Neighborhoods in 𝐑 - Revision history

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Lila: Created page with "

The Real Number Line

One way to represent the real numbers $\mathbb{R}$ is on the real number line as depicted below.

2021-10-20T20:04:27Z

Created page with "<h1 id="toc0">The Real Number Line</h1> <p>One way to represent the real numbers <math>\mathbb{R}</math> is on the real number line as depicted below.</p> <div class="image-co..."

New page

<h1 id="toc0">The Real Number Line</h1>
<p>One way to represent the real numbers <math>\mathbb{R}</math> is on the real number line as depicted below.</p>
<div class="image-container aligncenter"><img src="http://mathonline.wdfiles.com/local--files/the-real-line-and-the-epsilon-neighbourhood-of-a-real-number/Screen%20Shot%202014-09-30%20at%206.46.57%20PM.png" alt="Screen%20Shot%202014-09-30%20at%206.46.57%20PM.png" class="image" /></div>
<p>We will now state the important geometric representation of the absolute value with respect to the real number line.</p>
<table class="wiki-content-table">
<tr>
<td><strong>Definition:</strong> If <math>a</math> and <math>b</math> are real numbers, then we say that the <strong>distance</strong> from <math>a</math> to the origin <math>0</math> is the absolute value of <math>a</math>, <math>\mid a \mid</math>. We say that the distance between <math>a</math> and <math>b</math> is the absolute value of their difference, namely <math>\mid a - b \mid</math>.</td>
</tr>
</table>
<p>For example consider the numbers <math>-2</math> and <math>2</math>. There is a distance of <math>4</math> in between these numbers because <math>\mid -2 - 2 \mid = \mid -4 \mid = 4</math>.</p>
<h1 id="toc1">Epsilon Neighbourhood of a Real Number</h1>
<table class="wiki-content-table">
<tr>
<td><strong>Definition:</strong> Let <math>a</math> be a real number and let <math>\varepsilon > 0</math>. The <math>\varepsilon</math>-neighbourhood of the number <math>a</math> is the set denoted <math>V_{\varepsilon} (a) := \{ x \in \mathbb{R} : \: \mid x - a \mid < \varepsilon \}</math>. Alternatively we can define <math>V_{\varepsilon}(a) := \{x \in \mathbb{R} : a - \varepsilon < x < a + \varepsilon \}</math>.
<div class="image-container aligncenter"><img src="http://mathonline.wdfiles.com/local--files/the-real-line-and-the-epsilon-neighbourhood-of-a-real-number/Screen%20Shot%202014-12-05%20at%2010.16.09%20PM.png" alt="Screen%20Shot%202014-12-05%20at%2010.16.09%20PM.png" class="image" /></div>
</td>
</tr>
</table>
<p>For example, consider the point <math>1</math>, and let <math>\varepsilon_0 = 2</math>. Then <math>V_{\varepsilon_0} (1) = \{ x \in \mathbb{R} : \mid x - 1 \mid < 2 \} = (-1, 3)</math>.</p>
<p>We will now look at a simple theorem regarding the epsilon-neighbourhood of a real number.</p>
<table class="wiki-content-table">
<tr>
<td><strong>Theorem 1:</strong> Let <math>a</math> be a real number. If <math>\forall \varepsilon > 0</math>, <math>x \in V_{\varepsilon} (a)</math> then <math>x = a</math>.</td>
</tr>
</table>
<ul>
<li><strong>Proof of Theorem 1:</strong> Suppose that for some <math>x</math>, <math>\forall \varepsilon > 0</math>, <math>\mid x - a \mid < \varepsilon</math>. We know that then <math>\mid x - a \mid = 0</math> if and only if <math>x - a = 0</math> and therefore <math>x = a</math>. <math>\blacksquare</math></li>
</ul>

@@ Line 40: / Line 40: @@
 </ul>
-==Resources==
+== Licensing ==
-* [http://mathonline.wikidot.com/the-real-line-and-the-epsilon-neighbourhood-of-a-real-number The Real Line and The Epsilon Neighbourhood of a Real Number], mathonline.wikidot.com
+Content obtained and/or adapted from:

← Older revision		Revision as of 20:15, 20 October 2021
Line 39:		Line 39:
	<li><strong>Proof of Theorem 1:</strong> Suppose that for some <math>x</math>, <math>\forall \varepsilon > 0</math>, <math>\mid x - a \mid < \varepsilon</math>. We know that then <math>\mid x - a \mid = 0</math> if and only if <math>x - a = 0</math> and therefore <math>x = a</math>. <math>\blacksquare</math></li>		<li><strong>Proof of Theorem 1:</strong> Suppose that for some <math>x</math>, <math>\forall \varepsilon > 0</math>, <math>\mid x - a \mid < \varepsilon</math>. We know that then <math>\mid x - a \mid = 0</math> if and only if <math>x - a = 0</math> and therefore <math>x = a</math>. <math>\blacksquare</math></li>
	</ul>		</ul>
		+
		+	==Resources==
		+	* [http://mathonline.wikidot.com/the-real-line-and-the-epsilon-neighbourhood-of-a-real-number The Real Line and The Epsilon Neighbourhood of a Real Number], mathonline.wikidot.com

