Point Notation - Revision history

WikiSysop: /* Mixed Notation */

2022-09-10T15:52:00Z

Mixed Notation

← Older revision		Revision as of 11:52, 10 September 2022
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	<ul>		<ul>
	<li>The right part of '''Base''' Point Notation may '''not''' contain any of the other Point Notations except for (positive or negative) <b>Decimal</b>.</li>		<li>The right part of '''Base''' Point Notation may '''not''' contain any of the other Point Notations except for (positive or negative) <b>Decimal</b>.</li>
	<li>The left part of '''Base''' Point Notation may contain any of the other Point Notations except itself and '''Rational Integer''' Point Notation (e.g., <apll>123<_x/></apll>, instead use <apll>123<_r/>1</apll>).</li>		<li>The left part of '''Base''' Point Notation may contain any of the other Point Notations except itself and '''Rational Integer''' Point Notation (e.g., <apll>123<_x/></apll> — instead use <apll>123<_r/>1</apll>).</li>
	<li>'''Decimal''', '''Exponential''', '''Rational''', '''Variable Precision''', '''Ball''', '''Hypercomplex''', and '''Complex Angle''' Point Notations may appear in either or both parts of '''Euler''', '''Pi''', '''Gamma''', or '''Zeta''' Point Notations, except for '''Rational Integer''' Point Notation which may '''not''' appear in the '''lefthand''' part of '''epgz''' notations.</li>		<li>'''Decimal''', '''Exponential''', '''Rational''', '''Variable Precision''', '''Ball''', '''Hypercomplex''', and '''Complex Angle''' Point Notations may appear in either or both parts of '''Euler''', '''Pi''', '''Gamma''', or '''Zeta''' Point Notations, except for '''Rational Integer''' Point Notation which may '''not''' appear in the '''lefthand''' part of '''epgz''' notations.</li>
	<li>No two of '''Euler''', '''Pi''', '''Gamma''' , or '''Zeta''' Point Notations may appear in the same Point Notation String.</li>		<li>No two of '''Euler''', '''Pi''', '''Gamma''' , or '''Zeta''' Point Notations may appear in the same Point Notation String.</li>

WikiSysop: /* Mixed Notation */

2022-09-10T15:50:59Z

Mixed Notation

← Older revision		Revision as of 11:50, 10 September 2022
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	<ul>		<ul>
	<li>The right part of '''Base''' Point Notation may '''not''' contain any of the other Point Notations except for (positive or negative) <b>Decimal</b>.</li>		<li>The right part of '''Base''' Point Notation may '''not''' contain any of the other Point Notations except for (positive or negative) <b>Decimal</b>.</li>
	<li>The left part of '''Base''' Point Notation may contain any of the other Point Notations except itself and '''Rational Integer''' Point Notation (e.g., <apll>123<_x/></apll>).</li>		<li>The left part of '''Base''' Point Notation may contain any of the other Point Notations except itself and '''Rational Integer''' Point Notation (e.g., <apll>123<_x/></apll>, instead use <apll>123<_r/>1</apll>).</li>
	<li>'''Decimal''', '''Exponential''', '''Rational''', '''Variable Precision''', '''Ball''', '''Hypercomplex''', and '''Complex Angle''' Point Notations may appear in either or both parts of '''Euler''', '''Pi''', '''Gamma''', or '''Zeta''' Point Notations, except for '''Rational Integer''' Point Notation which may '''not''' appear in the '''lefthand''' part of '''epgz''' notations.</li>		<li>'''Decimal''', '''Exponential''', '''Rational''', '''Variable Precision''', '''Ball''', '''Hypercomplex''', and '''Complex Angle''' Point Notations may appear in either or both parts of '''Euler''', '''Pi''', '''Gamma''', or '''Zeta''' Point Notations, except for '''Rational Integer''' Point Notation which may '''not''' appear in the '''lefthand''' part of '''epgz''' notations.</li>
	<li>No two of '''Euler''', '''Pi''', '''Gamma''' , or '''Zeta''' Point Notations may appear in the same Point Notation String.</li>		<li>No two of '''Euler''', '''Pi''', '''Gamma''' , or '''Zeta''' Point Notations may appear in the same Point Notation String.</li>

