

R= (−∞,∞) =the set of all real numbers

3 Boundedness:

LetS be a set of real numbers.

(1)S is bounded above if∃M s.t. x ≤M ,∀x ∈ S. M is called an upper bound of S.

(2)S is bounded below if∃M s.t. x ≥M ,∀x ∈ S. M is called a lower bound ofS.

(3)S is bounded if bounded above and below.

2. Intervals

(a,b], [a,b), (a,b), [a,b], (−∞,b], (a,∞) open , closed

R= (−∞,∞) =the set of all real numbers

3 Boundedness:

LetS be a set of real numbers.

(1)S is bounded above if∃M s.t. x ≤M ,∀x ∈ S. M is called an upper bound of S.

(2)S is bounded below if∃M s.t. x ≥M ,∀x ∈ S. M is called a lower bound ofS.

(3)S is bounded if bounded above and below.

2. Intervals

(a,b], [a,b), (a,b), [a,b], (−∞,b], (a,∞) open , closed

R= (−∞,∞) =the set of all real numbers

3 Boundedness:

LetS be a set of real numbers.

(1)S is bounded above if∃M s.t. x ≤M ,∀x ∈ S. M is called an upper bound of S.

(2)S is bounded below if∃M s.t. x ≥M ,∀x ∈ S. M is called a lower bound ofS.

(3)S is bounded if bounded above and below.

2. Intervals

(a,b], [a,b), (a,b), [a,b], (−∞,b], (a,∞) open , closed

R= (−∞,∞) =the set of all real numbers

3 Boundedness:

LetS be a set of real numbers.

(1)S is bounded above if∃M s.t. x ≤M ,∀x ∈ S. M is called an upper bound of S.

(2)S is bounded below if∃M s.t. x ≥M ,∀x ∈ S. M is called a lower bound ofS.

(3)S is bounded if bounded above and below.

2. Intervals

(a,b], [a,b), (a,b), [a,b], (−∞,b], (a,∞) open , closed

R= (−∞,∞) =the set of all real numbers

3 Boundedness:

LetS be a set of real numbers.

(1)S is bounded above if∃M s.t. x ≤M ,∀x ∈ S. M is called an upper bound of S.

(2)S is bounded below if∃M s.t. x ≥M ,∀x ∈ S. M is called a lower bound ofS.

(3)S is bounded if bounded above and below.

2. Intervals

(a,b], [a,b), (a,b), [a,b], (−∞,b], (a,∞) open , closed

R= (−∞,∞) =the set of all real numbers

3 Boundedness:

LetS be a set of real numbers.

(1)S is bounded above if∃M s.t. x ≤M ,∀x ∈ S. M is called an upper bound of S.

(2)S is bounded below if∃M s.t. x ≥M ,∀x ∈ S. M is called a lower bound ofS.

(3)S is bounded if bounded above and below.

2. Intervals

(a,b], [a,b), (a,b), [a,b], (−∞,b], (a,∞) open , closed

R= (−∞,∞) =the set of all real numbers

3 Boundedness:

LetS be a set of real numbers.

(1)S is bounded above if∃M s.t. x ≤M ,∀x ∈ S. M is called an upper bound of S.

(2)S is bounded below if∃M s.t. x ≥M ,∀x ∈ S. M is called a lower bound ofS.

(3)S is bounded if bounded above and below.

2. Intervals

(a,b], [a,b), (a,b), [a,b], (−∞,b], (a,∞) open , closed

R= (−∞,∞) =the set of all real numbers

3 Boundedness:

LetS be a set of real numbers.

(1)S is bounded above if∃M s.t. x ≤M ,∀x ∈ S. M is called an upper bound of S.

(2)S is bounded below if∃M s.t. x ≥M ,∀x ∈ S. M is called a lower bound ofS.

(3)S is bounded if bounded above and below.

The continuum of numbers:

Integers=⇒Rational Numbers

↑

(rational operations+, −, ×, ÷) Consider the real line L:

We have that every point Pof L is either a rational point of the form ^p_q on lies between two successive rational point ^p_q and ^p+_q¹

⇒ |P − ^p_q| ≤ ¹_q

⇒ We can choose ¹_q as small as we want.

