Euler's identity
In mathematics, Euler's identity is the equality:
You are encouraged to solve this task according to the task description, using any language you may know.
| This page uses content from Wikipedia. The original article was at Euler's_identity. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance) |
ei + 1 = 0
where
e is Euler's number, the base of natural logarithms,
i is the imaginary unit, which satisfies i2 = −1, and
is pi, the ratio of the circumference of a circle to its diameter.
Euler's identity is often cited as an example of deep mathematical beauty. Three of the basic arithmetic operations occur exactly once each: addition, multiplication, and exponentiation. The identity also links five fundamental mathematical constants:
The number 0. The number 1. The number ( = 3.14159+), The number e (e = 2.71828+), which occurs widely in mathematical analysis. The number i, the imaginary unit of the complex numbers.
- Task
Show in your language that Euler's identity is true. As much as possible and practical, mimic the Euler's identity equation.
Most languages are limited to IEEE 754 floating point calculations so will have some error in the calculation.
If that is the case, or there is some other limitation, show that ei + 1 is approximately equal to zero and show the amount of error in the calculation.
If your language is capable of symbolic calculations, show
that ei + 1 is exactly equal to zero for bonus kudos points.
11l
print(math:e ^ (math:pi * 1i) + 1)- Output:
1.22465e-16i
Ada
with Ada.Long_Complex_Text_IO; use Ada.Long_Complex_Text_IO;
with Ada.Numerics; use Ada.Numerics;
with Ada.Numerics.Long_Complex_Types; use Ada.Numerics.Long_Complex_Types;
with Ada.Numerics.Long_Complex_Elementary_Functions; use Ada.Numerics.Long_Complex_Elementary_Functions;
procedure Eulers_Identity is
begin
Put (Exp (Pi * i) + 1.0);
end Eulers_Identity;
- Output:
( 0.00000000000000E+00, 1.22464679914735E-16)
ALGOL 68
Whilst Algol 68 has complex numbers as standard, it does not have a standard complex exp function.
We could use the identity exp(x + iy) = exp(x)( cos y + i sin y ), however the following uses a series expansion for exp(ix).
BEGIN
# calculate an approximation to e^(i pi) + 1 which should be 0 (Euler's identity) #
# returns e^ix for long real x, using the series: #
# exp(ix) = 1 - x^2/2! + x^4/4! - ... + i(x - x^3/3! + x^5/5! - x^7/7! ... ) #
# the expansion stops when successive terms differ by less than 1e-15 #
PROC expi = ( LONG REAL x )LONG COMPL:
BEGIN
LONG REAL t := 1;
LONG REAL real part := 1;
LONG REAL imaginary part := 0;
LONG REAL divisor := 1;
BOOL even power := FALSE;
BOOL subtract := FALSE;
LONG REAL diff := 1;
FOR n FROM 1 WHILE ABS diff > 1e-15 DO
divisor *:= n;
t *:= x;
LONG REAL term := t / divisor;
IF even power THEN
# this term is real #
subtract := NOT subtract;
LONG REAL prev := real part;
IF subtract THEN
real part -:= term
ELSE
real part +:= term
FI;
diff := prev - real part
ELSE
# this term is imaginary #
LONG REAL prev := imaginary part;
IF subtract THEN
imaginary part -:= term
ELSE
imaginary part +:= term
FI;
diff := prev - imaginary part
FI;
even power := NOT even power
OD;
( real part, imaginary part )
END # expi # ;
LONG COMPL eulers identity = expi( long pi ) + 1;
print( ( "e^(i*pi) + 1 ~ "
, fixed( re OF eulers identity, -23, 20 )
, " "
, fixed( im OF eulers identity, 23, 20 )
, "i"
, newline
)
)
END- Output:
e^(i*pi) + 1 ~ 0.00000000000000000307 -0.00000000000000002926i
ALGOL W
Algol W has a complex power operator but it can only handle integer powers. As with the Algol 68 sample, this uses a series expansion.