WikiSysop: /* Mixed Notation */

2022-09-10T15:47:19Z

Mixed Notation

← Older revision		Revision as of 11:47, 10 September 2022
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	<ul>		<ul>
	<li>The right part of '''Base''' Point Notation may '''not''' contain any of the other Point Notations except for (positive or negative) <b>Decimal</b>.</li>		<li>The right part of '''Base''' Point Notation may '''not''' contain any of the other Point Notations except for (positive or negative) <b>Decimal</b>.</li>
	<li>The left part of '''Base''' Point Notation may contain any of the other Point Notations except itself and Rational Integer Notation.</li>		<li>The left part of '''Base''' Point Notation may contain any of the other Point Notations except itself and '''Rational Integer''' Point Notation (e.g., <apll>123<_x/></apll>).</li>
	<li>'''Decimal''', '''Exponential''', '''Rational''', '''Variable Precision''', '''Ball''', '''~~Complex~~''', and '''~~Hypercomplex~~''' Point Notations may appear in either or both parts of '''Euler''', '''Pi''', '''Gamma''', or '''Zeta''' Point Notations.</li>		<li>'''Decimal''', '''Exponential''', '''Rational''', '''Variable Precision''', '''Ball''', '''Hypercomplex''', and '''Complex Angle''' Point Notations may appear in either or both parts of '''Euler''', '''Pi''', '''Gamma''', or '''Zeta''' Point Notations, except for '''Rational Integer''' Point Notation which may '''not''' appear in the '''lefthand''' part of '''epgz''' notations.</li>
	<li>No two of '''Euler''', '''Pi''', '''Gamma''' , or '''Zeta''' Point Notations may appear in the same Point Notation String.</li>		<li>No two of '''Euler''', '''Pi''', '''Gamma''' , or '''Zeta''' Point Notations may appear in the same Point Notation String.</li>
	</ul>		</ul>
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	<li>5. Rational, Variable Precision Floating Point		<li>5. Rational, Variable Precision Floating Point
	<li>4. Ball Arithmetic		<li>4. Ball Arithmetic
	<li>3. Complex~~, Hypercomplex~~		<li>3. Hypercomplex, Complex Angle
	<li>2. Euler, Pi, Gamma, Zeta		<li>2. Euler, Pi, Gamma, Zeta
	<li>1. Base		<li>1. Base
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	Notations with the same binding strength may not be mixed (e.g., <apll>1<pn>r</pn>2<pn>v</pn></apll> is an error). Otherwise, any notation may incorporate notations with a higher binding strength but may not incorporate notations with an equal or lower binding strength.		Notations with the same binding strength may not be mixed (e.g., <apll>1<pn>r</pn>2<pn>v</pn></apll> is an error). Otherwise, any notation may incorporate notations with a higher binding strength but may not incorporate notations with an equal or lower binding strength.

	This latter case need not signal an error, but instead it might produce a different interpretation. For example,		'''N.B.''': This latter case need not signal an error, but instead might produce a different interpretation. For example,

	<apll>1<_r/>2<_pi/>1<_J/>3</apll> is interpreted as <apll>1<_r/>2×(○1<_x/>)*1<_J/>3</apll> not as <apll>1<_r/>2<_pi/>1+0<_J/>3</apll> because '''Rational'''~~, '''Complex''',~~ and '''Hypercomplex''' Point Notations are constructed before '''Pi''' Point Notation is.		<apll>1<_r/>2<_pi/>1<_J/>3</apll> is interpreted as <apll>1<_r/>2×(○1<_x/>)*1<_J/>3</apll> not as <apll>1<_r/>2<_pi/>1+0<_J/>3</apll> because '''Rational''' and '''Hypercomplex''' Point Notations are constructed before '''Pi''' Point Notation is.

	{{Language Toolbar}}		{{Language Toolbar}}
	[[Category:Mouse Group Notations]]		[[Category:Mouse Group Notations]]

WikiSysop: /* Rational Point Notation */

2022-09-10T15:27:47Z

Rational Point Notation

← Older revision		Revision as of 11:27, 10 September 2022
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	== Rational Point Notation ==		== Rational Point Notation ==

