<link rel="shortcut icon" type="image/jpg" href="/favicon.jpeg"/>

<?xml version="1.0"?>
<feed xmlns="http://www.w3.org/2005/Atom" xml:lang="en">
	<id>http://64.23.165.198:80/index.php?action=history&amp;feed=atom&amp;title=E</id>
	<title>E - Revision history</title>
	<link rel="self" type="application/atom+xml" href="http://64.23.165.198:80/index.php?action=history&amp;feed=atom&amp;title=E"/>
	<link rel="alternate" type="text/html" href="http://64.23.165.198:80/index.php?title=E&amp;action=history"/>
	<updated>2026-04-17T15:11:59Z</updated>
	<subtitle>Revision history for this page on the wiki</subtitle>
	<generator>MediaWiki 1.41.0</generator>
	<entry>
		<id>http://64.23.165.198:80/index.php?title=E&amp;diff=469&amp;oldid=prev</id>
		<title>Lfox at 18:25, 8 July 2024</title>
		<link rel="alternate" type="text/html" href="http://64.23.165.198:80/index.php?title=E&amp;diff=469&amp;oldid=prev"/>
		<updated>2024-07-08T18:25:54Z</updated>

		<summary type="html">&lt;p&gt;&lt;/p&gt;
&lt;table style=&quot;background-color: #fff; color: #202122;&quot; data-mw=&quot;interface&quot;&gt;
				&lt;col class=&quot;diff-marker&quot; /&gt;
				&lt;col class=&quot;diff-content&quot; /&gt;
				&lt;col class=&quot;diff-marker&quot; /&gt;
				&lt;col class=&quot;diff-content&quot; /&gt;
				&lt;tr class=&quot;diff-title&quot; lang=&quot;en&quot;&gt;
				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;← Older revision&lt;/td&gt;
				&lt;td colspan=&quot;2&quot; style=&quot;background-color: #fff; color: #202122; text-align: center;&quot;&gt;Revision as of 18:25, 8 July 2024&lt;/td&gt;
				&lt;/tr&gt;&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot; id=&quot;mw-diff-left-l1&quot;&gt;Line 1:&lt;/td&gt;
&lt;td colspan=&quot;2&quot; class=&quot;diff-lineno&quot;&gt;Line 1:&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&#039;&#039;&#039;e&#039;&#039;&#039; is the  &lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&#039;&#039;&#039;e&#039;&#039;&#039; &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;is a [TODO] &lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;== Probabilistic interpretation ==&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;=== Bernoulli trials ===&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Suppose that something has a 1 in &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; chance of occurring. After &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; times, what &lt;/ins&gt;is the &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;probability that the event has not happened once? Each time, the probability that the event doesn&#039;t happen is &amp;lt;math&amp;gt;(1 - 1/n) &amp;lt;/math&amp;gt;, so the probability that the event doesn&#039;t happen &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; times in a row is &amp;lt;math&amp;gt;(1 - 1/n)^n &amp;lt;/math&amp;gt;. &lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;It turns out that this sequence is Cauchy [TODO], and &amp;lt;math&amp;gt;e&amp;lt;/math&amp;gt; is the inverse of its limit &amp;lt;math display=&quot;block&quot;&amp;gt;\frac{1}{e} := \lim_{n\rightarrow \infty } \left( 1 - \frac{1}{n} \right)^n. &amp;lt;/math&amp;gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Similarly, if something has a 1 in &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; chance of occurring, then after &amp;lt;math&amp;gt;n t&amp;lt;/math&amp;gt; times, what is the probability that the event has not happened once? For large &amp;lt;math&amp;gt;n&amp;lt;/math&amp;gt; it approaches some fixed value &amp;lt;math display=&quot;block&quot;&amp;gt;e^{-t}  = \lim_{n\rightarrow \infty } \left( 1 - \frac{1}{n} \right)^{n