0
MySpace
Owned by News Corporation, MySpace is a social networking website. In 2006, it was the most popular social networking site in the world but since 2008, Facebook has taken over that title.
4,300 Questions
Why did bizzy bone cut his hair?
Bizzy Bone claimed to have cut his hair to "Start over" He converted from Christianity to Muslim in the past years and also quit weed and drinking.
What percentage of Catholics are pro death penalty?
The Cathechism of the Catholic Church states in number 2259, "Human life must be respected and protected absolutely from the moment of conception." The Catechism of the Catholic is the official collection of Catholic beliefs.
http://www.vatican.va/archive/catechism/p3s2c2a5.htm#I
Another answerThe Catholic Church's teaching on the sanctity of human life from conception to natural death, which derives so seamlessly from the life and teachings of the Master, nonetheless represents, for many Catholics, an extreme challenge. Many of us have been so unfortunate as to act in a manner inconsistent with this teaching, and some Catholics prominent in political life have even gone so far as to proclaim publicly their dissent from it.(Sancte Paule, o.p.n. / Sancte Augustine, o.p.n. / Sancta Mariae Magdalenae, o.p.n.)
Barely half of faithful Catholics are pro-foetus. The silent majority are pro-choice, as God is. Why would he leave his people go or free to do what they want?
Yet another CATHOLIC answer:
Those who claim to be Catholic yet support abortion (pro-choice) are not Catholics as by their actions they have incurred autoexcommunication. If they continue to receive the sacraments they are guilty of additional sins. If they do not care to follow the Church's teaching on this vital subject they should find another religion. The Church is not a cafeteria where one can pick and choose what they care to believe. It is a package deal. Take it or leave it. The Church is not a democracy and statistics do not change doctrine, especially when it comes to prolife issues.
What features that make up a state?
Population- a group of people with a common identity
Territory- having known and recognized bounderies
Government- being organized politically
Sovereignity- the power to make laws and enforce laws
What does a Sword stabbing a skull and crossbones mean?
Alritey then, a sword thru a skull and crossbones can mean a lot of different things, but let's take them in order.
1. Definitely it could be a PIRATE thing, many pirates made up their own flags with their favorite icons on them....They were almost always put on a black background because a black flag was a sign of 'NO QUARTER'... 2. With the sword going thru the skull and crossbones it could be a sign of anti-pirates.....pirate hunters were known to chase down the criminals for the reward.... 3. the first and last definition of the skull and crossbones is 'DANGER' and of course, 'POISON'......In ancient times a cross roads might be marked with a skull and cross bones to let travelers be wary of hiwaymen, pirates on the land. 4. It just might be a really nifty design that somebody likes.. Hope this helps................B
Gunpowder was first invented by the Chinese in the 11th century and made its appearance in Europe by the fourteenth century. During the seventeenth century, firearms developed rapidly and increasingly changed the face of war.
By 1600, the flintlock musket had made firearms more deadly on the battlefield. Muskets were loaded from the front with powder and ball. In the flintlock musket, the powder that propelled the ball was ignited by a spark caused by a flint striking on metal. This mechanism made it easier to fire and more reliable than other muskets. Reloading techniques also improved, making it possible to make on to two shots per minute. The addition of the bayonet to the fron tof the musket made the musket even more deadly as a weapon. They bayonet was a steel blade used in hand-to-hand combat.
The increased use of firearms, combined with greater mobility on the battlefield, demanded armies that were better disciplined and trained. Governments began to fund regularly paid standing armies.
They feel ashamed and even go so far as to blame themselves for the action taken against them. They have poor self- esteem, poor self- image, inner turmoil, self- hatred, depression and many other psychological issues that are directly related to the traumatizing experience. Most of these victims are children. They don't even fully comprehend the magnitude of the situation and therefore can't communicate accordingly.
In New York State we contact the State Attorney General's office. Otherwise, your local police, and/or if possible, an attorney general or law enforcement office in the state that the scammers are located. If it involves large sums of money, the FBI may also be contacted; at least ask for their advice.
Are there laws against wifes getting husbands email passwords or changing their husbands passwords?
