OpenMath Content Dictionary: nums1

Canonical URL:

http://www.openmath.org/cd/nums1.ocd

CD Base:

http://www.openmath.org/cd

CD File:

nums1.ocd

CD as XML Encoded OpenMath:

nums1.omcd

Defines:

based_integer, e, gamma, i, infinity, NaN, pi, rational

Date:

2004-03-30

Version:

3

Review Date:

2006-03-30

Status:

official

     This document is distributed in the hope that it will be useful, 
     but WITHOUT ANY WARRANTY; without even the implied warranty of 
     MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

     The copyright holder grants you permission to redistribute this 
     document freely as a verbatim copy. Furthermore, the copyright
     holder permits you to develop any derived work from this document
     provided that the following conditions are met.
       a) The derived work acknowledges the fact that it is derived from
          this document, and maintains a prominent reference in the 
          work to the original source.
       b) The fact that the derived work is not the original OpenMath 
          document is stated prominently in the derived work.  Moreover if
          both this document and the derived work are Content Dictionaries
          then the derived work must include a different CDName element,
          chosen so that it cannot be confused with any works adopted by
          the OpenMath Society.  In particular, if there is a Content 
          Dictionary Group whose name is, for example, `math' containing
          Content Dictionaries named `math1', `math2' etc., then you should 
          not name a derived Content Dictionary `mathN' where N is an integer.
          However you are free to name it `private_mathN' or some such.  This
          is because the names `mathN' may be used by the OpenMath Society
          for future extensions.
       c) The derived work is distributed under terms that allow the
          compilation of derived works, but keep paragraphs a) and b)
          intact.  The simplest way to do this is to distribute the derived
          work under the OpenMath license, but this is not a requirement.
     If you have questions about this license please contact the OpenMath
     society at http://www.openmath.org.

This CD is intended to be `compatible' with the MathML view of constructors for numbers (integers to an arbitrary base, rationals) and symbols for some common numerical constants.

This CD holds a set of symbols for creating numbers, including some defined constants (i.e. nullary constructors).

based_integer

Role:

application

Description:

This symbol represents the constructor function for integers, specifying the base. It takes two arguments, the first is a positive integer to denote the base to which the number is represented, the second argument is a string which contains an optional sign and the digits of the integer, using 0-9a-z (as a consequence of this no radix greater than 35 is supported). Base 16 and base 10 are already covered in the encodings of integers.

Example:

A representation of 8 (radix 10) base 8

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd"><OMA>
  <OMS cd="relation1" name="eq"/>
  <OMI> 8 </OMI>
  <OMA>
    <OMS cd="nums1" name="based_integer"/>
    <OMI> 8 </OMI>
    <OMSTR> 10 </OMSTR>
  </OMA>

</OMA></OMOBJ>

eq ( 8 , based_integer ( 8 , " 10 " ) )

$$8=8_{\text{10}}$$

Signatures:

sts

rational

Role:

application

Description:

This symbol represents the constructor function for rational numbers. It takes two arguments, the first is an integer p to denote the numerator and the second a nonzero integer q to denote the denominator of the rational p/q.

Example:

A representation of the rational number 1/2

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd"><OMA>
  <OMS cd="nums1" name="rational"/>
  <OMI> 1 </OMI>
  <OMI> 2 </OMI>
</OMA></OMOBJ>

rational ( 1 , 2 )

$$\frac{1}{2}$$

Signatures:

sts

infinity

Role:

constant

Description:

A symbol to represent the notion of infinity.

Commented Mathematical property (CMP):

if x is a real number then x < infinity

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="logic1" name="implies"/>
  <OMA>
    <OMS cd="set1" name="in"/>
    <OMV name="x"/>
    <OMS cd="setname1" name="R"/>
  </OMA>
  <OMA>
    <OMS cd="relation1" name="lt"/>
    <OMV name="x"/>
    <OMS cd="nums1" name="infinity"/>
  </OMA>
</OMA>
</OMOBJ>

implies (in ( x, R) , lt ( x, infinity) )

$$x\in \mathbb{R}\Rightarrow x<\infty$$

Signatures:

sts

e

Role:

constant

Description:

This symbol represents the base of the natural logarithm, approximately 2.718. See Abramowitz and Stegun, Handbook of Mathematical Functions, section 4.1.