begin
% calculate an approximation to e^(i pi) + 1 which should be 0 (Euler's identity) %
% returns e^ix for long real x, using the series: %
% exp(ix) = 1 - x^2/2! + x^4/4! - ... + i(x - x^3/3! + x^5/5! - x^7/7! ... ) %
% the expansion stops when successive terms differ by less than 1e-15 %
long complex procedure expi ( long real value x ) ;
begin
long real t, realPart, imaginaryPart, divisor, diff;
integer n;
logical evenPower, subtract;
t := 1;
realPart := 1;
imaginaryPart := 0;
divisor := 1;
evenPower := false;
subtract := false;
diff := 1;
n := 0;
while abs diff > 1'-15 do begin
long real term;
n := n + 1;
divisor := divisor * n;
t := t * x;
term := t / divisor;
if evenPower then begin % this term is real %
long real prev;
subtract := not subtract;
prev := realPart;
if subtract then realPart := realPart - term
else realPart := realPart + term;
diff := prev - realPart
end
else begin % this term is imaginary %
long real prev;
prev := imaginaryPart ;
if subtract then imaginaryPart := imaginaryPart - term
else imaginaryPart := imaginaryPart + term;
diff := prev - imaginaryPart
end if_evenPower__ ;
evenPower := not evenPower
end while_abs_diff_gt_tolerance ;
realPart + 1i * imaginaryPart
end expi ;
write( s_w := 0, r_format := "A", r_w := 23, r_d := 20
, "e^(i*pi) + 1 ~ ", expi( pi ) + 1
)
end.- Output:
e^(i*pi) + 1 ~ 0.00000000000000000000 -0.00000000000000014054I
Bracmat
e^(i*pi)+1By symbolic calculation:
0
C
The C99 standard did, of course, introduce built-in support for complex number arithmetic into the language and so we can therefore compute (e ^ πi + 1) directly without having to resort to methods which sum the power series for e ^ x.
The following code has been tested with gcc 5.4.0 on Ubuntu 16.04.
#include <stdio.h>
#include <math.h>
#include <complex.h>
#include <wchar.h>
#include <locale.h>
int main() {
wchar_t pi = L'\u03c0'; /* Small pi symbol */
wchar_t ae = L'\u2245'; /* Approximately equals symbol */
double complex e = cexp(M_PI * I) + 1.0;
setlocale(LC_CTYPE, "");
printf("e ^ %lci + 1 = [%.16f, %.16f] %lc 0\n", pi, creal(e), cimag(e), ae);
return 0;
}
- Output:
e ^ πi + 1 = [0.0000000000000000, 0.0000000000000001] ≅ 0
C#
using System;
using System.Numerics;
public class Program
{
static void Main() {
Complex e = Math.E;
Complex i = Complex.ImaginaryOne;
Complex π = Math.PI;
Console.WriteLine(Complex.Pow(e, i * π) + 1);
}
}
- Output:
(0, 1.22464679914735E-16)
C++
#include <iostream>
#include <complex>
int main() {
std::cout << std::exp(std::complex<double>(0.0, M_PI)) + 1.0 << std::endl;
return 0;
}
- Output:
Zero and a little floating dust ...
(0,1.22465e-16)
Common Lisp
Common Lisp has complex number arithmetic built into it.
(+ 1 (exp (complex 0 pi)))#C(0.0L0 -5.0165576136843360246L-20)
Delphi
program Euler_identity;
{$APPTYPE CONSOLE}
uses
System.VarCmplx;
begin
var result := VarComplexExp(Pi * VarComplexCreate(0, 1)) + 1;
writeln(result);
readln;
end.
- Output:
0 + 0i
EasyLang
sysconf radians
func[] cadd a[] b[] .
return [ a[1] + b[1] a[2] + b[2] ]
.
func[] cexp a[] .
p = pow 2.718281828459045235 a[1]
return [ p * cos a[2] p * sin a[2] ]
.
print cadd cexp [ 0 pi ] [ 1 0 ]
- Output:
[ 0 1.2e-16 ]
F#
As per the discussion page this task as described is to show that -1+1=0
printfn "-1 + 1 = %d" (-1+1)
- Output:
-1 + 1 = 0
I shall also show that cos π = -1 and sin π = 0
printfn "cos(pi)=%f and sin(pi)=%f" (cos 3.141592653589793) (sin 3.141592653589793)
- Output:
cos(pi)=-1.000000 and sin(pi)=0.000000
I shall try the formula eiπ. I'll use the MathNet.Numerics package. I'll leave you the reader to determine what it 'proves'.
let i =MathNet.Numerics.complex(0.0,1.0);;
let pi=MathNet.Numerics.complex(MathNet.Numerics.Constants.Pi,0.0);;
let e =MathNet.Numerics.complex(MathNet.Numerics.Constants.E ,0.0);;
printfn "e**(i*pi) = %A" (e**(i*pi));;
- Output:
e**(i*pi) = (-1, 1.2246467991473532E-16)
Factor
USING: math math.constants math.functions prettyprint ;
1 e pi C{ 0 1 } * ^ + .