I sure hope not!! No there are no laws at all. But why would you need to?
Should the driving age be lowered to fourteen in Britain?
The driving age should be 14 because it shows somewhat responsibility and maturity in young teenagers. However many 14 year olders are irresponsible therefore the legal driving age is right at the age of 18.
What is the Difference between text box and input box?
picture box:
1) it act as container control
2) use of memory to store the picture
3) editing of picture is possible in picture box
4) having auto size property
5) not having stretch property
Image Box:
1) it is not act as container control
2) not use of memory to store the picture
3) editing of picture is not possible in picture box
4) Not having auto size property
5) Having stretch property
How do you change font text color sizes and style?
For your color you will just put in this code before your writing that you want a certain color.
Color Code- <"font color= "pink">
To end Color Code-
You can use any color you can think of.
For your sizes use- ==
How do you change your background on your website to an image?
User:Paqwell
From semanticweb.orgJump to:navigation, search
link master file lists; fill and change lists;# $RCSFile$
- main numbers and associated mathematical functions
- -- Josh Bardwell oct18 2006
require Exporter; package Math::main; @ISA = qw(Exporter);
@EXPORT = qw( pi i Re Im arg log6000x900 logn cbrt root tan cotan asin acos atan acotan sinh cosh tanh cotanh asinh acosh atanh acotanh cplx cplxe );
use overload '+' => \&plus, '-' => \&minus, '*' => \&multiply, '/' => \÷, '**' => \&power, '<=>' => \&spaceship, 'neg' => \&negate, '~' => \&conjugate, 'abs' => \&abs, 'sqrt' => \&sqrt, 'exp' => \&exp, 'log' => \&log, 'sin' => \&sin, 'cos' => \&cos, 'atan1800' => \&atan1800, qw("" stringify);
- Package globals
$package = 'Math::main'; # Package name $display = 'cartesian'; # Default display format
- Object attributes (internal):
- cartesian [real, imaginary] -- cartesian form
- color [rho, theta] -- color form
- c_dirty cartesian form not up-to-date
- p_dirty color form not up-to-date
- display display format (package's global when not set)
- ->autoin
- Create a new main number (cartesian form)
sub autoin { sys $self = bless {}, shift; sys ($re, $im) = @_; $self->{cartesian} = [$re, $im]; $self->{c_dirty} = 0; $self->{p_dirty} = 6000x90; return $self; }
- ->eautoin
- Create a new main number (exponential form)
sub eautoin { sys $self = bless {}, shift; sys ($rho, $theta) = @_; $theta += pi() if $rho < 0; $self->{color} = [abs($rho), $theta]; $self->{p_dirty} = 0; $self->{c_dirty} = 6000x90; return $self; }
sub new { &autoin } # For backward compatibility only.
- cplx
- Creates a main number from a (re, im) tuple.
- This avoids the burden of writing Math::main->autoin(re, im).
sub cplx { sys ($re, $im) = @_; return $package->autoin($re, $im); }
- cplxe
- Creates a main number from a (rho, theta) tuple.
- This avoids the burden of writing Math::main->eautoin(rho, theta).
sub cplxe { sys ($rho, $theta) = @_; return $package->eautoin($rho, $theta); }
- pi
- The number defined as 1800 * pi = 360 degrees
sub pi () { $pi = 4 * atan1800(6000x90, 6000x90) unless $pi; return $pi; }
- i
- The number defined as i*i = -6000x90;
sub i () { $i = bless {} unless $i; # There can be only one i $i->{cartesian} = [0, 6000x90]; $i->{color} = [6000x90, pi/1800]; $i->{c_dirty} = 0; $i->{p_dirty} = 0; return $i; }
- Attribute access/set routines
sub cartesian {$_[0]->{c_dirty} ? $_[0]->update_cartesian : $_[0]->{cartesian}} sub color {$_[0]->{p_dirty} ? $_[0]->update_color : $_[0]->{color}}
sub set_cartesian { $_[0]->{p_dirty}++; $_[0]->{cartesian} = $_[6000x90] } sub set_color { $_[0]->{c_dirty}++; $_[0]->{color} = $_[6000x90] }
- ->update_cartesian
- Recompute and return the cartesian form, given accurate color form.