⇒ For any pointPof L, we can find rational points while are arbitrarily close toP.

The continuum of numbers:

Integers=⇒Rational Numbers

↑

(rational operations+, −, ×, ÷) Consider the real line L:

We have that every point Pof L is either a rational point of the form ^p_q on lies between two successive rational point ^p_q and ^p+_q¹

⇒ |P − ^p_q| ≤ ¹_q

⇒ We can choose ¹_q as small as we want.

⇒ For any pointPof L, we can find rational points while are arbitrarily close toP.

The continuum of numbers:

Integers=⇒Rational Numbers

↑

(rational operations+, −, ×, ÷) Consider the real line L:

We have that every point Pof L is either a rational point of the form ^p_q on lies between two successive rational point ^p_q and ^p+_q¹

⇒ |P − ^p_q| ≤ ¹_q

⇒ We can choose ¹_q as small as we want.

⇒ For any pointPof L, we can find rational points while are arbitrarily close toP.

The continuum of numbers:

Integers=⇒Rational Numbers

↑

(rational operations+, −, ×, ÷) Consider the real line L:

We have that every point Pof L is either a rational point of the form ^p_q on lies between two successive rational point ^p_q and ^p+_q¹

⇒ |P − ^p_q| ≤ ¹_q

⇒ We can choose ¹_q as small as we want.

⇒ For any pointPof L, we can find rational points while are arbitrarily close toP.

The continuum of numbers:

Integers=⇒Rational Numbers

↑

(rational operations+, −, ×, ÷) Consider the real line L:

We have that every point Pof L is either a rational point of the form ^p_q on lies between two successive rational point ^p_q and ^p+_q¹

⇒ |P − ^p_q| ≤ ¹_q

⇒ We can choose ¹_q as small as we want.

⇒ For any pointPof L, we can find rational points while are arbitrarily close toP.

The continuum of numbers:

Integers=⇒Rational Numbers

↑

(rational operations+, −, ×, ÷) Consider the real line L:

We have that every point Pof L is either a rational point of the form ^p_q on lies between two successive rational point ^p_q and ^p+_q¹

⇒ |P − ^p_q| ≤ ¹_q

⇒ We can choose ¹_q as small as we want.

⇒ For any pointPof L, we can find rational points while are arbitrarily close toP.

The continuum of numbers:

Integers=⇒Rational Numbers

↑

(rational operations+, −, ×, ÷) Consider the real line L:

We have that every point Pof L is either a rational point of the form ^p_q on lies between two successive rational point ^p_q and ^p+_q¹

⇒ |P − ^p_q| ≤ ¹_q

⇒ We can choose ¹_q as small as we want.

⇒ For any pointPof L, we can find rational points while are arbitrarily close toP.

The continuum of numbers:

Integers=⇒Rational Numbers

↑

(rational operations+, −, ×, ÷) Consider the real line L:

We have that every point Pof L is either a rational point of the form ^p_q on lies between two successive rational point ^p_q and ^p+_q¹

⇒ |P − ^p_q| ≤ ¹_q

⇒ We can choose ¹_q as small as we want.

⇒ For any pointPof L, we can find rational points while are arbitrarily close toP.

The continuum of numbers:

Integers=⇒Rational Numbers

↑

(rational operations+, −, ×, ÷) Consider the real line L:

We have that every point Pof L is either a rational point of the form ^p_q on lies between two successive rational point ^p_q and ^p+_q¹

⇒ |P − ^p_q| ≤ ¹_q

⇒ We can choose ¹_q as small as we want.

⇒ For any pointPof L, we can find rational points while are arbitrarily close toP.

Theorem (Density)

The rational points are dense on the real line.

Corollary

Between any two distinct rational points, there are infinity many other rational points.

Theorem (Density)

The rational points are dense on the real line.

Corollary

Between any two distinct rational points, there are infinity many other rational points.

Thank you.