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# OpenMath Content Dictionary: arith1

Canonical URL:
http://www.openmath.org/cd/arith1.ocd
CD Base:
http://www.openmath.org/cd
CD File:
arith1.ocd
CD as XML Encoded OpenMath:
arith1.omcd
Defines:
abs, divide, gcd, lcm, minus, plus, power, product, root, sum, times, unary_minus
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.
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.
society at http://www.openmath.org.


This CD defines symbols for common arithmetic functions.

## lcm

Role:
application
Description:

The symbol to represent the n-ary function to return the least common multiple of its arguments.

Commented Mathematical property (CMP):
lcm(a,b) = a*b/gcd(a,b)
Formal Mathematical property (FMP):
$\mathrm{lcm}\left(a,b\right)=\frac{ab}{\mathrm{gcd}\left(a,b\right)}$
Commented Mathematical property (CMP):
for all integers a,b | There does not exist a c>0 such that c/a is an Integer and c/b is an Integer and lcm(a,b) > c.
Formal Mathematical property (FMP):
$\forall a,b.a\in \mathbb{Z}\wedge b\in \mathbb{Z}⇒¬\exists c.c>0\wedge \mathrm{factorof}\left(a,c\right)\wedge \mathrm{factorof}\left(b,c\right)\wedge c<\mathrm{lcm}\left(a,b\right)$
Signatures:
sts

 [Next: gcd] [Last: product] [Top]

## gcd

Role:
application
Description:

The symbol to represent the n-ary function to return the gcd (greatest common divisor) of its arguments.

Commented Mathematical property (CMP):
for all integers a,b | There does not exist a c such that a/c is an Integer and b/c is an Integer and c > gcd(a,b). Note that this implies that gcd(a,b) > 0
Formal Mathematical property (FMP):
$\forall a,b.a\in \mathbb{Z}\wedge b\in \mathbb{Z}⇒¬\exists c.\frac{a}{c}\in \mathbb{Z}\wedge \frac{b}{c}\in \mathbb{Z}\wedge c>\mathrm{gcd}\left(a,b\right)$
Example:
gcd(6,9) = 3
$\mathrm{gcd}\left(6,9\right)=3$
Signatures:
sts

 [Next: plus] [Previous: lcm] [Top]

## plus

Role:
application
Description:

The symbol representing an n-ary commutative function plus.

Commented Mathematical property (CMP):
for all a,b | a + b = b + a
Formal Mathematical property (FMP):
$\forall a,b.a+b=b+a$
Signatures:
sts

 [Next: unary_minus] [Previous: gcd] [Top]

## unary_minus

Role:
application
Description:

This symbol denotes unary minus, i.e. the additive inverse.

Commented Mathematical property (CMP):
for all a | a + (-a) = 0
Formal Mathematical property (FMP):
$\forall a.a-a=0$
Signatures:
sts

 [Next: minus] [Previous: plus] [Top]

## minus

Role:
application
Description:

The symbol representing a binary minus function. This is equivalent to adding the additive inverse.

Commented Mathematical property (CMP):
for all a,b | a - b = a + (-b)
Formal Mathematical property (FMP):
$\forall a,b.a-b=a-b$
Signatures:
sts

 [Next: times] [Previous: unary_minus] [Top]

## times

Role:
application
Description:

The symbol representing an n-ary multiplication function.

Example:
$\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)\left(\begin{array}{cc}5& 6\\ 7& 8\end{array}\right)=\left(\begin{array}{cc}19& 22\\ 43& 50\end{array}\right)$
Commented Mathematical property (CMP):
for all a,b | a * 0 = 0 and a * b = a * (b - 1) + a
Formal Mathematical property (FMP):
$\forall a,b.a0=0\wedge ab=a\left(b-1\right)+a$
Commented Mathematical property (CMP):
for all a,b,c | a*(b+c) = a*b + a*c
Formal Mathematical property (FMP):
$\forall a,b,c.a\left(b+c\right)=ab+ac$
Signatures:
sts

 [Next: divide] [Previous: minus] [Top]

## divide

Role:
application
Description:

This symbol represents a (binary) division function denoting the first argument right-divided by the second, i.e. divide(a,b)=a*inverse(b). It is the inverse of the multiplication function defined by the symbol times in this CD.