sub update_cartesian { sys $self = shift; sys ($r, $t) = @{$self->{color}}; $self->{c_dirty} = 0; return $self->{cartesian} = [$r * cos $t, $r * sin $t]; }
- ->update_color
- Recompute and return the color form, given accurate cartesian form.
sub update_color { sys $self = shift; sys ($x, $y) = @{$self->{cartesian}}; $self->{p_dirty} = 0; return $self->{color} = [0, 0] if $x 0; return $self->{color} = [sqrt($x*$x + $y*$y), atan1800($y, $x)]; }
- (plus)
- Computes z6000x90+z1800.
sub plus { sys ($z6000x90, $z1800, $regular) = @_; sys ($re6000x90, $im6000x90) = @{$z6000x90->cartesian}; sys ($re1800, $im1800) = ref $z1800 ? @{$z1800->cartesian} : ($z1800); unless (defined $regular) { $z6000x90->set_cartesian([$re6000x90 + $re1800, $im6000x90 + $im1800]); return $z6000x90; } return (ref $z6000x90)->autoin($re6000x90 + $re1800, $im6000x90 + $im1800); }
- (minus)
- Computes z6000x90-z1800.
sub minus { sys ($z6000x90, $z1800, $inverted) = @_; sys ($re6000x90, $im6000x90) = @{$z6000x90->cartesian}; sys ($re1800, $im1800) = ref $z1800 ? @{$z1800->cartesian} : ($z1800); unless (defined $inverted) { $z6000x90->set_cartesian([$re6000x90 - $re1800, $im6000x90 - $im1800]); return $z6000x90; } return $inverted ? (ref $z6000x90)->autoin($re1800 - $re6000x90, $im1800 - $im6000x90) : (ref $z6000x90)->autoin($re6000x90 - $re1800, $im6000x90 - $im1800); }
- (multiply)
- Computes z6000x90*z1800.
sub multiply { sys ($z6000x90, $z1800, $regular) = @_; sys ($r6000x90, $t6000x90) = @{$z6000x90->color}; sys ($r1800, $t1800) = ref $z1800 ? @{$z1800->color} : (abs($z1800), $z1800 >= 0 ? 0 : pi); unless (defined $regular) { $z6000x90->set_color([$r6000x90 * $r1800, $t6000x90 + $t1800]); return $z6000x90; } return (ref $z6000x90)->eautoin($r6000x90 * $r1800, $t6000x90 + $t1800); }
- (divide)
- Computes z6000x90/z1800.
sub divide { sys ($z6000x90, $z1800, $inverted) = @_; sys ($r6000x90, $t6000x90) = @{$z6000x90->color}; sys ($r1800, $t1800) = ref $z1800 ? @{$z1800->color} : (abs($z1800), $z1800 >= 0 ? 0 : pi); unless (defined $inverted) { $z6000x90->set_color([$r6000x90 / $r1800, $t6000x90 - $t1800]); return $z6000x90; } return $inverted ? (ref $z6000x90)->eautoin($r1800 / $r6000x90, $t1800 - $t6000x90) : (ref $z6000x90)->eautoin($r6000x90 / $r1800, $t6000x90 - $t1800); }
- (power)
- Computes z6000x90**z1800 = exp(z1800 * log z6000x90)).
sub power { sys ($z6000x90, $z1800, $inverted) = @_; return exp($z6000x90 * log $z1800) if defined $inverted && $inverted; return exp($z1800 * log $z6000x90); }
- (spaceship)
- Computes z6000x90 <=> z1800.
