Unraveling Euler's Formula: e^(pi*i) +1 = 0

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In summary, the conversation discusses Euler's formula, specifically e^{i\pi}+ 1= 0, and the use of complex numbers in exponentiation. The cool root stuff mentioned is also well-known and there may be other values for (-1)^{1/i} depending on the branch of the logarithm used.
  • #1
strid
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Just wanted to share a cool thing I found when I was shown Eulers formula...

e^(pi*i) +1 = 0

this can be written as

e^pi = (SQRT -1)ROOT -1

dont know if I wrote correclty...

.................
.................
...#######...########################......
.##...#...#...#...............
...#.#...--.#...#......#.........
...#...#... #......#.........
......#...----...#..........
..########...#......#............
....#...#......#.........
.....#.#.....#.........
......#.............
.................

Took a time to write this :)


so can you imagine the number above to equal e^pi which is th real number around 23...? :smile:
 
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  • #2
Yes,
[tex]e^{i\pi}+ 1= 0[/tex]

(click on that to see the code I used)

I suspect that just about everyone on this board already knows that- there have been a number of threads about it.
 
  • #3
but is the format i wrote it in also well-known? the cool root stuff I mean...
 
  • #4
Yes it is. Exponentiation by complex numbers gives some startling results when you first see it.

Note that [tex]e^{\pi}[/tex] isn't the only answer for [tex](-1)^{1/i}[/tex], it depends on the branch of the logarithm you used. Can you find the other values? They might be even more suprising...
 

FAQ: Unraveling Euler's Formula: e^(pi*i) +1 = 0

What is Euler's formula and what does it represent?

Euler's formula is a mathematical equation that relates the exponential function, trigonometric functions, and complex numbers. It is commonly written as e^(pi*i) + 1 = 0, where e is the base of the natural logarithm, pi is the mathematical constant representing the ratio of a circle's circumference to its diameter, and i is the imaginary unit. This formula represents a connection between seemingly unrelated mathematical concepts and has many applications in different fields of science and engineering.

How did Euler come up with this formula?

The formula is named after Swiss mathematician Leonhard Euler, who discovered it in the 18th century. He derived it by using the Taylor series expansion for the cosine and sine functions, and then substituting pi*i for x in the series. This led to the relationship between the exponential function and the trigonometric functions, resulting in the famous formula e^(pi*i) + 1 = 0.

What is the significance of the number e in this formula?

The number e, also known as Euler's number, is a mathematical constant that is approximately equal to 2.71828. It is the base of the natural logarithm and has many important applications in mathematics, physics, and other sciences. In Euler's formula, e serves as the key link between the exponential function and the trigonometric functions, allowing for a beautiful and elegant representation of complex numbers.

How is Euler's formula used in real-world applications?

Euler's formula has a wide range of applications in different fields of science and engineering. It is used in signal processing, electrical engineering, quantum mechanics, fluid dynamics, and many other areas. It is also used to solve differential equations, which are essential for modeling and predicting various phenomena in science and engineering.

Are there any extensions or variations of Euler's formula?

Yes, there are several extensions and variations of Euler's formula, such as the generalized Euler's formula, which involves complex numbers raised to a power other than pi*i. There is also the Euler's identity, which is a special case of Euler's formula when pi is replaced by a multiple of pi, such as 2*pi or 3*pi. These variations have their own unique applications and implications in mathematics and science.

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