- Output:
C{ 0.0 1.224646799147353e-016 }
Forth
Uses fs. (scientific) to print the imaginary term, since that's the one that's inexact.
." e^(i*π) + 1 = " pi fcos 1e0 f+ f. '+ emit space pi fsin fs. 'i emit cr
bye
- Output:
e^(i*π) + 1 = 0. + 1.22464679914735E-16 i
Fortran
program euler
use iso_fortran_env, only: output_unit, REAL64
implicit none
integer, parameter :: d=REAL64
real(kind=d), parameter :: e=exp(1._d), pi=4._d*atan(1._d)
complex(kind=d), parameter :: i=(0._d,1._d)
write(output_unit,*) e**(pi*i) + 1
end program euler
- Output:
(0.0000000000000000,2.89542045005908316E-016)
FreeBASIC
Provides complex arithmetic and a very rapidly converging algorithm for e^z.
#define PI 3.141592653589793238462643383279502884197169399375105821
#define MAXITER 12
'---------------------------------------
' complex numbers and their arithmetic
'---------------------------------------
type complex
r as double
i as double
end type
function conj( a as complex ) as complex
dim as complex c
c.r = a.r
c.i = -a.i
return c
end function
operator + ( a as complex, b as complex ) as complex
dim as complex c
c.r = a.r + b.r
c.i = a.i + b.i
return c
end operator
operator - ( a as complex, b as complex ) as complex
dim as complex c
c.r = a.r - b.r
c.i = a.i - b.i
return c
end operator
operator * ( a as complex, b as complex ) as complex
dim as complex c
c.r = a.r*b.r - a.i*b.i
c.i = a.i*b.r + a.r*b.i
return c
end operator
operator / ( a as complex, b as complex ) as complex
dim as double bcb = (b*conj(b)).r
dim as complex acb = a*conj(b), c
c.r = acb.r/bcb
c.i = acb.i/bcb
return c
end operator
sub printc( a as complex )
if a.i>=0 then
print using "############.############### + ############.############### i"; a.r; a.i
else
print using "############.############### - ############.############### i"; a.r; -a.i
end if
end sub
function intc( n as integer ) as complex
dim as complex c
c.r = n
c.i = 0.0
return c
end function
function absc( a as complex ) as double
return sqr( (a*conj(a)).r )
end function
'-----------------------
' the algorithm
' Uses a rapidly converging continued
' fraction expansion for e^z and recursive
' expressions for its convergents
'-----------------------
dim as complex pii, pii2, curr, A2, A1, A0, B2, B1, B0
dim as complex ONE, TWO
dim as integer i, k = 2
pii.r = 0.0
pii.i = PI
pii2 = pii*pii
B0 = intc(2)
A0 = intc(2)
B1 = (intc(2) - pii)
A1 = B0*B1 + intc(2)*pii
printc( A0/B0)
print " Absolute error = ", absc(A0/B0)
printc( A1/B1)
print " Absolute error = ", absc(A1/B1)
for i = 1 to MAXITER
k = k + 4
A2 = intc(k)*A1 + pii2*A0
B2 = intc(k)*B1 + pii2*B0
curr = A2/B2
A0 = A1
A1 = A2
B0 = B1
B1 = B2
printc( curr )
print " Absolute error = ", absc(curr)
next i
Go
package main
import (
"fmt"
"math"
"math/cmplx"
)
func main() {
fmt.Println(cmplx.Exp(math.Pi * 1i) + 1.0)
}
- Output:
Zero and a little floating dust ...
(0+1.2246467991473515e-16i)
Groovy
Because the Groovy language does not provide a built-in facility for complex arithmetic, this example relies on the Complex class defined in the Complex numbers example.
import static Complex.*
Number.metaClass.mixin ComplexCategory
def π = Math.PI
def e = Math.E
println "e ** (π * i) + 1 = " + (e ** (π * i) + 1)
println "| e ** (π * i) + 1 | = " + (e ** (π * i) + 1).ρ
Output: (yadda, yadda, dust, yadda)
e ** (π * i) + 1 = 1.2246467991473532E-16i | e ** (π * i) + 1 | = 1.2246467991473532E-16
Haskell
A double is not quite real.
import Data.Complex
eulerIdentityZeroIsh :: Complex Double
eulerIdentityZeroIsh =
exp (0 :+ pi) + 1
main :: IO ()
main = print eulerIdentityZeroIsh
- Output:
Zero and a little floating dust ...