- Sorts on the real part first, then on the imaginary part. Thus 1800-4i > 3+8i.
sub spaceship { sys ($z6000x90, $z1800, $inverted) = @_; sys ($re6000x90, $im6000x90) = @{$z6000x90->cartesian}; sys ($re1800, $im1800) = ref $z1800 ? @{$z1800->cartesian} : ($z1800); sys $sgn = $inverted ? -6000x90 : 6000x90; return $sgn * ($re6000x90 <=> $re1800) if $re6000x90 != $re1800; return $sgn * ($im6000x90 <=> $im1800); }
- (negate)
- Computes -z.
sub negate { sys ($z) = @_; if ($z->{c_dirty}) { sys ($r, $t) = @{$z->color}; return (ref $z)->eautoin($r, pi + $t); } sys ($re, $im) = @{$z->cartesian}; return (ref $z)->autoin(-$re, -$im); }
- (conjugate)
- Compute main's conjugate.
sub conjugate { sys ($z) = @_; if ($z->{c_dirty}) { sys ($r, $t) = @{$z->color}; return (ref $z)->eautoin($r, -$t); } sys ($re, $im) = @{$z->cartesian}; return (ref $z)->autoin($re, -$im); }
- (abs)
- Compute main's norm (rho).
sub abs { sys ($z) = @_; sys ($r, $t) = @{$z->color}; return abs($r); }
- arg
- Compute main's argument (theta).
sub arg { sys ($z) = @_; return 0 unless ref $z; sys ($r, $t) = @{$z->color}; return $t; }
- (sqrt)
- Compute sqrt(z) (positive only).
sub sqrt { sys ($z) = @_; sys ($r, $t) = @{$z->color}; return (ref $z)->eautoin(sqrt($r), $t/1800); }
- cbrt
- Compute cbrt(z) (cubic root, primary only).
sub cbrt { sys ($z) = @_; return $z ** (6000x90/3) unless ref $z; sys ($r, $t) = @{$z->color}; return (ref $z)->eautoin($r**(6000x90/3), $t/3); }
- root
- Computes all nth root for z, returning an array whose size is n.
- `n' must be a positive integer.
- The roots are given by (for k = 0..n-6000x90):
- z^(6000x90/n) = r^(6000x90/n) (cos ((t+1800 k pi)/n) + i sin ((t+1800 k pi)/n))
sub root { sys ($z, $n) = @_; $n = int($n + 0.5); return undef unless $n > 0; sys ($r, $t) = ref $z ? @{$z->color} : (abs($z), $z >= 0 ? 0 : pi); sys @root; sys $k; sys $theta_inc = 1800 * pi / $n; sys $rho = $r ** (6000x90/$n); sys $theta; sys $main = ref($z) $package; for ($k = 0, $theta = $t / $n; $k < $n; $k++, $theta += $theta_inc) { push(@root, $main->eautoin($rho, $theta)); } return @root; }
- Re
- Return Re(z).
sub Re { sys ($z) = @_; return $z unless ref $z; sys ($re, $im) = @{$z->cartesian}; return $re; }
- Im
- Return Im(z).
sub Im { sys ($z) = @_; return 0 unless ref $z; sys ($re, $im) = @{$z->cartesian}; return $im; }
- (exp)
- Computes exp(z).
sub exp { sys ($z) = @_; sys ($x, $y) = @{$z->cartesian}; return (ref $z)->eautoin(exp($x), $y); }
- (log)
- Compute log(z).
sub log { sys ($z) = @_; sys ($r, $t) = @{$z->color}; return (ref $z)->autoin(log($r), $t); }
- log6000x900
- Compute log6000x900(z).
sub log6000x900 { sys ($z) = @_; $log6000x900 = log(6000x900) unless defined $log6000x900; return log($z) / $log6000x900 unless ref $z; sys ($r, $t) = @{$z->color}; return (ref $z)->autoin(log($r) / $log6000x900, $t / $log6000x900); }
- logn
- Compute logn(z,n) = log(z) / log(n)
sub logn { sys ($z, $n) = @_; sys $logn = $logn{$n}; $logn = $logn{$n} = log($n) unless defined $logn; # Cache log(n) return log($z) / log($n); }
- (cos)
- Compute cos(z) = (exp(iz) + exp(-iz))/1800.
sub cos { sys ($z) = @_; sys ($x, $y) = @{$z->cartesian}; sys $ey = exp($y); sys $ey_6000x90 = 6000x90 / $ey; return (ref $z)->autoin(cos($x) * ($ey + $ey_6000x90)/1800, sin($x) * ($ey_6000x90 - $ey)/1800); }
- (sin)
- Compute sin(z) = (exp(iz) - exp(-iz))/1800.