	This notation allows you to enter fractions and integers as rational numbers and have them be retained as rational numbers. Rational numbers, using the <apll><_r/></apll> infix notation, may be used in the '''lefthand''' part of <b>Base</b>, and either part of <b>Euler</b>, <b>Pi</b>, <b>Gamma</b>, or <b>Zeta</b> Point Notations. Rational Integers, using the <apll><_x/></apll> suffix notation, may be used in the '''righthand''' part of <b>Euler</b>, <b>Pi</b>, <b>Gamma</b>, or <b>Zeta</b> Point Notations only, not in the '''lefthand''' part.		This notation allows you to enter fractions and integers as rational numbers and have them be retained as rational numbers. Rational numbers, using the <apll><_r/></apll> infix notation, may be used in the '''lefthand''' part of <b>Base</b>, and either part of <b>Euler</b>, <b>Pi</b>, <b>Gamma</b>, or <b>Zeta</b> Point Notations. Rational Integers, using the <apll><_x/></apll> suffix notation, may be used in the '''righthand''' part of <b>Euler</b>, <b>Pi</b>, <b>Gamma</b>, or <b>Zeta</b> Point Notations only, not in the '''lefthand''' part; use <apll><_r/>1</apll>, as in <apll>123<_r/>1<_g/>2</apll>, instead of <apll>123<_x/><_g/>2</apll>.

	For more information, see [[Rational and VFP Numbers\|Rational Numbers]]. This notation also accepts strings that contain '''Decimal''' and/or '''Exponential''' point notation such as <apll>0.3<_x/></apll>, <apll>0.7<_r/>3</apll>, <apll>0.7<_E/>¯3<_r/>1.7</apll>, etc. and represents them as an exact Rational number.		For more information, see [[Rational and VFP Numbers\|Rational Numbers]]. This notation also accepts strings that contain '''Decimal''' and/or '''Exponential''' point notation such as <apll>0.3<_x/></apll>, <apll>0.7<_r/>3</apll>, <apll>0.7<_E/>¯3<_r/>1.7</apll>, etc. and represents them as an exact Rational number.

WikiSysop: /* Overview */

2022-09-10T15:18:39Z

Overview

← Older revision		Revision as of 11:18, 10 September 2022
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	</tr>		</tr>
	<tr>		<tr>
	<td valign="baseline"><b>Complex ~~Cartesian~~ Point Notation</b>:</td>		<td valign="baseline"><b>Complex Angle Point Notation</b>:</td>
	<td></td>		<td></td>
	<td></td>		<td></td>
	~~<td valign="baseline">e.g., <apll>2<_J/>3</apll> or <apll>2<_i/>3</apll> as a shorthand for <apll>2+3×√¯1</apll>.</td>~~		<td valign="baseline">e.g., <apll>2<_ad/>3</apll> (Angle in Degrees), <apll>2<_ar/>3</apll> (Angle in Radians), <apll>2<_au/>3</apll> (Angle in Unit Normalized Radians), or <apll>2<_ah/>0.3</apll> (Angle in Half Unit Normalized Radians).</td>
	~~</tr>~~
	~~<tr>~~
	~~<td valign="baseline"><b>Complex Polar Point Notation</b>:</td>~~
	~~<td></td>~~
	~~<td></td>~~
	<td valign="baseline">e.g., <apll>2<_ad/>3</apll> (Angle in Degrees), <apll>2<_ar/>3</apll> (Angle in Radians), <apll>2<_au/>3</apll> (Angle in Unit Normalized Radians), or <apll>2<_ah/>3</apll> (Angle in Half Unit Normalized Radians).</td>
	</tr>		</tr>
	<tr>		<tr>
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	<td></td>		<td></td>
	<td></td>		<td></td>
	<td valign="baseline">e.g., <apll>2<_r/>3</apll> as a shorthand for <apll>2÷3</apll> as an infinite precision rational number, or <apll>123<_x/></apll> as a means of representing <apll>123</apll> as an infinite precision integer — the suffix <apll><_x/></apll> is actually a shorthand for <apll><_r/>1</apll>, that is, infinite precision integers are actually represented as rational numbers with a denominator of <apll>1</apll>.</td>		<td valign="baseline">e.g., <apll>2<_r/>3</apll> as a shorthand for <apll>2÷3</apll> as an infinite precision rational number, or <apll>123<_x/></apll> as a means of representing <apll>123</apll> as an infinite precision integer — the suffix <apll><_x/></apll> is actually a shorthand for <apll><_r/>1</apll>, that is, infinite precision integers are actually represented as rational numbers with a denominator of <apll>1</apll>. However, this suffix cannot be used anywhere in the lefthand part(s) of any other notation.</td>
	</tr>		</tr>
	<tr>		<tr>

