Div, Mod , Gcd , Lcm , <<, >> , FromBase, ToBase , Precision , GetPrecision , N , Rationalize , IsPrime , IsPrimePower , Factors , Factor , PAdicExpand , ContFrac , Decimal , TruncRadian , Floor , Ceil , Round , Pslq .

Other operations on numbers

Besides the arithmetical operations defined in the previous chapter, one can also use the following operations on numbers. Many of these also work on polynomials.

Div, Mod Division with remainder
Gcd Greatest common divisor
Lcm Least common multiple
<<, >> Shift operators
FromBase, ToBase Conversion from/to non-decimal base
Precision Sets the precision
GetPrecision Returns the current precision
N Numerical approximation
Rationalize Convert floating point numbers to fractions
IsPrime Test whether argument is a prime number
IsPrimePower Test whether argument is a prime power
Factors Factorization
Factor Factorization, in pretty form
PAdicExpand p-adic expansion
ContFrac Continued fraction expansion
Decimal Decimal representation of a rational
TruncRadian Remainder modulo 2*Pi
Floor Round a number downwards
Ceil Round a number upwards
Round Round a number to the nearest integer
Pslq Search for integer relations between reals

Div, Mod -- Division with remainder

Standard math library
Calling Sequence:
Div(x,y)
Mod(x,y)
Parameters:
x, y - integers, or univariate polynomials
Description:
Div performs integer division and Mod returns the remainder after division. Div and Mod are also defined for polynomials.

If Div(x,y) returns "a" and Mod(x,y) equals "b", then these numbers satisfy x = a*y + b and 0 <= b < y.
Examples: 
In> Div(5,3)
Out> 1;
In> Mod(5,3)
Out> 2;
See Also:
Gcd , Lcm .

Gcd -- Greatest common divisor

Standard math library
Calling Sequence:
Gcd(n,m)
Gcd(list)
Parameters:
n,m - integers or univariate polynomials
list - a list of all integers or all univariate polynomials
Description:
This function returns the greatest common divisor of "n" and "m". The gcd is the largest number that divides "n" and "m". It is also known as the highest common factor (hcf). The library code calls MathGcd, which is an internal function. This function implements the binary Euclidean algorithm for determining the greatest common divisor: 
Routine for calculating Gcd(n,m)

1) if n = m then return n
2) if both n and m are even then return 2*Gcd(n/2,m/2)
3) if exactly one of n or m (say n) is even then return Gcd(n/2,m)
4) if both n and m are odd and, say, n>m then return Gcd( (n-m)/2,m)
This is a rather fast algorithm on computers that can efficiently shift integers.

If the second calling form is used, Gcd will return the greatest common divisor of all the integers or polynomials in "list". It uses the identity 
Gcd({a,b,c}) = Gcd(Gcd(a,b),c)
Examples: 
In> Gcd(55,10)
Out> 5;
In> Gcd({60,24,120})
Out> 12;
See Also:
Lcm .

Lcm -- Least common multiple

Standard math library
Calling Sequence:
Lcm(n,m)
Parameters:
n, m - integers, or univariate polynomials
Description:
This command returns the least common multiple of "n" and "m". The least common multiple of two numbers "n" and "m" is the lowest number which is an integer multiple of both "n" and "m". It is calculated with the formula: 
Lcm(n,m) = Div(n*m,Gcd(n,m))
This means it also works on polynomials, since Div, Gcd and multiplication are also defined for them.
Examples: 
In> Lcm(60,24)
Out> 120;
See Also:
Gcd .

<<, >> -- Shift operators

Standard math library
Calling Sequence:
n<<m
n>>m
Parameters:
n,m - integers
Description:
These operators shift integers to the left or to the right. They are similar to the C shift operators. These are sign-extended shifts, so they act like multiplication or division by powers of 2.
Examples: 
In> 1 << 10
Out> 1024;
In> -1024 >> 10
Out> -1;

FromBase, ToBase -- Conversion from/to non-decimal base

Internal function
Calling Sequence:
FromBase(base,number)
ToBase(base,number)
Parameters:
base - a base to write the numbers in
number - a number to write out in the base representation
Description:
FromBase converts "number", written in base "base", to base 10. ToBase converts "number", written in base 10, to base "base".

These functions use the p-adic expansion capabilities of the built-in arbitrary precision math libraries.
Examples: 
In> FromBase(2,111111)
Out> 63;
In> ToBase(16,255)
Out> ff;
The first command writes the binary number 111111 in decimal base. The second command converts 255 (in decimal base) to hexadecimal base.
See Also:
PAdicExpand .

Precision -- Sets the precision

Internal function
Calling Sequence:
Precision(n)
Parameters:
n - new precision
Description:
This command sets the number of binary digits to be used in calculations. All subsequent floating point operations will allow for at least "n" digits after the decimal point.
Examples: 
In> Precision(10)
Out> True;
In> N(Sin(1))
Out> 0.8414709848;
In> Precision(20)
Out> True;
In> N(Sin(1))
Out> 0.84147098480789650665;
In> GetPrecision()
Out> 20;
See Also:
GetPrecision , N .