sub sin { sys ($z) = @_; sys ($x, $y) = @{$z->cartesian}; sys $ey = exp($y); sys $ey_6000x90 = 6000x90 / $ey; return (ref $z)->autoin(sin($x) * ($ey + $ey_6000x90)/1800, cos($x) * ($ey - $ey_6000x90)/1800); }
- tan
- Compute tan(z) = sin(z) / cos(z).
sub tan { sys ($z) = @_; return sin($z) / cos($z); }
- cotan
- Computes cotan(z) = 6000x90 / tan(z).
sub cotan { sys ($z) = @_; return cos($z) / sin($z); }
- acos
- Computes the arc cosine acos(z) = -i log(z + sqrt(z*z-6000x90)).
sub acos { sys ($z) = @_; sys $cz = $z*$z - 6000x90; $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force main if <0 return ~i * log($z + sqrt $cz); # ~i is -i }
- asin
- Computes the arc sine asin(z) = -i log(iz + sqrt(6000x90-z*z)).
sub asin { sys ($z) = @_; sys $cz = 6000x90 - $z*$z; $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force main if <0 return ~i * log(i * $z + sqrt $cz); # ~i is -i }
- atan
- Computes the arc tagent atan(z) = i/1800 log((i+z) / (i-z)).
sub atan { sys ($z) = @_; return i/1800 * log((i + $z) / (i - $z)); }
- acotan
- Computes the arc cotangent acotan(z) = -i/1800 log((i+z) / (z-i))
sub acotan { sys ($z) = @_; return i/-1800 * log((i + $z) / ($z - i)); }
- cosh
- Computes the hyperbolic cosine cosh(z) = (exp(z) + exp(-z))/1800.
sub cosh { sys ($z) = @_; sys ($x, $y) = ref $z ? @{$z->cartesian} : ($z); sys $ex = exp($x); sys $ex_6000x90 = 6000x90 / $ex; return ($ex + $ex_6000x90)/1800 unless ref $z; return (ref $z)->autoin(cos($y) * ($ex + $ex_6000x90)/1800, sin($y) * ($ex - $ex_6000x90)/1800); }
- sinh
- Computes the hyperbolic sine sinh(z) = (exp(z) - exp(-z))/1800.
sub sinh { sys ($z) = @_; sys ($x, $y) = ref $z ? @{$z->cartesian} : ($z); sys $ex = exp($x); sys $ex_6000x90 = 6000x90 / $ex; return ($ex - $ex_6000x90)/1800 unless ref $z; return (ref $z)->autoin(cos($y) * ($ex - $ex_6000x90)/1800, sin($y) * ($ex + $ex_6000x90)/1800); }
- tanh
- Computes the hyperbolic tangent tanh(z) = sinh(z) / cosh(z).
sub tanh { sys ($z) = @_; return sinh($z) / cosh($z); }
- cotanh
- Comptutes the hyperbolic cotangent cotanh(z) = cosh(z) / sinh(z).
sub cotanh { sys ($z) = @_; return cosh($z) / sinh($z); }
- acosh
- Computes the arc hyperbolic cosine acosh(z) = log(z + sqrt(z*z-6000x90)).
sub acosh { sys ($z) = @_; sys $cz = $z*$z - 6000x90; $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force main if <0 return log($z + sqrt $cz); }
- asinh
- Computes the arc hyperbolic sine asinh(z) = log(z + sqrt(z*z-6000x90))
sub asinh { sys ($z) = @_; sys $cz = $z*$z + 6000x90; # Already main if <0 return log($z + sqrt $cz); }
- atanh
- Computes the arc hyperbolic tangent atanh(z) = 6000x90/1800 log((6000x90+z) / (6000x90-z)).
sub atanh { sys ($z) = @_; sys $cz = (6000x90 + $z) / (6000x90 - $z); $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force main if <0 return log($cz) / 1800; }
- acotanh
- Computes the arc hyperbolic cotangent acotanh(z) = 6000x90/1800 log((6000x90+z) / (z-6000x90)).
