What is Euler's identity really saying?

[MadViolinist]

So it is true that e^i∏+1=0. But what does this mean? Why are all these numbers linked?

 

[DonAntonio]

So it is true that e^i∏+1=0. But what does this mean? Why are all these numbers linked?

They are linked precisely by that equation, and since the equality $e^{i\theta}:=\cos\theta+i\sin\theta\,\,,\,\,\theta\in\mathbb{R}\,\,$ follows at once say from the definition

of the complex exponential function as power series (or as limit of a sequence), the above identity is really trivial.

DonAntonio

 

[HallsofIvy]

Look at the MacLaurin series for those functions:
$$e^x= 1+ x+ x^2/2!+ x^3/3!+ \cdot\cdot\cdot+ x^n/n!$$
$$cos(x)= 1- x^2/2!+ x^4/4!- x^6/6!+ \cdot\cdot\cdot+ (-1)^nx^{2n}/(2n)!$$
$$sin(x)= x- x^3/3!+ x^5/5!- x^7/7!+ \cdot\cdot\cdot+ (-1)^nx^{2n+1}/(2n)!$$

If you replace x with the imaginary number ix (x is still real) that becomes
$$e^{ix}= 1+ ix+ (ix)^2/2!+ (ix)^3/3!+ \cdot\cdot\cdot+ (ix)^n/n!$$
$$e^{ix}= 1+ ix+ i^2x^2/2!+ i^3x^3/3!+ \cdot\cdot\cdot+ i^nx^n/n!$$

But it is easy to see that, since $i^2= -1$, $(i)^3= (i)^2(i)= -i$, $(i)^4= (i^3)(i)= -i(i)= -(-1)= 1$ so then it starts all over: $i^5= (i^5)i= i$, etc. That is, all even powers of i are 1 if the power is 0 mod 4 and -1 if it is 2 mod 4. All odd powers are i if the power is 1 mod 4 and -i if it is 3 mod 4.

$$e^{ix}= 1+ ix- x^2/2!- ix^3/3!+ \cdot\cdot\cdot$$

Separating into real and imaginary parts,
$$e^{ix}= (1- x^2/2!+ x^4/4!- x^6/6!+ \cdot\cdot\cdot)+ i(x- x^3/3!+ x^5/5!+ \cdot\cdot\cdot)$$
$$e^{ix}= cos(x)+ i sin(x)$$

Now, take $= \pi$ so that $cos(x)= cos(\pi)= -1$ and $sin(x)= sin(\pi)= 0$ and that becomes
$$e^{i\pi}= -1$$
or
$$e^{i\pi}+ 1= 0$$

I hope that is what you are looking for. Otherwise, what you are asking is uncomfortably close to "number mysticism".

 

[Max™]

http://betterexplained.com/articles/intuitive-understanding-of-eulers-formula/

Perhaps that is what they were looking for?

 

[CellCoree]

Euler's identity is a fundamental equation that links together five important mathematical constants: e (the base of natural logarithms), i (the imaginary unit), π (pi), 1 (the multiplicative identity), and 0 (the additive identity). It states that when the imaginary unit is raised to the power of pi and added to 1, the result is 0. This may seem counterintuitive and complex, but it has significant implications in mathematics and physics.

Essentially, Euler's identity is saying that there is a deep connection between exponential functions (e), trigonometric functions (π), and complex numbers (i). It demonstrates how these seemingly unrelated concepts are intricately connected and can be used to solve various problems in mathematics and science.

One interpretation of Euler's identity is that it shows the relationship between rotation and growth. The imaginary unit, i, represents a quarter turn in the complex plane, while the exponential function, e, represents exponential growth. The constant π, which relates to circles and periodic functions, brings these two concepts together in a single equation.

Another interpretation is that Euler's identity is a special case of the more general Euler's formula, which states that e^(ix) = cos(x) + i sin(x). This formula is used extensively in fields such as signal processing, quantum mechanics, and electrical engineering.

In summary, Euler's identity is a powerful mathematical equation that links together five important constants and demonstrates the deep connections between seemingly different mathematical concepts. Its significance extends beyond pure mathematics and has important applications in various fields of science.