Muon321

Is it mathematically correct to say √16=±4, or would it be correct to say that √16=4, unless specified ±√16? This isn't for any specific problem or anything, but I'm just curious if is correct to have ±, because I know every number has two square roots.

So, let's say we are talking about a photon. We can find E by the equation E=pc, which only works for light. But, when I take p=h/λ, then I fill in λ with v_p/f since λ=v_p/f, so then we get p=hf/v_p, then since E=hf we get p=E/v_p, then to p(v_p)=E, and E=pc for light so we get E=pc=p(v_p), which implies c=v_p, which in a vacuum is true for light but quite frequently isn't. I am confused! What am I doing wrong!

Let's say that we had some equation in physics that was v(m+c)/v, which isn't real I am just using it, would you be able to simplify it to m+c? You shouldn't be able to because v for velocity has a possibility of being 0, correct? You can't simplify if you would be cancelling out a variable that could be zero, is this true?

If a particle at rest has a momentum of 0, it still has a normal frequency, but it has a wavelength of undefined, so how does that work? What's going on with that wave?

Is frequency the number of oscillations every second or the number of wave crests that pass every second?

So, let's say we have a proton, and it is at rest, and we use the equation λ=h/p to find it has a wavelength of 0 because it has no momentum. Makes sense. Then we also use the equation E^2=m^2c^4+p^2c^2 to find its energy from its rest mass, then we plug that into E=hv to find that it still has frequency. But, shouldn't it not have frequency because it has no wavelength? If we use fλ=v_p, we get 0 for frequency. Someone help! I'm confused!

If the phase velocity of light isn't always c, then wouldn't the equation λf=c be more accurately λf=v_p?

If it is not always c, then λf=c would be incorrect, and it should be λf=v_p like matter waves, right?

How many photons correspond to a single electromagnetic wave? Is there more than one photon for each wave?

I am confused about group velocity, phase velocity, and signal velocity of waves. Which of these would be the velocity of the corresponding particle to that wave? And, what's the difference between signal and group velocity? Is there a difference?