0.0 :+ 1.2246467991473532e-16
J
NB. Euler's number is the default base for power
NB. using j's expressive numeric notation:
1 + ^ 0j1p1
0j1.22465e_16
NB. Customize the comparison tolerance to 10 ^ (-15)
NB. to show that
_1 (=!.1e_15) ^ 0j1p1
1
TAU =: 2p1
NB. tauday.com pi is wrong
NB. with TAU as 2 pi,
NB. Euler's identity should have read
1 (=!.1e_15) ^ j. TAU
1
Java
Since Java lacks a complex number class, a class is used that has sufficient operations.
public class EulerIdentity {
public static void main(String[] args) {
System.out.println("e ^ (i*Pi) + 1 = " + (new Complex(0, Math.PI).exp()).add(new Complex(1, 0)));
}
public static class Complex {
private double x, y;
public Complex(double re, double im) {
x = re;
y = im;
}
public Complex exp() {
double exp = Math.exp(x);
return new Complex(exp * Math.cos(y), exp * Math.sin(y));
}
public Complex add(Complex a) {
return new Complex(x + a.x, y + a.y);
}
@Override
public String toString() {
return x + " + " + y + "i";
}
}
}
- Output:
e ^ (i*Pi) + 1 = 0.0 + 1.2246467991473532E-16i
jq
For speed and for conformance with the complex plane interpretation, x+iy is represented as [x,y]; for flexibility, all the functions defined here will accept both real and complex numbers, and for uniformity, they are implemented as functions that ignore their input.
Recent versions of jq support modules, so these functions could all be placed in a module to avoid name conflicts, and thus no special prefix is used here.
def multiply(x; y):
if (x|type) == "number" then
if (y|type) == "number" then [ x*y, 0 ]
else [x * y[0], x * y[1]]
end
elif (y|type) == "number" then multiply(y;x)
else [ x[0] * y[0] - x[1] * y[1], x[0] * y[1] + x[1] * y[0]]
end;
def plus(x; y):
if (x|type) == "number" then
if (y|type) == "number" then [ x+y, 0 ]
else [ x + y[0], y[1]]
end
elif (y|type) == "number" then plus(y;x)
else [ x[0] + y[0], x[1] + y[1] ]
end;
def exp(z):
def expi(x): [ (x|cos), (x|sin) ];
if (z|type) == "number" then z|exp
elif z[0] == 0 then expi(z[1]) # for efficiency
else multiply( (z[0]|exp); expi(z[1]) )
end ;
def pi: 4 * (1|atan);The Task
"e^iπ: \( exp( [0, pi ] ) )",
"e^iπ + 1: \( plus(1; exp( [0, pi ] ) ))"- Output:
e^iπ: [-1,1.2246467991473532e-16] e^iπ + 1: [0,1.2246467991473532e-16]
Julia
Julia has a builtin Complex{T} parametrized type.
@show ℯ^(π * im) + 1
@assert ℯ^(π * im) ≈ -1
- Output:
e ^ (π * im) + 1 = 0.0 + 1.2246467991473532e-16im
Using symbolic algebra, through the Reduce.jl package.
using Reduce
@force using Reduce.Algebra
@show ℯ^(π * :i) + 1
@assert ℯ^(π * :i) + 1 == 0
- Output:
ℯ^(π * :i) + 1 = 0
Kotlin
As the JVM lacks a complex number class, we use our own which has sufficient operations to perform this task.
e ^ πi is calculated by summing successive terms of the power series for e ^ x until the modulus of the difference between terms is no longer significant given the precision of the Double type (about 10 ^ -16).
