Maxima/Numbers

Check

(%i1) load(to_poly);
(%o1)  home/a/maxima/share/to_poly_solve/to_poly.lisp
(%i2) r;
(%o2) r 
(%i3) isreal_p(r);
(%o3) true     /*  When I use unspecified variables, Maxima seems to assume that they are real */ 
(%i4) z:x+y*%i;
(%o4) %i y + x
(%i5) isreal_p(z);
(%o5) isreal_p(%i y) /* maxima can't check if it is real or not */
(%i6) isreal_p(x);
(%o6) true
(%i7) isreal_p(y);
(%o7) true
(%i8) complex_number_p(z); 
(%o8) false 
(%i9) declare(z, complex);
(%o9) done
(%i10) complex_number_p(z);
(%o10)   false /* still not complex */                                                                                                              true

Random numbers

  bfloat(random(10^fpprec) / 10^fpprec); /* random bfloat with fpprec decimal digits in the range 0 to 1 */

Number types

Binary numbers

(%i1) ibase;
(%o1) 10
(%i2) obase;
(%o2) 10
(%i3) ibase:2;
(%o3) 2
(%i4) x=1001110;
(%o4) x=78

Complex numbers

Argument

Principial value of argument of complex number in turns carg produces results in the range (-pi, pi] . It can be mapped to [0, 2*pi) by adding 2*pi to the negative values

carg_t(z):=
 block(
 [t],
 t:carg(z)/(2*%pi),  /* now in turns */
 if t<0 then t:t+1, /* map from (-1/2,1/2] to [0, 1) */
 return(t)
)$

On can order list of complex points according to it's argument :

l2_ordered: sort(l2, lambda([z1,z2], is(carg(z1) < carg(z2))))$

rational numbers

rat
ratp

Rationalize with limit denominator:^[1]

limit_denominator(x, max_denominator):=
block([p0, q0, p1, q1, n, d, a, q2, k, bound1, bound2, ratprint: false],
 [p0, q0, p1, q1]: [0, 1, 1, 0],
 [n, d]: ratexpand([ratnum(x), ratdenom(x)], 0),
 if d <= max_denominator then x else
 (catch(
   do block(
     a: quotient(n, d),
     q2: q0+a*q1,
     if q2 > max_denominator then throw('done),
     [p0, q0, p1, q1]: [p1, q1, p0+a*p1, q2],
     [n, d]: [d, n-a*d])),
 k: quotient(max_denominator-q0, q1),
 bound1: (p0+k*p1)/(q0+k*q1),
 bound2: p1/q1,
 if abs(bound2 - x) <= abs(bound1 - x) then bound2 else bound1))$

Predicate functions

(%i1) is(0=0.0);
(%o1) false

Compare :

(%i1) a:0.0$
(%i2)is(equal(a,0));
(%o2) true

List of predicate functions ( see p at the end ) :

abasep
alphacharp
alphanumericp
atom
bfloatp
blockmatrixp
cequal
cequalignore
cgreaterp
cgreaterpignore
charp
clessp
clesspignore
complex_number_p from to_poly package
constantp
constituent
diagmatrixp
digitcharp
disjointp
elementp
emptyp
evenp
featurep
floatnump ( compare : isreal_p)
if
integerp
intervalp
is
isreal_p from to_poly package
lcharp
listp
listp
lowercasep
mapatom
matrixp
matrixp
maybe
member
nonnegintegerp
nonscalarp
numberp
oddp
operatorp
ordergreatp
orderlessp
picture_equalp
picturep
poly_depends_p
poly_grobner_subsetp
polynomialp
prederror
primep
ratnump
ratp
scalarp
sequal
sequalignore
setequalp
setp
stringp
subsetp
subvarp
symbolp
symmetricp
taylorp
unknown
uppercasep
zeroequiv
zeromatrixp
zn_primroot_p

Number Theory

There are functions and operators useful with integer numbers

Elementary number theory

In Maxima there are some elementary functions like the factorial n! and the double factorial n!! defined as ${\frac {n!}{k!}}$ where $k$ is the greatest integer less than or equal to $n/2$

Divisibility

Some of the most important functions for integer numbers have to do with divisibility:

gcd, ifactor, mod, divisors...

all of them well documented in help. you can view it with the '?' command.

Function ifactors takes positive integer and returns a list of pairs : prime factor and its exponent. For example :

a:ifactors(8);
[[2,3]]

It means that : $8=2^{3}\,$

Other Functions

Continuus fractions :

(%i6) cfdisrep([1,1,1,1]);
(%o6) 1+1/(1+1/(1+1/1))
(%i7) float(%), numer;
(%o7) 1.666666666666667

[1] stackoverflow question : rationalize-number-with-limit-denominator

[2] xima manual : Category Declarations-and-inferences

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