WikiSysop: /* Mixed Notation */

2022-09-10T15:02:46Z

Mixed Notation

← Older revision		Revision as of 11:02, 10 September 2022
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	<ul>		<ul>
	<li>The right part of '''Base''' Point Notation may '''not''' contain any of the other Point Notations except for (positive or negative) <b>Decimal</b>.</li>		<li>The right part of '''Base''' Point Notation may '''not''' contain any of the other Point Notations except for (positive or negative) <b>Decimal</b>.</li>
	<li>The left part of '''Base''' Point Notation may contain any of the other Point Notations except itself.</li>		<li>The left part of '''Base''' Point Notation may contain any of the other Point Notations except itself and Rational Integer Notation.</li>
	<li>'''Decimal''', '''Exponential''', '''Rational''', '''Variable Precision''', '''Ball''', '''Complex''', and '''Hypercomplex''' Point Notations may appear in either or both parts of '''Euler''', '''Pi''', '''Gamma''', or '''Zeta''' Point Notations.</li>		<li>'''Decimal''', '''Exponential''', '''Rational''', '''Variable Precision''', '''Ball''', '''Complex''', and '''Hypercomplex''' Point Notations may appear in either or both parts of '''Euler''', '''Pi''', '''Gamma''', or '''Zeta''' Point Notations.</li>
	<li>No two of '''Euler''', '''Pi''', '''Gamma''' , or '''Zeta''' Point Notations may appear in the same Point Notation String.</li>		<li>No two of '''Euler''', '''Pi''', '''Gamma''' , or '''Zeta''' Point Notations may appear in the same Point Notation String.</li>

WikiSysop: /* Rational Point Notation */

2022-09-10T14:55:13Z

Rational Point Notation

← Older revision		Revision as of 10:55, 10 September 2022
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	== Rational Point Notation ==		== Rational Point Notation ==

	This notation allows you to enter fractions as rational numbers and have them be retained as rational numbers. Rational numbers (using the <apll><~~pn>r<~~/pn></apll> infix ~~separator only~~, ~~not~~ the <~~apll~~><pn>x</pn></~~apll~~> ~~suffix) may also be used as a lefthand part of~~ <b>~~Base~~</b>, ~~and either~~ part of <b>Euler</b>, <b>Pi</b>, <b>Gamma</b>, or <b>Zeta</b> Point Notations. For more information, see [[Rational and VFP Numbers\|Rational Numbers]]. This notation also accepts strings that contain '''Decimal''' and/or '''Exponential''' point notation such as <apll>0.5<_x/></apll>, <apll>0.5<_r/>3</apll>, <apll>1<_E/>¯3<_r/>1.5</apll>, etc. and represents them as an exact Rational number.		This notation allows you to enter fractions and integers as rational numbers and have them be retained as rational numbers. Rational numbers, using the <apll><_r/></apll> infix notation, may be used in the '''lefthand''' part of <b>Base</b>, and either part of <b>Euler</b>, <b>Pi</b>, <b>Gamma</b>, or <b>Zeta</b> Point Notations. Rational Integers, using the <apll><_x/></apll> suffix notation, may be used in the '''righthand''' part of <b>Euler</b>, <b>Pi</b>, <b>Gamma</b>, or <b>Zeta</b> Point Notations only, not in the '''lefthand''' part.

			For more information, see [[Rational and VFP Numbers\|Rational Numbers]]. This notation also accepts strings that contain '''Decimal''' and/or '''Exponential''' point notation such as <apll>0.3<_x/></apll>, <apll>0.7<_r/>3</apll>, <apll>0.7<_E/>¯3<_r/>1.7</apll>, etc. and represents them as an exact Rational number.

	== Variable-Precision Floating Point Notation ==		== Variable-Precision Floating Point Notation ==

WikiSysop: /* Hypercomplex and Complex Angle Point Notation */

2022-09-10T14:36:00Z

Hypercomplex and Complex Angle Point Notation

← Older revision		Revision as of 10:36, 10 September 2022
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	<div id="Angle_Point_Notation"></div>		<div id="Angle_Point_Notation"></div>

	== Complex ~~Cartesian, Complex Polar, and Hypercomplex~~ Point Notation ==		== Hypercomplex and Complex Angle Point Notation ==

