peteolcott
2018-11-23 15:13:19 UTC
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A newly discovered Infinitesimal number system is shown to<br>
be able to count adjacent geometric points on a number line.<br>
<br>
That adjacent points can be specifically identified is shown<br>
by a line segment that includes or excludes its endpoints.<br>
Line segments Q and R differ in length by a single geometric <br>
point, likewise for line segments R and S.<br>
<br>
<font face="Arial, sans-serif"><font size="4"><b><span
style="background: #ffff00">Infinitesimal numbers are
encoded as follows: </span></b></font></font> <br>
<b>BASE:</b> {real, rational, or irrational}<br>
<b>OFFSET: </b> {integer number of geometric points from the BASE}
<br>
<br>
<b><font size="+1">Point a (0.0) and Point b (1.0)</font></b><font
face="Arial, sans-serif"><span style="text-decoration: none"><span
style="text-decoration: none"><font size="4"><span
style="font-weight: normal"><br>
</span></font></span></span></font>
<table cellspacing="2" cellpadding="2" height="112" width="316"
border="1">
<tbody>
<tr>
<td valign="top"><b>Line Segment</b><br>
</td>
<td valign="top"><b>Start Point</b><br>
</td>
<td valign="top"><b>End Point</b><br>
</td>
</tr>
<tr>
<td valign="top">Q (a, b)<br>
</td>
<td valign="top">0.0(1)<br>
</td>
<td valign="top">1.0(-1)<br>
</td>
</tr>
<tr>
<td valign="top">R (a, b]<br>
</td>
<td valign="top">0.0(1)<br>
</td>
<td valign="top">1.0(0)<br>
</td>
</tr>
<tr>
<td valign="top">S [a, b]<br>
</td>
<td valign="top">0.0(0)<br>
</td>
<td valign="top">1.0(0)<br>
</td>
</tr>
</tbody>
</table>
<br>
Starting at the BASE of 0.0 we can count every geometric point on
the <br>
number line with elements of the set of integers. Since it is
impossible <br>
for any number to exist between geometric points of a number line
this <br>
bijective integer mapping must correspond to every element of the
set <br>
of real numbers. <br>
<br>
Copyright 2018 Pete Olcott<br>
<br>
Correction, points on a number line would not need to be specified<br>
in a Cartesian plane coordinate system. We only need to specify<br>
the x values, there would be no y values.
</body>
</html>
<head>
<meta http-equiv="content-type" content="text/html; charset=utf-8">
</head>
<body text="#000000" bgcolor="#FFFFFF">
A newly discovered Infinitesimal number system is shown to<br>
be able to count adjacent geometric points on a number line.<br>
<br>
That adjacent points can be specifically identified is shown<br>
by a line segment that includes or excludes its endpoints.<br>
Line segments Q and R differ in length by a single geometric <br>
point, likewise for line segments R and S.<br>
<br>
<font face="Arial, sans-serif"><font size="4"><b><span
style="background: #ffff00">Infinitesimal numbers are
encoded as follows: </span></b></font></font> <br>
<b>BASE:</b> {real, rational, or irrational}<br>
<b>OFFSET: </b> {integer number of geometric points from the BASE}
<br>
<br>
<b><font size="+1">Point a (0.0) and Point b (1.0)</font></b><font
face="Arial, sans-serif"><span style="text-decoration: none"><span
style="text-decoration: none"><font size="4"><span
style="font-weight: normal"><br>
</span></font></span></span></font>
<table cellspacing="2" cellpadding="2" height="112" width="316"
border="1">
<tbody>
<tr>
<td valign="top"><b>Line Segment</b><br>
</td>
<td valign="top"><b>Start Point</b><br>
</td>
<td valign="top"><b>End Point</b><br>
</td>
</tr>
<tr>
<td valign="top">Q (a, b)<br>
</td>
<td valign="top">0.0(1)<br>
</td>
<td valign="top">1.0(-1)<br>
</td>
</tr>
<tr>
<td valign="top">R (a, b]<br>
</td>
<td valign="top">0.0(1)<br>
</td>
<td valign="top">1.0(0)<br>
</td>
</tr>
<tr>
<td valign="top">S [a, b]<br>
</td>
<td valign="top">0.0(0)<br>
</td>
<td valign="top">1.0(0)<br>
</td>
</tr>
</tbody>
</table>
<br>
Starting at the BASE of 0.0 we can count every geometric point on
the <br>
number line with elements of the set of integers. Since it is
impossible <br>
for any number to exist between geometric points of a number line
this <br>
bijective integer mapping must correspond to every element of the
set <br>
of real numbers. <br>
<br>
Copyright 2018 Pete Olcott<br>
<br>
Correction, points on a number line would not need to be specified<br>
in a Cartesian plane coordinate system. We only need to specify<br>
the x values, there would be no y values.
</body>
</html>