## 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}*and*

_{q}

^{p}^{+}

_{q}^{1}

⇒ |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}*and*

_{q}

^{p}^{+}

_{q}^{1}

⇒ |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}*and*

_{q}

^{p}^{+}

_{q}^{1}

⇒ |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:

^{p}* _{q}* on lies between two successive rational point

^{p}*and*

_{q}

^{p}^{+}

_{q}^{1}

⇒ |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:

^{p}* _{q}* on lies between two successive rational point

^{p}*and*

_{q}

^{p}^{+}

_{q}^{1}

⇒ |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:

^{p}* _{q}* on lies between two successive rational point

^{p}*and*

_{q}

^{p}^{+}

_{q}^{1}

⇒ |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:

^{p}* _{q}* on lies between two successive rational point

^{p}*and*

_{q}

^{p}^{+}

_{q}^{1}

⇒ |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:

^{p}* _{q}* on lies between two successive rational point

^{p}*and*

_{q}

^{p}^{+}

_{q}^{1}

⇒ |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:

^{p}* _{q}* on lies between two successive rational point

^{p}*and*

_{q}

^{p}^{+}

_{q}^{1}

⇒ |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.*