Commented Mathematical property (CMP):
whenever not(a=0) then a/a = 1
Formal Mathematical property (FMP):
$\forall a.a\ne 0⇒\frac{a}{a}=1$
Signatures:
sts

 [Next: power] [Previous: times] [Top]

## power

Role:
application
Description:

This symbol represents a power function. The first argument is raised to the power of the second argument. When the second argument is not an integer, powering is defined in terms of exponentials and logarithms for the complex and real numbers. This operator can represent general powering.

Commented Mathematical property (CMP):
x\in C implies x^a = exp(a ln x)
Formal Mathematical property (FMP):
$x\in \mathbb{C}⇒{x}^{a}=\mathrm{exp}\left(a\mathrm{ln}\left(x\right)\right)$
Commented Mathematical property (CMP):
if n is an integer then x^0 = 1, x^n = x * x^(n-1)
Formal Mathematical property (FMP):
$n\in \mathbb{Z}⇒{x}^{0}=1\wedge {x}^{n}=x{x}^{\left(n-1\right)}$
Example:
${\left(\begin{array}{cc}1& 2\\ 3& 4\end{array}\right)}^{3}=\left(\begin{array}{cc}37& 54\\ 81& 118\end{array}\right)$
Example:
${e}^{\left(i\pi \right)}=-1$
Signatures:
sts

 [Next: abs] [Previous: divide] [Top]

## abs

Role:
application
Description:

A unary operator which represents the absolute value of its argument. The argument should be numerically valued. In the complex case this is often referred to as the modulus.

Commented Mathematical property (CMP):
for all x,y | abs(x) + abs(y) >= abs(x+y)
Formal Mathematical property (FMP):
$\forall x,y.|x|+|y|\ge |x+y|$
Signatures:
sts

 [Next: root] [Previous: power] [Top]

## root

Role:
application
Description:

A binary operator which represents its first argument "lowered" to its n'th root where n is the second argument. This is the inverse of the operation represented by the power symbol defined in this CD.

Care should be taken as to the precise meaning of this operator, in particular which root is represented, however it is here to represent the general notion of taking n'th roots. As inferred by the signature relevant to this symbol, the function represented by this symbol is the single valued function, the specific root returned is the one indicated by the first CMP. Note also that the converse of the second CMP is not valid in general.

Commented Mathematical property (CMP):
x\in C implies root(x,n) = exp(ln(x)/n)
Formal Mathematical property (FMP):
$x\in \mathbb{C}⇒\sqrt[n]{x}=\mathrm{exp}\left(\frac{\mathrm{ln}\left(x\right)}{n}\right)$
Commented Mathematical property (CMP):
for all a,n | power(root(a,n),n) = a (if the root exists!)
Formal Mathematical property (FMP):
$\forall a,n.{\sqrt[n]{a}}^{n}=a$
Signatures:
sts

 [Next: sum] [Previous: abs] [Top]

## sum

Role:
application
Description:

An operator taking two arguments, the first being the range of summation, e.g. an integral interval, the second being the function to be summed. Note that the sum may be over an infinite interval.

Example:
This represents the summation of the reciprocals of all the integers between 1 and 10 inclusive.
$\sum _{x=1}^{10}\frac{1}{x}$
Signatures:
sts

 [Next: product] [Previous: root] [Top]

## product

Role:
application
Description:

An operator taking two arguments, the first being the range of multiplication e.g. an integral interval, the second being the function to be multiplied. Note that the product may be over an infinite interval.

Example:
This represents the statement that the factorial of n is equal to the product of all the integers between 1 and n inclusive.
$n!=\prod _{i=1}^{n}i$
Signatures:
sts

 [First: lcm] [Previous: sum] [Top]

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