t} . &amp;lt;/math&amp;gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;The[TODO terminology] numerical value of &#039;&#039;e&#039;&#039; is useful to know, because if you&#039;re ever in a situation where you have to calculate one of these probabilities, it could be a major pain to multiply &amp;lt;math&amp;gt;(1 - 1/n) &amp;lt;/math&amp;gt; with itself &amp;lt;math&amp;gt;nt&amp;lt;/math&amp;gt; times, if you had no access to a calculator. Instead, you could save yourself some work by looking up this number[TODO it&#039;s not a single number; it&#039;s a sequence of numbers] &amp;lt;math&amp;gt;e&amp;lt;/math&amp;gt; in a table that someone has already computed. [TODO I don&#039;t find this explanation very convincing.] &lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;=== Central limit theorem [TODO] ===&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;Suppose you do &amp;lt;math&amp;gt;N&amp;lt;/math&amp;gt; experiments, where in each experiment you flip a coin 100 times and record the results. For &amp;lt;math&amp;gt;0 \leq k \leq 100 &amp;lt;/math&amp;gt;, what fraction of the experiments resulted in heads landing precisely &amp;lt;math&amp;gt;k&amp;lt;/math&amp;gt; times? &lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;For &amp;lt;math&amp;gt;N&amp;lt;/math&amp;gt; large, the answer is proportional to &amp;lt;math display=&quot;block&quot;&amp;gt;e^{ - (k - 50)^2 / \sigma  }&amp;lt;/math&amp;gt;for some constant &amp;lt;math&amp;gt;\sigma&amp;lt;/math&amp;gt;. &lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;== Compounding interest interpretation [TODO yuck] ==&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;If you have your money in a bank account, where you make a rate of interest &amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt;, which compounds &amp;lt;math&amp;gt;c&amp;lt;/math&amp;gt; times yearly, then after a year your bank account will contain &amp;lt;math display=&quot;block&quot;&amp;gt;(1 + r)^{c}&amp;lt;/math&amp;gt;times your principal. Now suppose that it compounds many times (&amp;lt;math&amp;gt;c&amp;lt;/math&amp;gt; is large), but the interest &amp;lt;math&amp;gt;r&amp;lt;/math&amp;gt; is small. Fix the rate to be &amp;lt;math&amp;gt;r = 1/c&amp;lt;/math&amp;gt;, then we are interested in what the multiplier &amp;lt;math&amp;gt;(1 + 1/c)^{c}&amp;lt;/math&amp;gt; is when &amp;lt;math&amp;gt;c&amp;lt;/math&amp;gt; is very large. It turns out to converge to some constant &amp;lt;math display=&quot;block&quot;&amp;gt;e := \lim_{c\rightarrow \infty} \left(1 + \frac{1}{c}\right)^{c}&amp;lt;/math&amp;gt;&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-deleted&quot;&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;This example, though it&#039;s how &#039;&#039;e&#039;&#039; is introduced, is rather disconnected from reality. No bank actually offers this as an option. And even if they did, why would they fix the rate to be &amp;lt;math&amp;gt;r = 1/c&amp;lt;/math&amp;gt;? Why not some other rate? [TODO the answer, I think, is that if &amp;lt;math&amp;gt;r(c)&amp;lt;/math&amp;gt; grows faster than &amp;lt;math&amp;gt;1/c&amp;lt;/math&amp;gt; then the limit will diverge, and if it grows slower than &amp;lt;math&amp;gt;1/c&amp;lt;/math&amp;gt; then the limit will be 1.]  &lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;br&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;== Geometric interpretation ==&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;== Geometric interpretation ==&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;[[File:1920px-Hyperbolic functions-2.svg.png|thumb|&amp;quot; A ray through the unit hyperbola &amp;#039;&amp;#039;x&amp;#039;&amp;#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; − &amp;#039;&amp;#039;y&amp;#039;&amp;#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; = 1 at the point (cosh &amp;#039;&amp;#039;a&amp;#039;&amp;#039;, sinh &amp;#039;&amp;#039;a&amp;#039;&amp;#039;), where &amp;#039;&amp;#039;a&amp;#039;&amp;#039; is twice the area between the ray, the hyperbola, and the &amp;#039;&amp;#039;x&amp;#039;&amp;#039;-axis. For points on the hyperbola below the &amp;#039;&amp;#039;x&amp;#039;&amp;#039;-axis, the area is considered negative&amp;quot; [TODO]]]&lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot;&gt;&lt;/td&gt;&lt;td style=&quot;background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;[[File:1920px-Hyperbolic functions-2.svg.png|thumb|&amp;quot; A ray through the unit hyperbola &amp;#039;&amp;#039;x&amp;#039;&amp;#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; − &amp;#039;&amp;#039;y&amp;#039;&amp;#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; = 1 at the point (cosh &amp;#039;&amp;#039;a&amp;#039;&amp;#039;, sinh &amp;#039;&amp;#039;a&amp;#039;&amp;#039;), where &amp;#039;&amp;#039;a&amp;#039;&amp;#039; is twice the area between the ray, the hyperbola, and the &amp;#039;&amp;#039;x&amp;#039;&amp;#039;-axis. For points on the hyperbola below the &amp;#039;&amp;#039;x&amp;#039;&amp;#039;-axis, the area is considered negative&amp;quot; [TODO]]]&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&amp;lt;math&amp;gt;\cosh(a)&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\sinh(a)&amp;lt;/math&amp;gt; are the &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;y&amp;lt;/math&amp;gt; coordinates on a unit hyperbola, at the place where the ray subtending an area of &amp;lt;math&amp;gt;a/2&amp;lt;/math&amp;gt; intersects the hyperbola. This description is not very clear; see the figure.  &lt;/div&gt;&lt;/td&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;+&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;&amp;lt;math&amp;gt;\cosh(a)&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\sinh(a)&amp;lt;/math&amp;gt; are the &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;y&amp;lt;/math&amp;gt; coordinates on a unit &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;[[&lt;/ins&gt;hyperbola&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;]]&lt;/ins&gt;, at the place where the ray subtending an area of &amp;lt;math&amp;gt;a/2&amp;lt;/math&amp;gt; intersects the hyperbola. &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;(&lt;/ins&gt;This description is not very clear; see the figure.&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;) &lt;/ins&gt;In this situation, &amp;lt;math&amp;gt;e^{a}&amp;lt;/math&amp;gt; is the &lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;simply &lt;/ins&gt;sum of &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;y&amp;lt;/math&amp;gt; coordinates&lt;ins style=&quot;font-weight: bold; text-decoration: none;&quot;&gt;.&lt;/ins&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt; &lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;tr&gt;&lt;td class=&quot;diff-marker&quot; data-marker=&quot;−&quot;&gt;&lt;/td&gt;&lt;td style=&quot;color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;&quot;&gt;&lt;div&gt;In this situation, &amp;lt;math&amp;gt;e^{a}&amp;lt;/math&amp;gt; is the sum of &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;y&amp;lt;/math&amp;gt; coordinates&lt;/div&gt;&lt;/td&gt;&lt;td colspan=&quot;2&quot; class=&quot;diff-side-added&quot;&gt;&lt;/td&gt;&lt;/tr&gt;
&lt;/table&gt;</summary>
		<author><name>Lfox</name></author>
	</entry>
	<entry>
