What is e^(pi*i) = - 1? Learn How to Solve it

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In summary, the conversation discusses the equation e^(pi*i) = -1 and its relevance and usage in calculus. It is also mentioned that e is a very important number in calculus and has various definitions. The result of e^(pi*i) = -1 comes from complex numbers and the formula e^(theta*i) = cos(theta) + i*sin(theta). A link is provided for further discussion on the topic.
  • #1
DennisG
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ok, so I saw this thing one time that looked like this:

e^(pi*i) = - 1

can anyone tell me what this is, what it's used for, and especially how it works? A friend showed me the equation one time, but neither of us knew a thing about it.

Thanks for any help you can offer,
Dennis
 
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  • #2
Its Euler formula coming from the expansions of sinx, cosx, e^x
e(PI^i)=cis(PI)=-1
 
  • #3
So as not to be to repetitive, see this thread
 
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  • #4
I'm going to assume from the way you asked the question that you don't know what e is in the 1st place? (Please don't take this as an offence if you do know it)

There are lots of definitions of e, it is probably the most important number in calculus. Here are a few definitions:

[tex]e = \sum_{n=0}^{\infty} \frac{1}{n!} = \lim_{x \rightarrow \infty} \left( 1 + \frac{1}{x} \right)^x[/tex]

Also:

[tex]\frac{d}{dx} \left( e^x \right) = e^x[/tex]

And:

[tex] \int_1^e \frac{dx}{x} = 1[/tex]

Edit: Amended thanks to below post.

As well as:

[tex]e = 2.718281828459045235326 \ldots[/tex]

As for the result:

[tex]e^{\pi i} = -1[/tex]

This comes from some maths orientated around complex numbers which yields the formulae:

[tex]e^{\theta i} = \cos \theta + i \sin \theta[/tex]

(More about the above result in a link given I believe)

Hope that helps.
 
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  • #5
I posted the link to the other thread in hopes that further posts on this topic would be in the existing thread.

EDIT: Oh yeah, I forgot to mention

[tex] \int^e _1 \frac {dx} x = 1 [/tex] not e.
 
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What is e^(pi*i) = -1?

e^(pi*i) = -1 is a mathematical equation that relates the five most important numbers in mathematics: e, pi, i, 1, and 0. It states that when the number e (the base of the natural logarithm) is raised to the power of pi (the ratio of a circle's circumference to its diameter), and then multiplied by i (the imaginary unit, equal to the square root of -1), the result is -1 (the additive inverse of the multiplicative identity).

Why is e^(pi*i) = -1 important?

This equation is important because it connects seemingly unrelated mathematical concepts, such as the exponential function, trigonometric functions, and imaginary numbers. It also has applications in physics, engineering, and other fields. Additionally, it has been called the most beautiful equation in mathematics due to its simplicity and elegance.

How do you solve e^(pi*i) = -1?

There are a few ways to solve this equation, depending on the mathematical tools and knowledge you have. One approach is to use the definition of e^(pi*i) as a complex exponential function and use Euler's formula to rewrite it in terms of sine and cosine. Another method is to use the Taylor series expansion of e^(pi*i) and evaluate it at the value of i.

What are the practical applications of e^(pi*i) = -1?

One practical application of this equation is in the study of alternating current circuits, where complex numbers are used to represent the relationship between voltage and current. It also has applications in signal processing, quantum mechanics, and Fourier analysis.

What is the history of e^(pi*i) = -1?

This equation was first discovered by the Swiss mathematician Leonhard Euler in the 18th century. However, the concept of complex numbers and their relationship to the exponential function can be traced back to the 16th century mathematician Rafael Bombelli. The equation has since been studied and expanded upon by many mathematicians and scientists, solidifying its importance in mathematics.

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