32 lines
1.5 KiB
Plaintext
32 lines
1.5 KiB
Plaintext
In mathematics, ''Euler's identity'' is the equality:
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<span style="font-size:150%;font-style:bold;"><span style="font-style:italic">e<sup>i<math>\pi</math></sup></span> + 1 = 0</span>
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where
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e is Euler's number, the base of natural logarithms,
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''i'' is the imaginary unit, which satisfies ''i''<sup>2</sup> = −1, and
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<math>\pi</math> is pi, the ratio of the circumference of a circle to its diameter.
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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:
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The number 0.
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The number 1.
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The number <math>\pi</math> (<math>\pi</math> = 3.14159<small>+</small>),
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The number e (e = 2.71828<small>+</small>), which occurs widely in mathematical analysis.
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The number ''i'', the imaginary unit of the complex numbers.
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;Task
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Show in your language that Euler's identity is true. As much as possible and practical, mimic the Euler's identity equation.
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Most languages are limited to IEEE 754 floating point calculations so will have some error in the calculation.
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If that is the case, or there is some other limitation, show
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that <big>e<sup>i<math>\pi</math></sup> + 1</big> is ''approximately'' equal to zero and
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show the amount of error in the calculation.
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If your language is capable of symbolic calculations, show
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that <big>e<sup>i<math>\pi</math></sup> + 1</big> is ''exactly'' equal to zero for bonus kudos points.
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<br><br>
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