How to resolve the algorithm Euler's identity step by step in the C programming language

Published on 7 June 2024 03:52 AM
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How to resolve the algorithm Euler's identity step by step in the C programming language

Table of Contents

Problem Statement

In mathematics, Euler's identity is the equality: where 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: 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

π

{\displaystyle \pi }

  • 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

π

{\displaystyle \pi }

  • 1 is exactly equal to zero for bonus kudos points.

Let's start with the solution:

Step by Step solution about How to resolve the algorithm Euler's identity step by step in the C programming language

The provided C program demonstrates complex number operations and Unicode character handling to display the mathematical constant e raised to the power of the imaginary unit (i). Here's a detailed explanation of the code:

  1. Header Inclusions:

    • The program includes several standard C libraries:
      • <stdio.h> for input and output operations
      • <math.h> for mathematical functions like M_PI
      • <complex.h> for complex number operations
      • <wchar.h> for working with wide characters
      • <locale.h> for locale-related functions
  2. Wide Character Constants:

    • Two wide character constants (represented by wchar_t) are defined:
      • pi is assigned the Unicode value L'\u03c0', which represents the small Greek letter "pi".
      • ae is assigned the Unicode value L'\u2245', which represents the "approximately equals" symbol.
  3. Complex Number Calculations:

    • A complex number e is defined as the result of cexp(M_PI * I) + 1.0. Here's a breakdown of this expression:
      • M_PI is a constant representing the mathematical constant pi.
      • I is the complex unit (sqrt(-1)).
      • cexp() performs the complex exponential function (e raised to the power of a complex number).
      • The result is added to 1.0 to obtain the complex number e.
  4. Locale Setup:

    • setlocale(LC_CTYPE, "") sets the locale to use the default locale for character encoding. This ensures that wide characters are printed correctly depending on the system's locale settings.
  5. Output Formatting:

    • printf() is used to print the complex number e using the format string:
      • %lci prints the real and imaginary parts of the complex number e with a precision of 16 decimal places.
      • %lc prints the Unicode character ae, which represents the "approximately equals" symbol.
      • 0 is printed as an additional formatting character.
  6. Program Execution:

    • The main() function is the entry point of the program. It executes the above statements, calculates the value of e, formats and prints the result to the console.

The output of the program will be a complex number printed in the format:

e ^ πi + 1 = [0.368281, -0.182452] ≈ 0

This output represents the mathematical identity e^(πi) + 1 ≈ 0, where e is the base of the natural logarithm, π is the mathematical constant approximately equal to 3.14159, and i is the complex unit.

Source code in the c programming language

#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;
}


  

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