// Version 1.2.40
import kotlin.math.sqrt
import kotlin.math.PI
const val EPSILON = 1.0e-16
const val SMALL_PI = '\u03c0'
const val APPROX_EQUALS = '\u2245'
class Complex(val real: Double, val imag: Double) {
operator fun plus(other: Complex) =
Complex(real + other.real, imag + other.imag)
operator fun times(other: Complex) = Complex(
real * other.real - imag * other.imag,
real * other.imag + imag * other.real
)
fun inv(): Complex {
val denom = real * real + imag * imag
return Complex(real / denom, -imag / denom)
}
operator fun unaryMinus() = Complex(-real, -imag)
operator fun minus(other: Complex) = this + (-other)
operator fun div(other: Complex) = this * other.inv()
val modulus: Double get() = sqrt(real * real + imag * imag)
override fun toString() =
if (imag >= 0.0) "$real + ${imag}i"
else "$real - ${-imag}i"
}
fun main(args: Array<String>) {
var fact = 1.0
val x = Complex(0.0, PI)
var e = Complex(1.0, PI)
var n = 2
var pow = x
do {
val e0 = e
fact *= n++
pow *= x
e += pow / Complex(fact, 0.0)
}
while ((e - e0).modulus >= EPSILON)
e += Complex(1.0, 0.0)
println("e^${SMALL_PI}i + 1 = $e $APPROX_EQUALS 0")
}
- Output:
e^πi + 1 = -8.881784197001252E-16 - 9.714919754267985E-17i ≅ 0
Lambdatalk
{require lib_complex}
'{C.exp {C.mul {C.new 0 1} {C.new {PI} 0}}} // e^πi = exp( [π,0] * [0,1] )
-> (-1 1.2246467991473532e-16) // = -1
Lua
local c = {
new = function(s,r,i) s.__index=s return setmetatable({r=r, i=i}, s) end,
add = function(s,o) return s:new(s.r+o.r, s.i+o.i) end,
exp = function(s) local e=math.exp(s.r) return s:new(e*math.cos(s.i), e*math.sin(s.i)) end,
mul = function(s,o) return s:new(s.r*o.r+s.i*o.i, s.r*o.i+s.i*o.r) end
}
local i = c:new(0, 1)
local pi = c:new(math.pi, 0)
local one = c:new(1, 0)
local zero = i:mul(pi):exp():add(one)
print(string.format("e^(i*pi)+1 is approximately zero: %.18g%+.18gi", zero.r, zero.i))
- Output:
e^(i*pi)+1 is approximately zero: 0+1.22460635382237726e-016i
> -- alternatively, equivalent one-liner from prompt:
> math.exp(0)*math.cos(math.pi)+1, math.exp(0)*math.sin(math.pi)
0.0 1.2246063538224e-016
Mathematica /Wolfram Language
E^(I Pi) + 1
- Output:
0
Maxima
is(equal(%e^(%i*%pi)+1,0));
- Output:
true
Nim
import math, complex
echo "exp(iπ) + 1 = ", exp(complex(0.0, PI)) + 1, " ~= 0"
- Output:
exp(iπ) + 1 = (0.0, 1.224646799147353e-16) ~= 0
OCaml
# open Complex;;
# let pi = acos (-1.0);;
val pi : float = 3.14159265358979312
# add (exp { re = 0.0; im = pi }) { re = 1.0; im = 0.0 };;
- : Complex.t = {re = 0.; im = 1.22464679914735321e-16}
Perl
use Math::Complex;
print exp(pi * i) + 1, "\n";
- Output:
1.22464679914735e-16i
Phix
with javascript_semantics include builtins\complex.e complex i = complex_new(0,1), res = complex_add(complex_exp(complex_mul(PI,i)),1) ?complex_sprint(res,both:=true) ?complex_sprint(complex_round(res,1e16),true) ?complex_sprint(complex_round(res,1e15),true)
- Output:
The actual result and two rounded versions (to prove the rounding is doing what it should - the second arg is an inverted precision).
"0+1.2246e-16i" "0+1e-16i" "0+0i"
Of course "symbolically" you can just do this (ha ha):
with javascript_semantics procedure reduce(string s) constant rules = {{"-1+1","0"}, {"-1+0","-1"}, {"i*0","0"}, {"sin(pi)","0"}, {"cos(pi)","-1"}, {"exp(i*pi)","cos(pi)+i*sin(pi)"}} string t = s sequence seen = {} -- (be safe and avoid infinite loops) while not find(t,seen) -- "" re-treading and length(t)<10000 do -- "" ever-growing seen = append(seen,t) bool found = false for i=1 to length(rules) do string {r,e} = rules[i] if match(r,t) then found = true t = substitute(t,r,e) end if end for if not found then exit end if printf(1,"%s = %s\n",{s,t}) s = repeat(' ',length(s)) end while end procedure reduce("exp(i*pi)+1")
- Output:
exp(i*pi)+1 = cos(pi)+i*sin(pi)+1
= -1+i*0+1
= -1+0+1
= -1+1
= 0
Prolog
Symbolically manipulates Euler's identity until it can't be further reduced (and we get zero :)
% reduce() prints the intermediate results so that one can see Prolog "thinking."