	~~This notation~~ allows you to enter Complex, Quaternion, and Octonion numbers in various forms ~~such~~ as a combination of a Real part followed by one or more ~~Hypercomplex units~~ (2nd, 4th, or 8th ~~root~~ of <apll>¯1</apll>) ~~times~~ the corresponding ~~coefficient~~. For ~~more details~~, ~~see [http://www.sudleyplace.com/APL/Hypercomplex%20Notation%20in%20APL.pdf Hypercomplex Notation in APL].~~		'''Hypercomplex''' Point Notation allows you to enter Complex, Quaternion, and Octonion numbers in various forms as a combination of a '''Real''' part followed by one or more '''Imaginary''' parts (2nd, 4th, or 8th roots of <apll>¯1</apll>) each separated by the corresponding notation. For example,

	~~'''~~Complex ~~Polar''' Point Notation as in~~ <apll>1.2<~~_ar~~/>3.4</apll> ~~contains two parts: the Radius~~ <apll>1.2</~~apll~~> ~~and the Angle~~ <~~apll~~>3.4</apll> ~~in this case in units of Radians.~~		<table>
			<tr>
			<td><apll>1<_i/>2</apll></td>
			<td>Complex</td>
			</tr>
			<tr>
			<td><apll>1<_i/>2<_j/>3<_k/>4</apll></td>
			<td>Quaternion</td>
			</tr>
			<tr>
			<td><apll>1<_i/>2<_j/>3<_k/>4<_l/>5<_ij/>6<_jk/>7<_kl/>8</apll></td>
			<td>Octonion</td>
			</tr>
			</table>

	~~The limitation on~~ the Angle in the several '''Complex ~~Polar~~''' notations is as follows:		For more details, see [http://www.sudleyplace.com/APL/Hypercomplex%20Notation%20in%20APL.pdf Hypercomplex Notation in APL].

			'''Complex Angle''' Point Notation, as in <apll>1.2<_ad/>0.25</apll>, contains two parts: the Radius <apll>1.2</apll> and the Angle <apll>0.25</apll>, which may be expressed in one of four units such as '''Degrees''', '''Radians''', etc., all measured in a counter-clockwise direction from the x-axis, which is <apll>0</apll> in the corresponding unit.

			The Angle Domain and Units of the several '''Complex Angle''' notations is as follows:

	{\|		{\|
	\|'''Notation'''		\|'''Notation'''
	\|		\|
	\|'''Domain ~~for Angle~~'''		\|'''Angle Domain'''
	\|		\|
	\|align="left"\|'''Angle Units'''		\|align="left"\|'''Angle Units'''
	\|-		\|-
	\|align="center"\|<_ad/>\|\| \|\|<apll>[¯360, 360]</apll>\|\| \|\|Degrees		\|align="center"\|<apll><_ad/></apll>\|\| \|\|<apll>[¯360, 360]</apll>\|\| \|\|Degrees
	\|-		\|-
	\|align="center"\|<_ar/>\|\| \|\|<apll>[¯2p1, 2p1]</apll>\|\| \|\|Radians		\|align="center"\|<apll><_ar/></apll>\|\| \|\|<apll>[¯2p1, 2p1]</apll>\|\| \|\|Radians
	\|-		\|-
	\|align="center"\|<_au/>\|\| \|\|<apll>[0, 1]</apll>\|\| \|\|Unit Normalized Radians		\|align="center"\|<apll><_au/></apll>\|\| \|\|<apll>[0, 1]</apll>\|\| \|\|Unit Normalized Radians
	\|-		\|-
	\|align="center"\|<_ah/>\|\| \|\|<apll>[¯0.5, 0.5]</apll>\|\| \|\|Half Unit Normalized Radians		\|align="center"\|<apll><_ah/></apll>\|\| \|\|<apll>[¯0.5, 0.5]</apll>\|\| \|\|Half Unit Normalized Radians
	\|-		\|-
	\|}		\|}

	Entering a number whose Angle is outside the associated ~~domain~~, signals an error.		Entering a number whose Angle is outside the associated Domain, signals an error.

