Skip to main content ☰ Contents You! < Prev ^ Up Next > \(\def\ann{\operatorname{ann}}
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Section 1.3 The Real Number Field
Subsection Existence and Uniqueness
“It's always better to be real.”
―Selena Gomez
Definition 1.22 .
A Dedekind cut in \(\mathbb{Q}\) is a set \(x\subseteq \mathbb{Q}\) (we're thinking of \(x\) as a real number) such that
\(\displaystyle x\neq\emptyset, x\neq \mathbb{Q}\)
If \(p\in\mathbb{Q},q\in x\) with \(p<q\text{,}\) then \(p\in x\)
\(\nexists\) maximal element in \(x\text{,}\) ie if \(p\in x\,\exists\, q\in x\) with \(p<q\text{.}\)
Example 1.23 .
If \(p\in \mathbb{Q}\text{,}\) then \(\hat{p}=\{q\in\mathbb{Q}\,|\,q<p\}\subseteq \mathbb{Q}\) is a Dedekind cut
\(\{p\in \mathbb{Q}\,|\,p<0\text{ or }p^2<2\}(=\{p\in\mathbb{Q}\,|\,p<\sqrt{2}\}\) but we don't technically have square root language yet)
Theorem 1.24 . Existence of Real Numbers.
There exists an ordered field \(\R\) which has the least-upper-bound property.
The elements of \(\R\) are called real numbers .
Subsection Properties of the Real Numbers
“Never apologize for what you feel. It's like being sorry for being real.”
―Lil Wayne
Theorem 1.26 . Archimedian Property.
For every real number \(x\text{,}\) there exists a natural number \(n\) such that \(n > x\text{.}\)
Corollary 1.27 . Archimedian Corrollary.
For every positive number \(\vep\text{,}\) there exists a natural number \(n\) such that \(\frac 1n < \vep\text{.}\)
Theorem 1.28 . Existence of Positive Square Roots.
If \(a\) is a nonnegative real number and \(n\) is a positive integer, there exists \(a\) real number \(b ≥ 0\) such that \(b^n = a\text{.}\)
Corollary 1.29 . Existence of Odd Square Roots.
If \(a\) is a real number and \(n\) is an odd positive integer, there exists a real number \(b\) such that \(b^n = a\text{.}\)
Definition 1.30 . \(n\th\) Root. Let \(x\) be a nonnegative real number, and let \(n\) be a positive integer. We define \(x^{\frac1/n}\) to be the nonnegative real number \(y\) such that \(y^n = x\text{.}\)
Theorem 1.31 . Density of the Rationals.
If \(a\) and \(b\) are real numbers with \(a < b\text{,}\) there exists \(a\) rational number \(r\) such that \(a < r < b\text{.}\)
Subsection Absolute Values and \(\C\)
“True faith is belief in the reality of absolute values.”
―William Ralph Inge
Definition 1.32 .
Consider complex numbers \(\mathbb{C}=\{a+ib\,|\, a,b\in\mathbb{R}\}\) with mult by \(i^2=-1\text{.}\)
Definition 1.33 .
Write \(\overline{a+ib}=a-ib\)
Re\((z)=\frac{z+\overline{z}}{z}\text{,}\) Im\((z)=\frac{z-\overline{z}}{zi}\) for all \(z\in\mathbb{C}\)
Or equiv: Re\((a+ib)=a\text{,}\) Im\((a+ib)=b\) for all \(a,b\in\mathbb{R}\text{.}\)
Proposition 1.34 .
\(\mathbb{C}\) is not an ordered field with any order
Proof. In any ordered field \(F\text{,}\) know \(-1<0, x^2\geq 0\) for all \(x\in F\) since \(i^2=-1\) in \(\mathbb{C}, \mathbb{C}\) is not an ordered field.
Definition 1.35 . Absolute Value.
For \(z\in\mathbb{C}\text{,}\) \(|z|=\sqrt{\text{Re}(z)^2+\text{Im}(z)^2}=\sqrt{z\overline{z}}=\) “distance from \(z\) to \(0\) ”.
Theorem 1.36 . Properties of Absolute Values.
\(|z|>0\) iff \(z\neq 0\)
\(\displaystyle |z|=|\overline{z}|\)
\(\displaystyle |\text{Re}(z)|,|\text{Im}(z)|\leq |z|\)
\(|zw|=|z||w|,|\frac{z}{w}|=\frac{|z|}{|w|}\) if \(w\neq 0\)
triangle inequality \(|z+w|\leq |z|+|w|\)