		<id>http://64.23.165.198:80/index.php?title=E&amp;diff=468&amp;oldid=prev</id>
		<title>Lfox: Created page with &quot;&#039;&#039;&#039;e&#039;&#039;&#039; is the   == Geometric interpretation == &quot; A ray through the unit hyperbola &#039;&#039;x&#039;&#039;&lt;sup&gt;2&lt;/sup&gt; − &#039;&#039;y&#039;&#039;&lt;sup&gt;2&lt;/sup&gt; = 1 at the point (cosh &#039;&#039;a&#039;&#039;, sinh &#039;&#039;a&#039;&#039;), where &#039;&#039;a&#039;&#039; is twice the area between the ray, the hyperbola, and the &#039;&#039;x&#039;&#039;-axis. For points on the hyperbola below the &#039;&#039;x&#039;&#039;-axis, the area is considered negative&quot; [TODO] &lt;math&gt;\cosh(a)&lt;/math&gt; and &lt;math&gt;\sinh(a)&lt;/math&gt; are the &lt;math&gt;x&lt;/math&gt; and &lt;math&gt;y&lt;/...&quot;</title>
		<link rel="alternate" type="text/html" href="http://64.23.165.198:80/index.php?title=E&amp;diff=468&amp;oldid=prev"/>
		<updated>2024-07-08T17:22:09Z</updated>

		<summary type="html">&lt;p&gt;Created page with &amp;quot;&amp;#039;&amp;#039;&amp;#039;e&amp;#039;&amp;#039;&amp;#039; is the   == Geometric interpretation == &lt;a href=&quot;/index.php/File:1920px-Hyperbolic_functions-2.svg.png&quot; title=&quot;File:1920px-Hyperbolic functions-2.svg.png&quot;&gt;thumb|&amp;quot; A ray through the unit hyperbola &amp;#039;&amp;#039;x&amp;#039;&amp;#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; − &amp;#039;&amp;#039;y&amp;#039;&amp;#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; = 1 at the point (cosh &amp;#039;&amp;#039;a&amp;#039;&amp;#039;, sinh &amp;#039;&amp;#039;a&amp;#039;&amp;#039;), where &amp;#039;&amp;#039;a&amp;#039;&amp;#039; is twice the area between the ray, the hyperbola, and the &amp;#039;&amp;#039;x&amp;#039;&amp;#039;-axis. For points on the hyperbola below the &amp;#039;&amp;#039;x&amp;#039;&amp;#039;-axis, the area is considered negative&amp;quot; [TODO&lt;/a&gt;] &amp;lt;math&amp;gt;\cosh(a)&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\sinh(a)&amp;lt;/math&amp;gt; are the &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;y&amp;lt;/...&amp;quot;&lt;/p&gt;
&lt;p&gt;&lt;b&gt;New page&lt;/b&gt;&lt;/p&gt;&lt;div&gt;&amp;#039;&amp;#039;&amp;#039;e&amp;#039;&amp;#039;&amp;#039; is the &lt;br /&gt;
&lt;br /&gt;
== Geometric interpretation ==&lt;br /&gt;
[[File:1920px-Hyperbolic functions-2.svg.png|thumb|&amp;quot; A ray through the unit hyperbola &amp;#039;&amp;#039;x&amp;#039;&amp;#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; − &amp;#039;&amp;#039;y&amp;#039;&amp;#039;&amp;lt;sup&amp;gt;2&amp;lt;/sup&amp;gt; = 1 at the point (cosh &amp;#039;&amp;#039;a&amp;#039;&amp;#039;, sinh &amp;#039;&amp;#039;a&amp;#039;&amp;#039;), where &amp;#039;&amp;#039;a&amp;#039;&amp;#039; is twice the area between the ray, the hyperbola, and the &amp;#039;&amp;#039;x&amp;#039;&amp;#039;-axis. For points on the hyperbola below the &amp;#039;&amp;#039;x&amp;#039;&amp;#039;-axis, the area is considered negative&amp;quot; [TODO]]]&lt;br /&gt;
&amp;lt;math&amp;gt;\cosh(a)&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;\sinh(a)&amp;lt;/math&amp;gt; are the &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;y&amp;lt;/math&amp;gt; coordinates on a unit hyperbola, at the place where the ray subtending an area of &amp;lt;math&amp;gt;a/2&amp;lt;/math&amp;gt; intersects the hyperbola. This description is not very clear; see the figure. &lt;br /&gt;
&lt;br /&gt;
In this situation, &amp;lt;math&amp;gt;e^{a}&amp;lt;/math&amp;gt; is the sum of &amp;lt;math&amp;gt;x&amp;lt;/math&amp;gt; and &amp;lt;math&amp;gt;y&amp;lt;/math&amp;gt; coordinates&lt;/div&gt;</summary>
		<author><name>Lfox</name></author>
	</entry>
</feed>