Commented Mathematical property (CMP):

e = the sum as j ranges from 0 to infinity of 1/(j!)

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMS cd="nums1" name="e"/>
  <OMA>
    <OMS cd="arith1" name="sum"/>
    <OMA>
      <OMS cd="interval1" name="integer_interval"/>
      <OMS cd="alg1" name="zero"/>
      <OMS cd="nums1" name="infinity"/>
    </OMA>
    <OMBIND>
      <OMS cd="fns1" name="lambda"/>
      <OMBVAR>
        <OMV name="j"/>
      </OMBVAR>
      <OMA>
        <OMS cd="arith1" name="divide"/>
	<OMS cd="alg1" name="one"/>
	<OMA>
	  <OMS cd="integer1" name="factorial"/>
	  <OMV name="j"/>
	</OMA>
      </OMA>
    </OMBIND>
  </OMA>
</OMA>
</OMOBJ>

eq (e, sum (integer_interval (zero, infinity) , lambda [ j ] . (divide (one, factorial ( j) ) ) ) )

$$e=\sum _{j=0}^{\infty }\frac{1}{j!}$$

Example:

2.718 = The decimal approximation to 3 significant places of e

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
    <OMA>
      <OMS cd="relation1" name="approx"/>
      <OMF dec="2.718"/>
      <OMS cd="nums1" name="e"/>
    </OMA>
  </OMOBJ>

approx ( 2.718 , e)

$$2.718\approx e$$

Signatures:

sts

i

Role:

constant

Description:

This symbol represents the square root of -1.

Commented Mathematical property (CMP):

i^2 = -1

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
    <OMA>
      <OMS cd="relation1" name="eq"/>
      <OMA>
        <OMS cd="arith1" name="power"/>
        <OMS cd="nums1" name="i"/>
        <OMI> 2 </OMI>
      </OMA>
      <OMA>
        <OMS cd="arith1" name="unary_minus"/>
        <OMS cd="alg1" name="one"/>
      </OMA>
    </OMA>
  </OMOBJ>

eq (power (i, 2 ) , unary_minus (one) )

$$i^2=-1$$

Signatures:

sts

pi

Role:

constant

Description:

A symbol to convey the notion of pi, approximately 3.142. The ratio of the circumference of a circle to its diameter.

Commented Mathematical property (CMP):

pi = 4 * the sum as j ranges from 0 to infinity of ((1/(4j+1))-(1/(4j+3)))

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMS cd="nums1" name="pi"/>
  <OMA>
    <OMS cd="arith1" name="times"/>
    <OMI> 4 </OMI>
    <OMA>
      <OMS cd="arith1" name="sum"/>
      <OMA>
        <OMS cd="interval1" name="integer_interval"/>
        <OMS cd="alg1" name="zero"/>
        <OMS cd="nums1" name="infinity"/>
      </OMA>
      <OMBIND>
        <OMS cd="fns1" name="lambda"/>
        <OMBVAR>
          <OMV name="j"/>
        </OMBVAR>
        <OMA>
          <OMS cd="arith1" name="minus"/>
	  <OMA>
	    <OMS cd="arith1" name="divide"/>
	    <OMS cd="alg1" name="one"/>
	    <OMA>
	      <OMS cd="arith1" name="plus"/>
	      <OMA>
	        <OMS cd="arith1" name="times"/>
	        <OMI> 4 </OMI>
	        <OMV name="j"/>
	      </OMA>
	      <OMS cd="alg1" name="one"/>
	    </OMA>
	  </OMA>
	  <OMA>
	    <OMS cd="arith1" name="divide"/>
	    <OMS cd="alg1" name="one"/>
	    <OMA>
	      <OMS cd="arith1" name="plus"/>
	      <OMA>
	        <OMS cd="arith1" name="times"/>
  	        <OMI> 4 </OMI>
	        <OMV name="j"/>
	      </OMA>
	      <OMI> 3 </OMI>
	    </OMA>
	  </OMA>
        </OMA>
      </OMBIND>
    </OMA>
  </OMA>
</OMA>
</OMOBJ>

eq (pi, times ( 4 , sum (integer_interval (zero, infinity) , lambda [ j ] . (minus (divide (one, plus (times ( 4 , j) , one) ) , divide (one, plus (times ( 4 , j) , 3 ) ) ) ) ) ) )

$$\pi =4\sum _{j=0}^{\infty }\frac{1}{4j+1}-\frac{1}{4j+3}$$

Example:

3.142 = The decimal approximation to 3 significant places of pi

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
    <OMA>
      <OMS cd="relation1" name="approx"/>
      <OMF dec="3.142"/>
      <OMS cd="nums1" name="pi"/>
    </OMA>
  </OMOBJ>

approx ( 3.142 , pi)

$$3.142\approx \pi$$

Signatures:

sts

gamma

Role:

constant

Description:

A symbol to convey the notion of the gamma constant as defined in Abramowitz and Stegun, Handbook of Mathematical Functions, section 6.1.3. It is the limit of 1 + 1/2 + 1/3 + ... + 1/m - ln m as m tends to infinity, this is approximately 0.5772 15664.

Commented Mathematical property (CMP):

gamma = limit_(m -> infinity)(sum_(j ranges from 1 to m)(1/j) - ln m)

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
<OMA>
  <OMS cd="relation1" name="eq"/>
  <OMS cd="nums1" name="gamma"/>
  <OMA>
    <OMS cd="limit1" name="limit"/>
    <OMS cd="nums1" name="infinity"/>
    <OMS cd="limit1" name="below"/>
    <OMBIND>
      <OMS cd="fns1" name="lambda"/>
      <OMBVAR>
        <OMV name="m"/>
      </OMBVAR>
      <OMA>
        <OMS cd="arith1" name="minus"/>
        <OMA>
          <OMS cd="arith1" name="sum"/>
	  <OMA>
	    <OMS cd="interval1" name="integer_interval"/>
	    <OMS cd="alg1" name="one"/>
	    <OMV name="m"/>
	  </OMA>
	  <OMBIND>
            <OMS cd="fns1" name="lambda"/>
            <OMBVAR>
              <OMV name="j"/>
            </OMBVAR>
	    <OMA>
              <OMS cd="arith1" name="divide"/>
              <OMI> 1 </OMI>
              <OMV name="j"/>
	    </OMA>
	  </OMBIND>
        </OMA>
        <OMA>
          <OMS cd="transc1" name="ln"/>
          <OMV name="m"/>
        </OMA>
      </OMA>
    </OMBIND>
  </OMA>
</OMA>
</OMOBJ>

eq (gamma, limit (infinity, below, lambda [ m ] . (minus (sum (integer_interval (one, m) , lambda [ j ] . (divide ( 1 , j) ) ) , ln ( m) ) ) ) )

$$\gamma =\underset{m\to {\infty }_{-}}{\mathrm{limit}}\sum _{j=1}^m\frac{1}{j}-\ln \left(m\right)$$

Example:

0.577 = The decimal approximation to 3 significant places of gamma

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
    <OMA>
      <OMS cd="relation1" name="approx"/>
      <OMF dec="0.577"/>
      <OMS cd="nums1" name="gamma"/>
    </OMA>
  </OMOBJ>

approx ( 0.577 , gamma)

$$0.577\approx \gamma$$

Signatures:

sts

NaN

Role:

constant

Description:

A symbol to convey the notion of not-a-number. The result of an ill-posed floating computation. See IEEE standard for floating point representations.

Commented Mathematical property (CMP):

NaN is not equal to NaN

Formal Mathematical property (FMP):

xml prefix mathml

<OMOBJ xmlns="http://www.openmath.org/OpenMath" version="2.0" cdbase="http://www.openmath.org/cd">
  <OMA>
    <OMS cd="relation1" name="neq"/>
    <OMS cd="nums1" name="NaN"/>
    <OMS cd="nums1" name="NaN"/>
  </OMA>
</OMOBJ>

neq (NaN, NaN)

$$\mathrm{NaN}\ne \mathrm{NaN}$$

Signatures:

sts