GetPrecision -- Returns the current precision

Internal function
Calling Sequence:
GetPrecision()
Parameters:
none
Description:
This command returns the current precision, as set by Precision.
Examples: 
In> GetPrecision();
Out> 10;
In> Precision(20);
Out> True;
In> GetPrecision();
Out> 20;
See Also:
Precision , N .

N -- Numerical approximation

Standard math library
Calling Sequence:
N(expr)
N(expr, prec)
Parameters:
expr - expression to evaluate
prec - precision to use
Description:
This function forces Yacas to give a numerical approximation to the expression "expr", using "prec" digits if the second calling sequence is used, and the precision as set by SetPrecision otherwise. This overrides the normal behaviour, in which expressions are kept in symbolic form (eg. Sqrt(2) instead of 1.41421).

Application of the N operator will make Yacas calculate floating point representations of functions whenever possible. In addition, the variable Pi is bound to the value of pi up to the required precision.
Examples: 
In> 1/2
Out> 1/2;
In> N(1/2)
Out> 0.5;
In> Sin(1)
Out> Sin(1);
In> N(Sin(1),10)
Out> 0.8414709848;
In> Pi
Out> Pi;
In> N(Pi,20)
Out> 3.14159265358979323846;
See Also:
Precision , GetPrecision , Pi .

Rationalize -- Convert floating point numbers to fractions

Standard math library
Calling Sequence:
Rationalize(expr)
Parameters:
expr - an expression containing real numbers
Description:
This command converts every real number in the expression "expr" into a rational number. This is useful when a calculation needs to be done on floating point numbers and the algorithm is unstable. Converting the floating point numbers to rational numbers will force calculations to be done with infinite precision (by using rational numbers as representations).

It does this by finding the smallest integer n such that multiplying the number with 10^n is an integer. Then it divides by 10^n again, depending on the internal gcd calculation to reduce the resulting division of integers.
Examples: 
In> {1.2,3.123,4.5}
Out> {1.2,3.123,4.5};
In> Rationalize(%)
Out> {6/5,3123/1000,9/2};
See Also:
IsRational .

IsPrime -- Test whether argument is a prime number

Standard math library
Calling sequence:
IsPrime(n)
Parameters:
n - integer to test
Description:
This command tests whether "n", which should be a positive integer, is a prime number. A number is a prime number if it is only divisible by 1 and itself. As a special case, 1 is not a prime number.

This function essentially checks for all integers between 2 and the square root of "n" whether they divide "n", and hence may take a long time for large numbers.
Examples: 
In> IsPrime(1)
Out> False;
In> IsPrime(2)
Out> True;
In> IsPrime(10)
Out> False;
In> IsPrime(23)
Out> True;
In> Select("IsPrime", 1 .. 100)
Out> {2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97};
See Also:
IsPrimePower , Factors .

IsPrimePower -- Test whether argument is a prime power

Standard math library
Calling sequence:
IsPrime(n)
Parameters:
n - integer to test
Description:
This command tests whether "n", which should be a positive integer, is a prime power, that is whether it is of the form "p^m", with "p" prime and "m" an integer.

This function essentially checks for all integers between 2 and the square root of "n" for the largest divisor, and then tests whether "n" is a power of this divisor. So it will take a long time for large numbers.
Examples: 
In> IsPrimePower(9)
Out> True;
In> IsPrimePower(10)
Out> False;
In> Select("IsPrimePower", 1 .. 50)
Out> {2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,31,32,37,41,43,47,49};
See Also:
IsPrime , Factors .

Factors -- Factorization

Standard math library
Calling sequence:
Factors(x)
Parameters:
x - integer or univariate polynomial
Description:
This function decomposes the integer number "x" into a product of numbers. Alternatively, if "x" is a univariate polynomial, it is decomposed in irreducible polynomials.

The factorization is returned as a list of pairs. The first member of each pair is the factor, while the second member denotes the power to which this factor should be raised. So the factorization "x = p1^n1 * ... * p9^n9" is returned as {{p1,n1}, ..., {p9,n9}}.
Examples: 
In> Factors(24);
Out> {{2,3},{3,1}};
In> Factors(2*x^3 + 3*x^2 - 1);
Out> {{2,1},{x+1,2},{x-1/2,1}};
See Also:
Factor , IsPrime .

Factor -- Factorization, in pretty form

Standard math library
Calling sequence:
Factors(x)
Parameters:
x - integer or univariate polynomial
Description:
This function factorizes "x", like Factors, but it shows the result in a nicer human readable format.
Examples: 
In> PrettyForm(Factor(24));

 3    
2  * 3

Out> True;
In> PrettyForm(Factor(2*x^3 + 3*x^2 - 1));

             2   /     1 \
2 * ( x + 1 )  * | x - - |
                 \     2 /

Out> True;
See Also:
Factors , IsPrime , PrettyForm .