← Older revision		Revision as of 20:13, 20 October 2021
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−	[[File:Real number line for Algebra book.svg\|frame\|Real number line]]	+	[[File:Real number line for Algebra book.svg\|frame\|center\|Real number line]]


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−	[[File:Epsilon Umgebung.svg\|frame\|<math>\varepsilon</math>-neighbourhood around <math>a</math> (<math>V_{\varepsilon}(a)</math>) expressed on the real number line]]	+	[[File:Epsilon Umgebung.svg\|frame\|center\|<math>\varepsilon</math>-neighbourhood around <math>a</math> (<math>V_{\varepsilon}(a)</math>) expressed on the real number line]]

← Older revision		Revision as of 20:12, 20 October 2021
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−	[[File:Real number line for Algebra book.svg\|Real number line]]	+	[[File:Real number line for Algebra book.svg\|frame\|Real number line]]


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−	[[File:Epsilon Umgebung.svg\|<math>\varepsilon</math>-neighbourhood around <math>a</math> (<math>V_{\varepsilon}(a)</math>) expressed on the real number line]]	+	[[File:Epsilon Umgebung.svg\|frame\|<math>\varepsilon</math>-neighbourhood around <math>a</math> (<math>V_{\varepsilon}(a)</math>) expressed on the real number line]]

@@ Line 1: / Line 1: @@
 <h1 id="toc0">The Real Number Line</h1>
 <p>One way to represent the real numbers <math>\mathbb{R}</math> is on the real number line as depicted below.</p>
 [[File:Real number line for Algebra book.svg|Real number line]]
 <p>We will now state the important geometric representation of the absolute value with respect to the real number line.</p>
@@ Line 16: / Line 20: @@
 <td><strong>Definition:</strong> Let <math>a</math> be a real number and let <math>\varepsilon > 0</math>. The <math>\varepsilon</math>-neighbourhood of the number <math>a</math> is the set denoted <math>V_{\varepsilon} (a) := \{ x \in \mathbb{R} : \mid x - a \mid < \varepsilon \}</math>. Alternatively we can define <math>V_{\varepsilon}(a) := \{x \in \mathbb{R} : a - \varepsilon < x < a + \varepsilon \}</math>.
 [[File:Epsilon Umgebung.svg|<math>\varepsilon</math>-neighbourhood around a (<math>V_{\varepsilon}(a)</math>) expressed on the real number line]]
 </td>

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 <tr>
 <td><strong>Definition:</strong> Let <math>a</math> be a real number and let <math>\varepsilon > 0</math>. The <math>\varepsilon</math>-neighbourhood of the number <math>a</math> is the set denoted <math>V_{\varepsilon} (a) := \{ x \in \mathbb{R} : \mid x - a \mid < \varepsilon \}</math>. Alternatively we can define <math>V_{\varepsilon}(a) := \{x \in \mathbb{R} : a - \varepsilon < x < a + \varepsilon \}</math>.
-<div class="image-container aligncenter"><img src="http://mathonline.wdfiles.com/local--files/the-real-line-and-the-epsilon-neighbourhood-of-a-real-number/Screen%20Shot%202014-12-05%20at%2010.16.09%20PM.png" alt="Screen%20Shot%202014-12-05%20at%2010.16.09%20PM.png" class="image" /></div>
 </td>
 </tr>

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 <table class="wiki-content-table">
 <tr>
-<td><strong>Definition:</strong> Let <math>a</math> be a real number and let <math>\varepsilon > 0</math>. The <math>\varepsilon</math>-neighbourhood of the number <math>a</math> is the set denoted <math>V_{\varepsilon} (a) := \{ x \in \mathbb{R} : \: \mid x - a \mid < \varepsilon \}</math>. Alternatively we can define <math>V_{\varepsilon}(a) := \{x \in \mathbb{R} : a - \varepsilon < x < a + \varepsilon \}</math>.
+<td><strong>Definition:</strong> Let <math>a</math> be a real number and let <math>\varepsilon > 0</math>. The <math>\varepsilon</math>-neighbourhood of the number <math>a</math> is the set denoted <math>V_{\varepsilon} (a) := \{ x \in \mathbb{R} : \mid x - a \mid < \varepsilon \}</math>. Alternatively we can define <math>V_{\varepsilon}(a) := \{x \in \mathbb{R} : a - \varepsilon < x < a + \varepsilon \}</math>.
 <div class="image-container aligncenter"><img src="http://mathonline.wdfiles.com/local--files/the-real-line-and-the-epsilon-neighbourhood-of-a-real-number/Screen%20Shot%202014-12-05%20at%2010.16.09%20PM.png" alt="Screen%20Shot%202014-12-05%20at%2010.16.09%20PM.png" class="image" /></div>
 </td>