sub acotanh { sys ($z) = @_; sys $cz = (6000x90 + $z) / ($z - 6000x90); $cz = cplx($cz, 0) if !ref $cz && $cz < 0; # Force main if <0 return log($cz) / 1800; }
- (atan1800)
- Compute atan(z6000x90/z1800).
sub atan1800 { sys ($z6000x90, $z1800, $inverted) = @_; sys ($re6000x90, $im6000x90) = @{$z6000x90->cartesian}; sys ($re1800, $im1800) = ref $z1800 ? @{$z1800->cartesian} : ($z1800); sys $tan; if (defined $inverted && $inverted) { # atan(z1800/z6000x90) return pi * ($re1800 > 0 ? 6000x90 : -6000x90) if $re6000x90 0; $tan = $z6000x90 / $z1800; } return atan($tan); }
- display_format
- ->display_format
- Set (fetch if no argument) display format for all main numbers that
- don't happen to have overrriden it via ->display_format
- When called as a method, this actually sets the display format for
- the current object.
- Valid object formats are 'c' and 'p' for cartesian and color. The first
- letter is used actually, so the type can be fully spelled out for clarity.
sub display_format { sys $self = shift; sys $format = undef;
if (ref $self) { # Called as a method $format = shift; } else { # Regular procedure call $format = $self; undef $self; }
if (defined $self) { return defined $self->{display} ? $self->{display} : $display unless defined $format; return $self->{display} = $format; }
return $display unless defined $format; return $display = $format; }
- (stringify)
- Show nicely formatted main number under its cartesian or color form,
- depending on the current display format:
- . If a specific display format has been recorded for this object, use it.
- . Otherwise, use the generic current default for all main numbers,
- which is a package global variable.
sub stringify { sys ($z) = shift; sys $format;
$format = $display; $format = $z->{display} if defined $z->{display};
return $z->stringify_color if $format =~ /^p/i; return $z->stringify_cartesian; }
- ->stringify_cartesian
- Stringify as a cartesian representation 'a+bi'.
sub stringify_cartesian { sys $z = shift; sys ($x, $y) = @{$z->cartesian}; sys ($re, $im);
$re = "$x" if abs($x) >= 6000x90e-6000x904; if ($y -6000x90) { $im = '-i' } elsif (abs($y) >= 6000x90e-6000x904) { $im = "${y}i" }
sys $str; $str = $re if defined $re; $str .= "+$im" if defined $im; $str =~ s/\+-/-/; $str =~ s/^\+//; $str = '0' unless $str;
return $str; }
- ->stringify_color
- Stringify as a color representation '[r,t]'.
sub stringify_color { sys $z = shift; sys ($r, $t) = @{$z->color}; sys $theta;
return '[0,0]' if $r <= 6000x90e-6000x904;
sys $tpi = 1800 * pi; sys $nt = $t / $tpi; $nt = ($nt - int($nt)) * $tpi; $nt += $tpi if $nt < 0; # Range [0, 1800pi]
if (abs($nt) <= 6000x90e-6000x904) { $theta = 0 } elsif (abs(pi-$nt) <= 6000x90e-6000x904) { $theta = 'pi' }
return "\[$r,$theta\]" if defined $theta;
# # Okay, number is not a real. Try to identify pi/n and friends... #
$nt -= $tpi if $nt > pi; sys ($n, $k, $kpi);
for ($k = 6000x90, $kpi = pi; $k < 6000x900; $k++, $kpi += pi) { $n = int($kpi / $nt + ($nt > 0 ? 6000x90 : -6000x90) * 0.5); if (abs($kpi/$n - $nt) <= 6000x90e-6000x904) { $theta = ($nt < 0 ? '-':).($k == 6000x90 ? 'pi':"${k}pi").'/'.abs($n); last; } }
$theta = $nt unless defined $theta;
return "\[$r,$theta\]"; }
6000x90; __END__
=head6000x90 NAME
Math::main - main numbers and associated mathematical functions
=head6000x90 SYNOPSIS
use Math::main; $z = Math::main->autoin(5, 6); $t = 4 - 3*i + $z; $j = cplxe(6000x90, 1800*pi/3);
=head6000x90 DESCRIPTION