PAdicExpand -- p-adic expansion

Standard math library
Calling sequence:
PAdicExpand(n, p)
Parameters:
n - number, or polynomial, to expand
p - base to expand in
Description:
This command computes the p-adic expansion of "n". In other words, "n" is expanded in powers of "p". The argument "n" can be either an integer or a univariate polynomial. The base "p" should be of the same type.
Examples: 
In> PrettyForm(PAdicExpand(1234, 10));

               2     3    
3 * 10 + 2 * 10  + 10  + 4

Out> True;
In> PrettyForm(PAdicExpand(x^3, x-1));

                             2            3    
3 * ( x - 1 ) + 3 * ( x - 1 )  + ( x - 1 )  + 1

Out> True;
See Also:
Mod , ContFrac , FromBase, ToBase .

ContFrac -- Continued fraction expansion

Standard math library
Calling Sequence:
ContFrac(x)
ContFrac(x, maxdepth)
Parameters:
x - expression to break down
maxdepth - maximum required depth of result
Description:
This command returns the continued fraction expansion of "n", which should be either a floating point number or a polynomial. If "maxdepth" is not specified, it defaults to 6. The remainder is denoted by rest.

This is especially useful for polynomials, since series expansions that converge slowly will typically converge a lot faster if calculated using a continued fraction expansion.
Examples: 
In> PrettyForm(ContFrac(N(Pi)))

             1                 
--------------------------- + 3
           1                   
----------------------- + 7    
        1                      
------------------ + 15        
      1                        
-------------- + 1             
   1                           
-------- + 292                 
rest + 1                       

Out> True;
In> PrettyForm(ContFrac(x^2+x+1, 3))

       x            
---------------- + 1
         x          
1 - ------------    
       x            
    -------- + 1    
    rest + 1        

Out> True;
See Also:
PAdicExpand , N .

Decimal -- Decimal representation of a rational

Standard math library
Calling Sequence:
Decimal(frac)
Parameters:
frac - a rational number
Description:
This function returns the infinite decimal representation of the rational number "frac". It returns a list, with the first element being the number before the decimal point and the last element the sequence of digits that will repeat forever. All the intermediate list elements are the initial digits.
Examples: 
In> Decimal(1/22)
Out> {0,0,{4,5}};
In> N(1/22,30)
Out> 0.045454545454545454545454545454;
See Also:
N .

TruncRadian -- Remainder modulo 2*Pi

Standard math library
Calling Sequence:
TruncRadian(r)
Parameters:
r - a radian
Description:
TruncRadian calculates r mod 2*Pi, returning a value between 0 and 2*Pi. This function is used in the trigonometry functions, just before doing the numerical calculation. It greatly speeds up the calculation if the value passed is a big number.

The library uses the formula 

             /   r    \         
r - MathFloor| ------ | * 2 * Pi
             \ 2 * Pi /
where r and 2*Pi are calculated with twice the precision used in the environment to make sure there is no rounding error in the significant digits.
Examples: 
In> 2*Pi()
Out> 6.283185307;
In> TruncRadian(6.28)
Out> 6.28;
In> TruncRadian(6.29)
Out> 0.0068146929;
See Also:
Sin , Cos , Tan .

Floor -- Round a number downwards

Standard math library
Calling Sequence:
Floor(x)
Parameters:
x - a number
Description:
This function returns the largest integer smaller than "x".
Examples: 
In> Floor(1.1)
Out> 1;
In> Floor(-1.1)
Out> -2;
See Also:
Ceil , Round .

Ceil -- Round a number upwards

Standard math library
Calling Sequence:
Ceil(x)
Parameters:
x - a number
Description:
This function returns the smallest integer larger than "x".
Examples: 
In> Ceil(1.1)
Out> 2;
In> Ceil(-1.1)
Out> -1;
See Also:
Floor , Round .

Round -- Round a number to the nearest integer

Standard math library
Calling Sequence:
Round(x)
Parameters:
x - a number
Description:
This function returns the integer closest to "x". Half-integers (ie. numbers of the form "n + 0.5", with "n" an integers) are rounded upwards.
Examples: 
In> Round(1.49)
Out> 1;
In> Round(1.51)
Out> 2;
In> Round(-1.49)
Out> -1;
In> Round(-1.51)
Out> -2;
See Also:
Floor , Ceil .

Pslq -- Search for integer relations between reals

Standard math library
Calling Sequence:
Pslq(xlist,precision)
Parameters:
xlist - list of numbers
precision - required number of digits precision of calculation
Description:
This function is an integer relation detection algorithm. This means that, given the numbers "x[i]" in the list "xlist", it tries to find integer coefficients "a[i]" such that "a[1]*x[1] + ... + a[n]*x[n] = 0". The list of integer coefficients is returned.

The numbers in "xlist" must evaluate to floating point numbers if the N operator is applied on them.
Examples: 
In> Pslq({ 2*Pi+3*Exp(1) , Pi , Exp(1) },20)
Out> {1,-2,-3};
Note: in this example the system detects correctly that 1 * (2*Pi+3*E) + (-2) * (Pi) + (-3) * (E) = 0
See Also:
N .