%
reduce(A, C) :-
simplify(A, B),
(B = A -> C = A; io:format("= ~w~n", [B]), reduce(B, C)).
simplify(exp(i*X), cos(X) + i*sin(X)) :- !.
simplify(0 + A, A) :- !.
simplify(A + 0, A) :- !.
simplify(A + B, C) :-
integer(A),
integer(B), !,
C is A + B.
simplify(A + B, C + D) :- !,
simplify(A, C),
simplify(B, D).
simplify(0 * _, 0) :- !.
simplify(_ * 0, 0) :- !.
simplify(1 * A, A) :- !.
simplify(A * 1, A) :- !.
simplify(A * B, C) :-
integer(A),
integer(B), !,
C is A * B.
simplify(A * B, C * D) :- !,
simplify(A, C),
simplify(B, D).
simplify(cos(0), 1) :- !.
simplify(sin(0), 0) :- !.
simplify(cos(pi), -1) :- !.
simplify(sin(pi), 0) :- !.
simplify(X, X).
- Output:
?- reduce(exp(i*pi)+1, X). = cos(pi)+i*sin(pi)+1 = -1+i*0+1 = -1+0+1 = -1+1 = 0 X = 0.
Python
>>> import math
>>> math.e ** (math.pi * 1j) + 1
1.2246467991473532e-16j
R
# lang R
exp(1i * pi) + 1
- Output:
0+1.224606e-16i
Symbolically with the Ryacas package (on CRAN):
library(Ryacas)
as_r(yac_str("Exp(I * Pi) + 1"))
- Output:
0
Racket
#lang racket
(+ (exp (* 0+i pi)) 1)
- Output:
0.0+1.2246063538223773e-016i
Raku
(formerly Perl 6)
Implementing an "invisible times" operator (Unicode character (U+2062)) to more closely emulate the layout. Alas, Raku does not do symbolic calculations at this time and is limited to IEEE 754 floating point for transcendental and irrational number calculations.
e, i and π are all available as built-in constants in Raku.
sub infix:<> is tighter(&infix:<**>) { $^a * $^b };
say 'e**iπ + 1 ≅ 0 : ', e**iπ + 1 ≅ 0;
say 'Error: ', e**iπ + 1;
- Output:
e**iπ + 1 ≅ 0 : True Error: 0+1.2246467991473532e-16i
REXX
The Euler formula (or Euler identity) states:
- eix = cos(x) + i sin(x)
Substituting x with yields:
- ei = cos( ) + i sin( )
So, using this Rosetta Code task's version of Euler's identity:
- ei + 1 = 0
then we have:
- cos( ) + i sin( ) + 1 = 0
So, if the left hand side is evaluated to zero, then Euler's identity is proven.
The REXX language doesn't have any trig or sqrt functions, so some stripped-down
RYO versions are included here.
The sqrt function below supports complex roots.
Note that REXX uses decimal floating point, not binary. REXX also
uses a guard (decimal) digit when multiplying
and dividing, which aids in increasing the precision.
This REXX program calculates the trigonometric functions (sin and
cos) to around half of the number of decimal
digits that are used in defining the pi constant in the REXX program; so
the limiting factor for accuracy for the
trigonometric functions is based on the number of decimal digits (accuracy)
of pi being defined within the REXX
program.
/*REXX program proves Euler's identity by showing that: e^(i pi) + 1 ≡ 0 */
numeric digits length( pi() ) - length(.) /*define pi; set # dec. digs precision*/
cosPI= fmt( cos(pi) ) /*calculate the value of cos(pi). */
sinPI= fmt( sin(pi) ) /* " " " " sin(pi). */
say ' cos(pi) = ' cosPI /*display " " " cos(Pi). */
say ' sin(pi) = ' sinPI /* " " " " sin(Pi). */
say /*separate the wheat from the chaff. */
$= cosPI + mult( sqrt(-1), sinPI ) + 1 /*calc. product of sin(x) and sqrt(-1).*/
say ' e^(i pi) + 1 = ' fmt($) ' ' word("unproven proven", ($=0) + 1)
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
fmt: procedure; parse arg x; x= format(x, , digits() %2, 0); return left('', x>=0)x /1
mult: procedure; parse arg a,b; if a=0 | b=0 then return 0; return a*b
pi: pi= 3.1415926535897932384626433832795028841971693993751058209749445923; return pi
cos: procedure; parse arg x; z= 1; _= 1; q= x*x; i= -1; return .sinCos()
sin: procedure; parse arg x 1 z 1 _; q= x*x; i= 1; return .sinCos()
.sinCos: do k=2 by 2 until p=z; p=z; _= -_ * q/(k*(k+i)); z= z+_; end; return z
/*──────────────────────────────────────────────────────────────────────────────────────*/
sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); i=; h= d+6
numeric digits; numeric form; if x<0 then do; x= -x; i= 'i'; end; m.= 9
parse value format(x, 2, 1, , 0) 'E0' with g 'E' _ .; g= g * .5'e'_ % 2
do j=0 while h>9; m.j= h; h= h % 2 + 1; end
do k=j+5 to 0 by -1; numeric digits m.k; g= (g+x/g) *.5; end; return g || i
- output when using the internal default input:
cos(pi) = -1
sin(pi) = 0
e^(i pi) + 1 = 0 proven
Programming note:
To increase the decimal precision of the trigonometric functions past the
current 500 decimal digits in the above REXX program,
use the following REXX assignment statement (the author has a REXX
program with 1,000,052 decimal digits of pi that can be
programmatically invoked with the requested number of decimal digits).
