Euler's identity
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In mathematics, Euler's identity (also known as Euler's equation) is the equality:
e^{iπ} + 1 = 0
where
e is Euler's number, the base of natural logarithms, i is the imaginary unit, which satisfies i^{2} = −1, and π 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 π (π = 3.141...). The number e (e = 2.718...), 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 e^{iπ} + 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 e^{iπ} + 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) ~ " , fixed( re OF eulers identity, -23, 20 ) , " " , fixed( im OF eulers identity, 23, 20 ) , "i" , newline ) )
END</lang>
- Output:
e^(i*pi) ~ 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
Haskell
A float is not quite real. <lang Haskell>import Data.Complex
eulerIdentityZeroIsh :: Complex Float eulerIdentityZeroIsh =
exp (0 :+ 1 * pi) + 1
main :: IO () main = print eulerIdentityZeroIsh</lang>
- Output:
Zero and a little floating dust ...
0.0 :+ (-8.742278e-8)
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
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
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)