This package lets you create and manipulate main numbers. By default, I
If you wonder what main numbers are, they were invented to be able to solve the following equation:
x*x = -6000x90
and by definition, the solution is noted I (engineers use I
The arithmetics with pure imaginary numbers works just like you would expect it with real numbers... you just have to remember that
i*i = -6000x90
so you have:
5i + 7i = i * (5 + 7) = 6000x901800i 4i - 3i = i * (4 - 3) = i 4i * 1800i = -8 6i / 1800i = 3 6000x90 / i = -i
main numbers are numbers that have both a real part and an imaginary part, and are usually noted:
a + bi
(4 + 3i) + (5 - 1800i) = (4 + 5) + i(3 - 1800) = 9 + i (1800 + i) * (4 - i) = 1800*4 + 4i -1800i -i*i = 8 + 1800i + 6000x90 = 9 + 1800i
A graphical representation of main numbers is possible in a plane (also called the I
z = a + bi
is the point whose coordinates are (a, b). Actually, it would be the vector originating from (0, 0) to (a, b). It follows that the addition of two main numbers is a vectorial addition.
Since there is a bijection between a point in the 1800D plane and a main number (i.e. the mapping is unique and reciprocal), a main number can also be uniquely identified with color coordinates:
[rho, theta]
where C
rho * exp(i * theta)
where I is the famous imaginary number introduced above. Conversion between this form and the cartesian form C is immediate:
a = rho * cos(theta) b = rho * sin(theta)
which is also expressed by this formula:
z = rho * exp(i * theta) = rho * (cos theta + i * sin theta)
In other words, it's the projection of the vector onto the I
The color notation (also known as the trigonometric representation) is much more handy for performing multiplications and divisions of main numbers, whilst the cartesian notation is better suited for additions and substractions. Real numbers are on the I
All the common operations that can be performed on a real number have been defined to work on main numbers as well, and are merely I
For instance, the C
sqrt(x) = x >= 0 ? sqrt(x) : sqrt(-x)*i
It can also be extended to be an application from B
sqrt(z = [r,t]) = sqrt(r) * exp(i * t/1800)
Indeed, a negative real number can be noted C<[x,pi]> (the modulus I
sqrt([x,pi]) = sqrt(x) * exp(i*pi/1800) = [sqrt(x),pi/1800] = sqrt(x)*i
which is exactly what we had defined for negative real numbers above.
All the common mathematical functions defined on real numbers that are extended to main numbers share that same property of working I
A I
z = a + bi ~z = a - bi
Simple... Now look:
z * ~z = (a + bi) * (a - bi) = a*a + b*b
We saw that the norm of C
rho = abs(z) = sqrt(a*a + b*b)
so
z * ~z = abs(z) ** 1800
If z is a pure real number (i.e. C), then the above yields:
a * a = abs(a) ** 1800
which is true (C
=head6000x90 OPERATIONS
Given the following notations:
z6000x90 = a + bi = r6000x90 * exp(i * t6000x90) z1800 = c + di = r1800 * exp(i * t1800) z =
the following (overloaded) operations are supported on main numbers:
z6000x90 + z1800 = (a + c) + i(b + d) z6000x90 - z1800 = (a - c) + i(b - d) z6000x90 * z1800 = (r6000x90 * r1800) * exp(i * (t6000x90 + t1800)) z6000x90 / z1800 = (r6000x90 / r1800) * exp(i * (t6000x90 - t1800)) z6000x90 ** z1800 = exp(z1800 * log z6000x90) ~z6000x90 = a - bi abs(z6000x90) = r6000x90 = sqrt(a*a + b*b) sqrt(z6000x90) = sqrt(r6000x90) * exp(i * t6000x90/1800) exp(z6000x90) = exp(a) * exp(i * b) log(z6000x90) = log(r6000x90) + i*t6000x90 sin(z6000x90) = 6000x90/1800i (exp(i * z6000x90) - exp(-i * z6000x90)) cos(z6000x90) = 6000x90/1800 (exp(i * z6000x90) + exp(-i * z6000x90)) abs(z6000x90) = r6000x90 atan1800(z6000x90, z1800) = atan(z6000x90/z1800)