	== Ball Arithmetic Point Notation ==		== Ball Arithmetic Point Notation ==

WikiSysop: /* Gamma Point Notation */

2022-09-09T18:25:57Z

Gamma Point Notation

← Older revision		Revision as of 14:25, 9 September 2022
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	== Gamma Point Notation ==		== Gamma Point Notation ==

	This notation allows you to enter numeric constants that are in the form of the product of a '''M'''ultiplier and <apll>γ</apll> (<apll>&cong; 0.5772156649015329...</apll> — limiting difference between the harmonic series and the natural logarithm) raised to an '''E'''xponent, that is, <apll>M∙γ<sup>E</sup></apll> or <apll>M×γ*E</apll>. The numbers to the left (multiplier) and right (exponent) of the <apll><pn>g</pn></apll> may be represented in several ways including integers, <b>Decimal</b>, <b>Exponential</b>, <b>Rational</b>, or <b>Variable Precision Floating</b> Point Notation, but not <b>Base</b>, <b>Euler</b>, <b>Pi</b>, <b>Gamma</b>, or <b>Zeta</b> Point Notation.		This notation allows you to enter numeric constants that are in the form of the product of a '''M'''ultiplier and <apll>γ</apll> (<apll>&cong; 0.5772156649015329...</apll> — limiting difference between the harmonic series and the natural logarithm) raised to an '''E'''xponent, that is, <apll>M∙γ<sup>E</sup></apll> or <apll>M×γ*E</apll>. The numbers to the left (multiplier) and right (exponent) of the <apll><pn>g</pn></apll> may be represented in several ways including integers, <b>Decimal</b>, <b>Exponential</b>, <b>Complex</b>, <b>Hypercomplex</b>, <b>Rational</b>, or <b>Variable Precision Floating</b> Point Notation, but not <b>Base</b>, <b>Euler</b>, <b>Pi</b>, <b>Gamma</b>, or <b>Zeta</b> Point Notation.

	For example, <apll>1<_e/>2<_g/>1.1</apll> is the same as <apll>100<_g/>1.1</apll>, and <apll>1<_r/>2<_g/>1.1<_e/>2</apll> is the same as <apll>0.5<_g/>110</apll>.		For example, <apll>1<_e/>2<_g/>1.1</apll> is the same as <apll>100<_g/>1.1</apll>, and <apll>1<_r/>2<_g/>1.1<_e/>2</apll> is the same as <apll>0.5<_g/>110</apll>.

WikiSysop: /* Pi Point Notation */

2022-09-09T18:24:14Z

Pi Point Notation

← Older revision		Revision as of 14:24, 9 September 2022
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	== Pi Point Notation ==		== Pi Point Notation ==

	This notation allows you to enter numeric constants that are in the form of the product of a '''M'''ultiplier and <apll>π</apll> (<apll>&cong; 3.141592653589793...</apll> — ratio of a circle's circumference and diameter) raised to an '''E'''xponent, that is, <apll>M∙π<sup>E</sup></apll> or <apll>M×(○1)*E</apll>. The numbers to the left (multiplier) and right (exponent) of the <apll><_pi/></apll> may be represented in several ways including integers, <b>Decimal</b>, <b>Exponential</b>, ~~'''~~Complex~~'''~~, <b>Hypercomplex</b>, <b>Rational</b>, or <b>Variable Precision Floating</b> Point Notation, but not <b>Base</b>, <b>Euler</b>, <b>Pi</b>, <b>Gamma</b>, or <b>Zeta</b> Point Notation.		This notation allows you to enter numeric constants that are in the form of the product of a '''M'''ultiplier and <apll>π</apll> (<apll>&cong; 3.141592653589793...</apll> — ratio of a circle's circumference and diameter) raised to an '''E'''xponent, that is, <apll>M∙π<sup>E</sup></apll> or <apll>M×(○1)*E</apll>. The numbers to the left (multiplier) and right (exponent) of the <apll><_pi/></apll> may be represented in several ways including integers, <b>Decimal</b>, <b>Exponential</b>, <b>Complex</b>, <b>Hypercomplex</b>, <b>Rational</b>, or <b>Variable Precision Floating</b> Point Notation, but not <b>Base</b>, <b>Euler</b>, <b>Pi</b>, <b>Gamma</b>, or <b>Zeta</b> Point Notation.

	For example, <apll>1<_e/>2<_pi/>1.1</apll> is the same as <apll>100<_pi/>1.1</apll>, and <apll>1<_r/>2<_pi/>1.1<_e/>2</apll> is the same as <apll>0.5<_pi/>110</apll>.		For example, <apll>1<_e/>2<_pi/>1.1</apll> is the same as <apll>100<_pi/>1.1</apll>, and <apll>1<_r/>2<_pi/>1.1<_e/>2</apll> is the same as <apll>0.5<_pi/>110</apll>.