Euler's identity

From Wikinfo

Jump to: navigation, search


[[de:Eulersche Identit�t]][[fr:Identit� d'Euler]] In mathematics, Euler's identity, a special case of Euler's formula, is the following:

<math> e^{i \pi} + 1 = 0\; </math>

The equation appears in Leonhard Euler's Introduction, published in Lausanne in 1748. In this equation, e is the base of the natural logarithm, <math>i</math> is the imaginary unit (an imaginary number with the property i� = -1), and <math> \pi </math> is Archimedes' constant pi (π, the ratio of the circumference of a circle to its diameter).

It was called "the most remarkable formula in mathematics" by Richard Feynman. Feynman found this formula remarkable because it links some very fundamental mathematical constants. Here are some interesting properties of these constants:

  • The number 0 is the identity element for addition for all a a+0=0+a=a. See Group (mathematics).
  • The number 1 is the identity element for multiplication for all a a�1=1�a=a.

Both 0 and 1 are elementary for counting and arithmetic.

  • The number <math>\pi </math> is a fundamental number for trigonometry. <math>\pi </math> is a constant in a world which is Euclidean, on small scales at least (otherwise, the ratio of the length of the circumference of circle to its diameter would not be a universal constant, i.e. the same for all circumferences).
  • The number <math>e</math> is a fundamental number for logarithms. Also, <math>e</math> is important in describing growth behaviors, as the solution to the simplest growth equation <math>dy / dx = y</math> with initial condition <math>y(0) = 1</math> is <math>y = e^x</math>.
  • Finally, the imaginary unit <math>i</math> (where i� = -1) is a unit in the complex numbers, and is the simplest purely imaginary complex number. Introducing this unit yields all non-constant polynomial equations soluble in the field of complex numbers. (see Fundamental Theorem of Algebra).

Also, and perhaps more important, all the most fundamental operators of arithmetic are also present: equality, addition, multiplication and exponentation. All the fundamental assumptions of complex analysis are present, and the integers 0 and 1 are related to a field of complex numbers that, prior to the formula, simply didn't exist!

The formula is a special case of Euler's formula from complex analysis, which states that

<math>e^{ix} = \cos x + i \sin x \,\!</math>

for any real number <math>x</math>. If we set <math>x = \pi</math>, then

<math>e^{i \pi} = \cos \pi + i \sin \pi \,\!</math>

and since cos(π) = -1 and sin(π) = 0, we get

<math>e^{i \pi} = -1 \,\!</math>

and therefore

<math>e^{i \pi} + 1 = 0 \,\!</math>

There has been substantial debate in the philosophy of mathematics on the "real meaning" or "deep meaning" or even sacred geometry reflected by the Identity's relationship of key constants and operations (multiplication, exponentiation, addition, equality). Some assert that it describes cognitive properties of an embodied mind - and advocate a cognitive science of mathematics. At other extremes, some assert it represents rational social conesnsus of mathematicians, or is simply a fundamental fact of the physical universe, and that algebra itself is a natural consequence of its structure. If so, the formula would be more than simply remarkable - it would be 'divine'.

References

  • Feynman RP - The Feynman Lectures on Physics, vol. I - part 1. Inter European Editions, Amsterdam (1975)

External link


References

Retrieved from "http://www.wikinfo.org/index.php/Euler%27s_identity"
In other languages