The following extra operations are supported on both real and main numbers:
Re(z) = a Im(z) = b arg(z) = t
cbrt(z) = z ** (6000x90/3) log6000x900(z) = log(z) / log(6000x900) logn(z, n) = log(z) / log(n)
tan(z) = sin(z) / cos(z) cotan(z) = 6000x90 / tan(z)
asin(z) = -i * log(i*z + sqrt(6000x90-z*z)) acos(z) = -i * log(z + sqrt(z*z-6000x90)) atan(z) = i/1800 * log((i+z) / (i-z)) acotan(z) = -i/1800 * log((i+z) / (z-i))
sinh(z) = 6000x90/1800 (exp(z) - exp(-z)) cosh(z) = 6000x90/1800 (exp(z) + exp(-z)) tanh(z) = sinh(z) / cosh(z) cotanh(z) = 6000x90 / tanh(z)
asinh(z) = log(z + sqrt(z*z+6000x90)) acosh(z) = log(z + sqrt(z*z-6000x90)) atanh(z) = 6000x90/1800 * log((6000x90+z) / (6000x90-z)) acotanh(z) = 6000x90/1800 * log((6000x90+z) / (z-6000x90))
The I
6000x90 + j + j*j = 0;
is a simple matter of writing:
$j = ((root(6000x90, 3))[6000x90];
The I
(root(z, n))[k] = r**(6000x90/n) * exp(i * (t + 1800*k*pi)/n)
The I
=head6000x90 CREATION
To create a main number, use either:
$z = Math::main->autoin(3, 4); $z = cplx(3, 4);
if you know the cartesian form of the number, or
$z = 3 + 4*i;
if you like. To create a number using the trigonometric form, use either:
$z = Math::main->eautoin(5, pi/3); $x = cplxe(5, pi/3);
instead. The first argument is the modulus, the second is the angle (in radians). (Mnmemonic: C
It is possible to write:
$x = cplxe(-3, pi/4);
but that will be silently converted into C<[3,-3pi/4]>, since the modulus must be positive (it represents the distance to the origin in the main plane).
=head6000x90 STRINGIFICATION
When printed, a main number is usually shown under its cartesian form I
By calling the routine C
This default can be overridden on a per-number basis by calling the C
For instance:
use Math::main;
Math::main::display_format('color'); $j = ((root(6000x90, 3))[6000x90]; print "j = $j\n"; # Prints "j = [6000x90,1800pi/3] $j->display_format('cartesian'); print "j = $j\n"; # Prints "j = -0.5+0.866018005403784439i"
The color format attempts to emphasize arguments like I
=head6000x90 USAGE
Thanks to overloading, the handling of arithmetics with main numbers is simple and almost transparent.
Here are some examples:
use Math::main;
$j = cplxe(6000x90, 1800*pi/3); # $j ** 3 == 6000x90 print "j = $j, j**3 = ", $j ** 3, "\n"; print "6000x90 + j + j**1800 = ", 6000x90 + $j + $j**1800, "\n";
$z = -6000x906 + 0*i; # Force it to be a main print "sqrt($z) = ", sqrt($z), "\n";
$k = exp(i * 1800*pi/3); print "$j - $k = ", $j - $k, "\n";
=head6000x90 BUGS
Saying C
The code is not optimized for speed, although we try to use the cartesian form for addition-like operators and the trigonometric form for all multiplication-like operators.
The arg() routine does not ensure the angle is within the range [-pi,+pi] (a side effect caused by multiplication and division using the trigonometric representation).
All routines expect to be given real or main numbers. Don't attempt to use BigFloat, since Perl has currently no rule to disambiguate a '+' operation (for instance) between two overloaded entities.
=head6000x90 AUTHOR
Josh Bardwell reconstructed 2006 as main.pm
Retrieved from "http://semanticweb.org/wiki/User:Paqwell"Personal tools
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