NAME

CayleyDickson - create and operate with hypercomplex numbers

SYNOPSIS

use Tangle;
my $q1 = Tangle->new(1,0);
print "q1 = $q1\n";
$q1->x_gate;
print "X(q1) = $q1\n";
$q1->hadamard;
print "H(X(q1)) = $q1\n";

my $q2 = Tangle->new(1,0);
print "q2 = $q2\n";

# perform CNOT($q1 ⊗ $q2)
$q1->cnot($q2);

print "q1 = $q1\n";
print "q2 = $q2\n";

$q1->x_gate;
print "X(q1) = $q1\n";
print "entanglement causes q2 to automatically changed: $q2\n";

DESCRIPTION

Cayley-Dickson construction and operations are defined here: https://en.wikipedia.org/wiki/Cayley–Dickson_construction

This object provides natural and intuitive operations on these numbers by overriding the native numeric operations (+,-,/,*)

USAGE

new()

# create a new CayleyDickson number "i" ...
my $q1 = CayleyDickson->new(0,1);


# create a new CayleyDickson number "1+2i+3j+4k" ...
my $q2 = CayleyDickson->new(1,2,3,4);


# create a bigger CayleyDickson number (a Sedenion) ...
my $q3 = CayleyDickson->new(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16);


# create a CayleyDickson number from others ...
my $one  = CayleyDickson->new(0,1);
my $zero = CayleyDickson->new(1,0);
my $quaternion = CayleyDickson->new($one,$zero);

conjugate()

  if z = (a,b)
then conjugate z = z* = (a,b)* = (a*,-b)
  or conjugate(number) = number

printf "The conjugate of q1 is: %s\n", $q1->conjugate;

inverse()

  if z = (a,b)
then inverse z = z⁻¹ = (a,b)⁻¹ = (a,b)*/(norm(a,b)²)
  or inverse(number) = number

printf "The inverse of q1 is: %s\n", $q1->inverse;

norm()

  if z = (a,b)
then norm z = norm(a,b) = √(norm(a)²+norm(b)²)
  or norm(number) = number

printf "The norm of q1 is: %s\n", $q1->norm;

add()

# ass z1 + z2 = (a,b)+(c,d) = (a+c,b+d)

printf "The sum of q1 + q2 is: %s\n", $q1 + $q2;

subtract()

# subtract z1 - z2 =  (a,b)-(c,d) = (a-c,b-d)

printf "The difference of q1 - q2 is: %s\n", $q1 - $q2;

divide()

# divide z1 / z2 = z1 × inverse(z2)

printf "The division of q1 / q2 is: %s\n", $q1 / $q2;

multiply()

# Multiply: (a,b)×(c,d) = (a×c - d*×b, d×a + b×c*) where x* = conjugate(x) or x if x is a number

printf "The product of q1 * q2 is: %s\n", $q1 * $q2;

new()

create a new CayleyDickson number of any size ...

# create the number 1+j-k ...
my $c = CayleyDickson->new( -1, 0, 1, -1 );

# create an octonion ...
my $c = CayleyDickson->new( 3, 7, -2, 8, 0, 3, 3, 5 );

# create a representation of the Horne bell state |+-> ...
my $c = CayleyDickson->new( 1/2, 1/2, 1/2 ,-1/2 );

# create a 128 number construction: 1+2i+3j+4k+ .. + 128 ....
my $c = CayleyDickson->new(1 .. 128);

tensor()

Tensors two Cayley Dickson numbers to calculate a new number of higher dimensional construction.
reference: https://en.wikipedia.org/wiki/Tensor_product

# calculate the tensor of c1 ⊗  c2 ...
$d = $c1->tensor($c2);

$d will be a number of the product of the dimensions of c1 and c2.

a()

b()

returns the two objects or numbers held by this object

flat()

return all the coefficients of the number as an array

printf "[%s]\n", join( ', ', $q1->flat); 

as_string()

called automatically when this object is requested in a string form.
if you want to force the object to be resolved as a string ...

printf "q1 as a string = %s\n", $q1->as_string;

i_squared()

returns the square of i: i² = -1

normally this will be -1, but you can change it to +1 or 0 using the constant I_SQUARED

doubling_product()

something

is_complex()

is_quaternion()

is_octonion()

is_sedenion()

is_trigintaduonions()

is_sexagintaquatronions()

is_centumduodetrigintanions()

is_ducentiquinquagintasexions()

returns true if the given object has depth equal to the function name

if ($q1->is_octionion) {
   print "q1 is an Octonion\n";
}
else {
   print "q1 is NOT an Octonion\n";
}

SUMMARY

This object holds Cayley Dickson numbers and provides math operations on them.

=back

AUTHOR

Jeff Anderson
truejeffanderson@gmail.com

cpanm CayleyDickson

perl -MCPAN -e shell
install CayleyDickson