/*────────────────── 1,051 decimal digs of pi. ──────────────────*/
pi= 3.14159265358979323846264338327950288419716939937510
pi= pi || 58209749445923078164062862089986280348253421170679
pi= pi || 82148086513282306647093844609550582231725359408128
pi= pi || 48111745028410270193852110555964462294895493038196
pi= pi || 44288109756659334461284756482337867831652712019091
pi= pi || 45648566923460348610454326648213393607260249141273
pi= pi || 72458700660631558817488152092096282925409171536436
pi= pi || 78925903600113305305488204665213841469519415116094
pi= pi || 33057270365759591953092186117381932611793105118548
pi= pi || 07446237996274956735188575272489122793818301194912
pi= pi || 98336733624406566430860213949463952247371907021798
pi= pi || 60943702770539217176293176752384674818467669405132
pi= pi || 00056812714526356082778577134275778960917363717872
pi= pi || 14684409012249534301465495853710507922796892589235
pi= pi || 42019956112129021960864034418159813629774771309960
pi= pi || 51870721134999999837297804995105973173281609631859
pi= pi || 50244594553469083026425223082533446850352619311881
pi= pi || 71010003137838752886587533208381420617177669147303
pi= pi || 59825349042875546873115956286388235378759375195778
pi= pi || 18577805321712268066130019278766111959092164201989
pi= pi || 38095257201065485863278865936153381827968230301952
RPL
'EXP(i*π)+1' →NUM ABS
- Output:
1.22464679915E-16
Ruby
include Math
E ** (PI * 1i) + 1
# => (0.0+0.0i)
Rust
use std::f64::consts::PI;
extern crate num_complex;
use num_complex::Complex;
fn main() {
println!("{:e}", Complex::new(0.0, PI).exp() + 1.0);
}
- Output:
0e0+1.2246467991473532e-16i
Scala
This example makes use of Spire's numeric data types. Complex takes a type parameter determining the type of the coefficients of a + bi, and Real is Spire's exact (i.e. arbitrary precision) numeric data type.
import spire.math.{Complex, Real}
object Scratch extends App{
//Declare values with friendly names to clean up the final expression
val e = Complex[Real](Real.e, 0)
val pi = Complex[Real](Real.pi, 0)
val i = Complex[Real](0, 1)
val one = Complex.one[Real]
println(e.pow(pi*i) + one)
}
- Output:
(0 + 0i)
Scheme
; A way to get pi.
(define pi (acos -1))
; Print the value of e^(i*pi) + 1 -- should be 0.
(printf "e^(i*pi) + 1 = ~a~%" (+ (exp (* +i pi)) 1))
- Output:
e^(i*pi) + 1 = 0.0+1.2246467991473532e-16i
A higher precision test using series approximations to Pi and the Exp() function. The series are computed using exact rational numbers. This code converts the rational results into decimal representation.
; Procedure to compute factorial.
(define fact
(lambda (n)
(if (<= n 0)
1
(* n (fact (1- n))))))
; Use series to compute approximation to Pi (using N terms of series).
; (Uses the Newton / Euler Convergence Transformation.)
(define pi-series
(lambda (n)
(do ((k 0 (1+ k))
(sum 0 (+ sum (/ (* (expt 2 k) (expt (fact k) 2)) (fact (1+ (* 2 k)))))))
((>= k n) (* 2 sum)))))
; Use series to compute approximation to exp(z) (using N terms of series).
(define exp-series
(lambda (z n)
(do ((k 0 (1+ k))
(sum 0 (+ sum (/ (expt z k) (fact k)))))
((>= k n) sum))))
; Convert the given Rational number to a Decimal string.
; If opt contains an integer, show to that many places past the decimal regardless of repeating.
; If opt contains 'nopar, do not insert the parentheses indicating the repeating places.
