Interpreting eiπ in Equation: Radians or Number?

[forcefield]

Consider the following equation:

e^iπ = -1

Is π in that equation just a number or do I need to interpret it as radians ?

I mean, if I defined a 'diameterian' so that π diameterians would make the full circle, would the above equation still be correct ?

Thanks

 

[micromass]

Radians are just number. There is no difference between "the number 1" and "1 radian". People like to use the notation radian because it makes it clear that you are dealing with an angle. But the radian symbol is often omitted since a radian is just a number.

 

[Mark44]

Also, "radians" are "unitless" dimensions, due to the fact that a radian is defined as the ratio between two lengths -- the length along an arc of a circle and the length of the radius. These length units cancel.

 

[forcefield]

So I guess I need to interpret π as an angle in that equation.

This is going beyond high school level I guess and is only somewhat related but I was reading Feynman Lectures (http://feynmanlectures.caltech.edu/III_06.html#Ch6-S3) and it looks like a 360 degrees rotation in physical space corresponds to a 180 degrees rotation of a phase of a complex amplitude. I also find it interesting that if I square i, -1 or -i, I get the rotation doubled.

 

[micromass]

So I guess I need to interpret π as an angle in that equation.

No, $\pi$ is a number.

 

[forcefield]

No, $\pi$ is a number.

Then what do you say about my second question in the first post ?

 

[SteamKing]

Consider the following equation:

e^iπ = -1

Is π in that equation just a number or do I need to interpret it as radians ?

If you are familiar with the complex plane, the following diagram should establish a connection between e and any complex number, a + ib:

​

Also, r² = a² + b²

 

[lavinia]

Consider the following equation:

e^iπ = -1

Is π in that equation just a number or do I need to interpret it as radians ?

I mean, if I defined a 'diameterian' so that π diameterians would make the full circle, would the above equation still be correct ?

Thanks

π is a just a number here. It has no units. If you interpret it as an angle then the units are radians. but in the complex plane it is just another number.

In fact you may define π as the smallest positive number that satisfies the equation, e^iπ = -1

 

[FactChecker]

e^i*3.14159... = -1. Call it what you want, as long as it equals 3.14159... To fit in with the rest of complex analysis, it should be thought of as the radian measure of the angle in the complex plane.

 

[forcefield]

e^i*3.14159... = -1

Yes, that's approximately what one gets when one types it in Wolfram Alpha. But isn't it because Wolfram Alpha silently assumes radians ?

 

[FactChecker]

Yes. In the context of complex numbers, the imaginary part of the exponent of e^z should be thought of as the radian measure of the angle in the complex plane. See @SteamKing post #7. So yes, it is the radian measure of the angle in the complex plane. And in the complex plane, that means e^iπ = -1. The equation e^iθ = cos(θ) + i sin(θ) is called Euler's formula. It is considered by many to be the most significant equation in mathematics.

 

[forcefield]

Well, e^iπ is also the limit (1 + iπ/N)^N where N goes to infinity. There it is clear that π is just a number.

Wikipedia has a nice animation: https://en.wikipedia.org/wiki/Euler's_identity

 

[lavinia]

Yes, that's approximately what one gets when one types it in Wolfram Alpha. But isn't it because Wolfram Alpha silently assumes radians ?

no. the exponential function can be defined for any complex number. there are no units, just numbers. iπ is just another complex number.