Euler's identity
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) |
In mathematics, Euler's identity (also known as Euler's equation) is the equality:
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.
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).
<lang algol68>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</lang>
- Output:
e^(i*pi) + 1 ~ 0.00000000000000000307 -0.00000000000000002926i
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. <lang c>#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;
}</lang>
- Output:
e ^ πi + 1 = [0.0000000000000000, 0.0000000000000001] ≅ 0
C++
<lang cpp>#include <iostream>
- include <complex>
int main() {
std::cout << std::exp(std::complex<double>(0.0, M_PI)) + 1.0 << std::endl; return 0;
}</lang>
- Output:
Zero and a little floating dust ...
(0,1.22465e-16)
C#
<lang csharp>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);
}
}</lang>
- Output:
(0, 1.22464679914735E-16)
Common Lisp
Common Lisp has complex number arithmetic built into it. <lang Common Lisp> (+ 1 (exp (complex 0 pi))) </lang>
#C(0.0L0 -5.0165576136843360246L-20)
Factor
<lang factor>USING: math math.constants math.functions prettyprint ; 1 e pi C{ 0 1 } * ^ + .</lang>
- Output:
C{ 0.0 1.224646799147353e-016 }
Go
<lang go>package main
import (
"fmt" "math" "math/cmplx"
)
func main() {
fmt.Println(cmplx.Exp(math.Pi * 1i) + 1.0)
}</lang>
- 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. <lang groovy>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</lang> 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. <lang Haskell>import Data.Complex
eulerIdentityZeroIsh :: Complex Double eulerIdentityZeroIsh =
exp (0 :+ pi) + 1
main :: IO () main = print eulerIdentityZeroIsh</lang>
- Output:
Zero and a little floating dust ...
0.0 :+ 1.2246467991473532e-16
J
<lang 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 </lang>
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. <lang jq>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); </lang>
The Task
<lang jq>"e^iπ: \( exp( [0, pi ] ) )", "e^iπ + 1: \( plus(1; exp( [0, pi ] ) ))"</lang>
- Output:
e^iπ: [-1,1.2246467991473532e-16] e^iπ + 1: [0,1.2246467991473532e-16]
Julia
Julia has a builtin Complex{T} parametrized type.
<lang julia>@show ℯ^(π * im) + 1 @assert ℯ^(π * im) ≈ -1</lang>
- Output:
e ^ (π * im) + 1 = 0.0 + 1.2246467991473532e-16im
Using symbolic algebra, through the Reduce.jl package.
<lang julia>using Reduce @force using Reduce.Algebra
@show e ^ (π * :i) + 1 @assert e ^ (π * :i) + 1 == 0</lang>
- 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). <lang scala>// 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")
}</lang>
- Output:
e^πi + 1 = -8.881784197001252E-16 - 9.714919754267985E-17i ≅ 0
OCaml
<lang 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}</lang>
Perl
<lang perl>use Math::Complex; print exp(pi * i) + 1, "\n";</lang>
- Output:
1.22464679914735e-16i
Perl 6
Implementing an "invisible times" operator (Unicode character (U+2062)) to more closely emulate the layout. Alas, Perl 6 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 Perl 6.
<lang perl6>sub infix:<> is tighter(&infix:<**>) { $^a * $^b };
say 'e**iπ + 1 ≅ 0 : ', e**iπ + 1 ≅ 0; say 'Error: ', e**iπ + 1;</lang>
- Output:
e**iπ + 1 ≅ 0 : True Error: 0+1.2246467991473532e-16i
Phix
<lang Phix>include builtins\complex.e -- (0.8.0+) complex 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))</lang>
- 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"
Python
<lang python>>>> import math >>> math.e ** (math.pi * 1j) + 1 1.2246467991473532e-16j</lang>
Racket
<lang racket>#lang racket (+ (exp (* 0+i pi)) 1)</lang>
- Output:
0.0+1.2246063538223773e-016i
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.
<lang rexx>/*REXX program proves Euler's identity by showing that: e^(i pi) + 1 ≡ 0 */
numeric digits length( pi() ) - 1 /*set number of decimal 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. */
$= cosPi + mult( sqrt(-1), sinPi) + 1 /*calc. product of sin(x) and sqrt(-1).*/
say 'e^(i pi) + 1 = ' fmt($) proof($ = 0) /*display both sides of the equation. */ 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 proof: procedure; parse arg ?; return ' ' word("unproven proven", ? + 1) cos: procedure; parse arg x; return .sinCos(1, -1) sin: procedure; parse arg x; return .sinCos(x, 1) /*──────────────────────────────────────────────────────────────────────────────────────*/ .sinCos: parse arg z 1 _,i; q=x*x; do k=2 by 2 until p=z; p=z; _=-_*q/(k*(k+i)); z=z+_
end /*k*'; return z
/*──────────────────────────────────────────────────────────────────────────────────────*/ sqrt: procedure; parse arg x; if x=0 then return 0; d=digits(); i=; m.=9; h=d+6
numeric digits; numeric form; if x<0 then do; x= -x; i= 'i'; end
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 /*j*/
do k=j+5 to 0 by -1; numeric digits m.k; g= (g+x/g) * .5; end /*k*/
numeric digits d; return (g/1)i /*make complex if X < 0.*/</lang>
- output:
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).
<lang rexx>/*────────────────── 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 </lang>
Ruby
<lang ruby>> require 'complex' > Math::E ** (Math::PI * Complex::I) + 1 => (0.0+0.0i)</lang>
Rust
<lang 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);
}</lang>
- 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.
<lang scala>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)
}</lang>
- Output:
(0 + 0i)
Sidef
<lang ruby>say ('e**iπ + 1 ≅ 0 : ', Num.e**Num.pi.i + 1 ≅ 0) say ('Error: ', Num.e**Num.pi.i + 1)</lang>
- Output:
e**iπ + 1 ≅ 0 : true Error: -2.42661922624586582047028764157944836122122513308e-58i
zkl
<lang 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"));</lang>
- Output:
e^(πi) + 1 = (0.00+0.00i) ≅ 0 TMI: (0+1.224646799147353207173764e-16i)