The Real Number Field

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If $p \in Q,$ then $\hat{p} = {q \in Q | q < p} \subseteq Q$ is a Dedekind cut
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${p \in Q | p < 0 or p^{2} < 2} (= {p \in Q | p < \sqrt{2}}$ but we don't technically have square root language yet)

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Theorem 1.24. Existence of Real Numbers.

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There exists an ordered field

R

which has the least-upper-bound property.

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The elements of

R

are called real numbers.

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Convention 1.25.

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From now on

R

is “the” complete ordered field. We'll always view

Q \subseteq R .

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Subsection Properties of the Real Numbers

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“Never apologize for what you feel. It's like being sorry for being real.”
―Lil Wayne

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Theorem 1.26. Archimedian Property.

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For every real number

x,

there exists a natural number

n

such that

n > x .

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Corollary 1.27. Archimedian Corrollary.

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For every positive number

\vep,

there exists a natural number

n

such that

\frac{1}{n} < \vep .

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Theorem 1.28. Existence of Positive Square Roots.

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a

is a nonnegative real number and

n

is a positive integer, there exists

a

real number

b \geq 0

such that

b^{n} = a .

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Corollary 1.29. Existence of Odd Square Roots.

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a

is a real number and

n

is an odd positive integer, there exists a real number

b

such that

b^{n} = a .

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Definition 1.30. $n^{t h}$ Root.

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Let

x

be a nonnegative real number, and let

n

be a positive integer. We define

x^{\frac{1}{/} n}

to be the nonnegative real number

y

such that

y^{n} = x .

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Theorem 1.31. Density of the Rationals.

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a

and

b

are real numbers with

a < b,

there exists

a

rational number

r

such that

a < r < b .

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Subsection Absolute Values and $C$

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“True faith is belief in the reality of absolute values.”
―William Ralph Inge

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Definition 1.32.

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Consider complex numbers

C = {a + i b | a, b \in R}

with mult by

i^{2} = - 1 .

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Definition 1.33.

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Write $\overset{―}{a + i b} = a - i b$
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Re $(z) = \frac{z + \overset{―}{z}}{z},$ Im $(z) = \frac{z - \overset{―}{z}}{z i}$ for all $z \in C$
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Or equiv: Re $(a + i b) = a,$ Im $(a + i b) = b$ for all $a, b \in R .$

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Proposition 1.34.

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C

is not an ordered field with any order

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Proof.

In any ordered field

F,

know

- 1 < 0, x^{2} \geq 0

for all

x \in F

since

i^{2} = - 1

C, C

is not an ordered field.

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Definition 1.35. Absolute Value.

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For

z \in C,

| z | = \sqrt{Re (z)^{2} + Im (z)^{2}} = \sqrt{z \overset{―}{z}} =

“distance from

z

0

”.

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Theorem 1.36. Properties of Absolute Values.

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$| z | > 0$ iff $z \neq 0$
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$| z | = | \overset{―}{z} |$
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$| Re (z) |, | Im (z) | \leq | z |$
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$| z w | = | z | | w |, | \frac{z}{w} | = \frac{| z |}{| w |}$ if $w \neq 0$
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triangle inequality $| z + w | \leq | z | + | w |$

Mathematical Analysis

Search Results:

Section 1.3 The Real Number Field

Subsection Existence and Uniqueness

Definition 1.22.

Example 1.23.

Theorem 1.24. Existence of Real Numbers.

Convention 1.25.

Subsection Properties of the Real Numbers

Theorem 1.26. Archimedian Property.

Corollary 1.27. Archimedian Corrollary.

Theorem 1.28. Existence of Positive Square Roots.

Corollary 1.29. Existence of Odd Square Roots.

Definition 1.30. $n^{t h}$ Root.

Theorem 1.31. Density of the Rationals.

Subsection Absolute Values and $C$

Definition 1.32.

Definition 1.33.

Proposition 1.34.

Proof.

Definition 1.35. Absolute Value.

Theorem 1.36. Properties of Absolute Values.

Section 1.3 The Real Number Field

Subsection Existence and Uniqueness

Definition 1.22.

Example 1.23.

Theorem 1.24. Existence of Real Numbers.

Convention 1.25.

Subsection Properties of the Real Numbers

Theorem 1.26. Archimedian Property.

Corollary 1.27. Archimedian Corrollary.

Theorem 1.28. Existence of Positive Square Roots.

Corollary 1.29. Existence of Odd Square Roots.

Definition 1.30. nth Root.

Theorem 1.31. Density of the Rationals.

Subsection Absolute Values and C

Definition 1.32.

Definition 1.33.

Proposition 1.34.

Proof.

Definition 1.35. Absolute Value.

Theorem 1.36. Properties of Absolute Values.

Definition 1.30. $n^{t h}$ Root.

Subsection Absolute Values and $C$