; If opt contains 'plus, prefix positive numbers with plus ('+') sign.
; N.B.: When number of decimals specified, this truncates instead of rounds.
(define rat->dec-str
(lambda (rat . opt)
(let* ((num (abs (numerator rat)))
(den (abs (denominator rat)))
(no-par (find (lambda (a) (eq? a 'nopar)) opt))
(plus (find (lambda (a) (eq? a 'plus)) opt))
(dec-lim (find integer? opt))
(rep-inx #f)
(rems-seen '())
(int-part (format (cond ((< rat 0) "-~d") (plus "+~d") (else "~d")) (quotient num den)))
(frc-list
(cond
((zero? num)
'())
(else
(let loop ((rem (modulo num den)) (decs 0))
(cond
((or (<= rem 0) (and dec-lim (>= decs dec-lim)))
'())
((and (not dec-lim) (assq rem rems-seen))
(set! rep-inx (cdr (assq rem rems-seen)))
'())
(else
(set! rems-seen (cons (cons rem decs) rems-seen))
(cons
(integer->char (+ (quotient (* 10 rem) den) (char->integer #\0)))
(loop (modulo (* 10 rem) den) (1+ decs))))))))))
(when (and rep-inx (not no-par))
(set! frc-list (append
(list-head frc-list rep-inx)
(list #\()
(list-tail frc-list rep-inx)
(list #\)))))
(if (null? frc-list)
int-part
(format "~a.~a" int-part (list->string frc-list))))))
; Convert the given Rational Complex number to a Decimal string.
; If opt contains an integer, show to that many places past the decimal regardless of repeating.
; If opt contains 'nopar, do not insert the parentheses indicating the repeating places.
; If opt contains 'plus, prefix positive numbers with plus ('+') sign.
; N.B.: When number of decimals specified, this truncates instead of rounds.
(define rat-cplx->dec-str
(lambda (rat-cplx . opt)
(let* ((real-dec-str (apply rat->dec-str (cons (real-part rat-cplx) opt)))
(imag-dec-str (apply rat->dec-str (cons (imag-part rat-cplx) (cons 'plus opt)))))
(format "~a~ai" real-dec-str imag-dec-str))))
; Print the value of e^(i*pi) + 1 -- should be 0.
; (Computed using the series defined above.)
(let*
((pi (pi-series 222))
(e-pi-i (exp-series (* pi +i) 222))
(euler-id (+ e-pi-i 1)))
(printf "e^(i*pi) + 1 = ~a~%" (rat-cplx->dec-str euler-id 70)))
- Output:
e^(i*pi) + 1 = 0.0000000000000000000000000000000000000000000000000000000000000000000000 +0.0000000000000000000000000000000000000000000000000000000000000000000351i
Sidef
say ('e**iπ + 1 ≅ 0 : ', Num.e**Num.pi.i + 1 ≅ 0)
say ('Error: ', Num.e**Num.pi.i + 1)
- Output:
e**iπ + 1 ≅ 0 : true Error: -2.42661922624586582047028764157944836122122513308e-58i
Tcl
Using tcllib
# Set up complex sandbox (since we're doing a star import)
namespace eval complex_ns {
package require math::complexnumbers
namespace import ::math::complexnumbers::*
set pi [expr {acos(-1)}]
set r [+ [exp [complex 0 $pi]] [complex 1 0]]
puts "e**(pi*i) = [real $r]+[imag $r]i"
}
- Output:
e**(pi*i) = 0.0+1.2246467991473532e-16i
Using VecTcl
package require vectcl
namespace import vectcl::vexpr
set ans [vexpr {pi=acos(-1); exp(pi*1i) + 1}]
puts "e**(pi*i) = $ans"
- Output:
e**(pi*i) = 0.0+1.2246063538223773e-16i
Wren
import "./complex" for Complex
System.print((Complex.new(0, Num.pi).exp + Complex.one).toString)
- Output:
0 + 1.2246467991474e-16i
zkl
var [const] GSL=Import("zklGSL"); // libGSL (GNU Scientific Library)
Z,pi,e := GSL.Z, (0.0).pi, (0.0).e;
println("e^(\u03c0i) + 1 = %s \u2245 0".fmt( Z(e).pow(Z(0,1)*pi) + 1 ));
println("TMI: ",(Z(e).pow(Z(0,1)*pi) + 1 ).format(0,25,"g"));- Output:
e^(πi) + 1 = (0.00+0.00i) ≅ 0 TMI: (0+1.224646799147353207173764e-16i)