Why does (e^{i\alpha})^2 always equal 1?

[Piano man]

In solutions to a problem I was working on, I saw that when an expression such as $$e^{i\alpha}$$ , alpha being an angle (in polar coordinates), was squared, the expression goes to unity, ie $$(e^{i\alpha})^2=1$$
But I see no reason to think that $$\alpha$$ is a multiple of $$\pi$$.
Could there be any other reason why it would be 1?

 

[Mark44]

In solutions to a problem I was working on, I saw that when an expression such as $$e^{i\alpha}$$ , alpha being an angle (in polar coordinates), was squared, the expression goes to unity, ie $$(e^{i\alpha})^2=1$$
But I see no reason to think that $$\alpha$$ is a multiple of $$\pi$$.
Could there be any other reason why it would be 1?

Since $$(e^{i\alpha})^2=1$$
$$e^{i\alpha} = \pm 1$$
so $\alpha = k \pi~ \text{for some integer k}$

 

[SpectraCat]

In solutions to a problem I was working on, I saw that when an expression such as $$e^{i\alpha}$$ , alpha being an angle (in polar coordinates), was squared, the expression goes to unity, ie $$(e^{i\alpha})^2=1$$
But I see no reason to think that $$\alpha$$ is a multiple of $$\pi$$.
Could there be any other reason why it would be 1?

I don't think it actually said: $$(e^{i\alpha})^2=1$$

I'll bet what it said was: $$|e^{i\alpha}|^2=1$$

That notation indicates the square-modulus of a complex quantity, rather than just the square. In other words:

$$|e^{i\alpha}|^2=(e^{i\alpha})(e^{i\alpha})^\star=e^{i\alpha}e^{-i\alpha}=1$$

Thus $$\alpha$$ can still be an arbitrary real number (it cannot be an imaginary number).

 

[Piano man]

Ah! That's probably right - thank you :D

 

[CellCoree]

The reason why the expression (e^{i\alpha})^2=1 is equal to 1 is due to the properties of complex numbers and the definition of the imaginary unit i. In complex numbers, the imaginary unit i is defined as the square root of -1, which means that i^2=-1. Therefore, when we square e^{i\alpha}, we get (e^{i\alpha})^2=e^{i\alpha} \cdot e^{i\alpha}=e^{2i\alpha}. Using the definition of i, we can rewrite this as e^{2i\alpha}=(e^{i\alpha})^2=(-1)^2=1. This is true for any value of \alpha, whether it is a multiple of \pi or not.

In other words, the squaring of e^{i\alpha} is not related to the value of \alpha being a multiple of \pi, but rather it is a consequence of the properties of complex numbers. This result can also be seen geometrically, as squaring a complex number corresponds to rotating it by twice the angle \alpha, which results in a full rotation of 360 degrees or 2\pi radians, bringing the complex number back to its original position.

Therefore, there is no other reason why the expression (e^{i\alpha})^2 would be equal to 1 other than the properties of complex numbers and the definition of i. This result is fundamental in complex analysis and has many applications in mathematics and physics.