# If "c" is a constant, doesn't "c2" (ie. e = mc2) equal "c"?

curiousat61
I’m obviously looking at this the wrong way but…..with reference to the formula e = mc2 isn’t “c” (the speed of light) a constant? So if that is true doesn’t c2 (or any other multiple of c) equal c?

##2## is a constant, but ##2^2## doesn't equal ##2##...

Edit: although you're right in natural units lol

• Grasshopper, russ_watters and Ibix
curiousat61
This is true but if I have 2 things I can add two more things and now have four things. In the case of light speed I cannot add multiples of light speed to get more light speed - the speed of light will always be the same. (I will not belabor this question - I know I am wrong. Just curious as to the answer)

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If you have a 10m length at right angles to another 10m length that defines a 100m2 area. You have to multiply the units as well as the values.

So ##c^2## isn't a speed. What it actually is here is a scale factor relating our units of mass to our units of energy (which we thought were unrelated concepts pre-Einstein).

• russ_watters and etotheipi
##c## is just a number (with dimensions), and you can add, subtract, multiply, exponentiate, or otherwise do whatever you want to it.

If you have something moving at ##0.8c## with respect to something else, which is itself moving at ##0.8c## with respect to a third thing, you might reasonably ask what speed the first thing is going with respect to the third thing. Relativistically, this is given by the velocity addition formula, and it is not ##1.6c##.

But, that doesn't change the fact that ##0.8c + 0.8c = 1.6c##, that is still clearly true. It's just that the '+' operation is no longer the operation that represents adding two velocities relativistically.

• Delta2 and Dale
curiousat61
Thank you for the replies. I'm beginning to see the "light"...

• • Delta2, berkeman and etotheipi
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Thank you for the replies. I'm beginning to see the "light"...
##c^2## isn't a speed, it's a speed squared, and has different units. You cannot compare the two, because ##c## and ##c^2## have different physical units.

Something like ##c^2 = c, \ c^2 < c## or ##c^2 > c \ ## would all be physically meaningless.

• etotheipi
I’m obviously looking at this the wrong way
Yes.
…..with reference to the formula e = mc2 isn’t “c” (the speed of light) a constant? So if that is true doesn’t c2 (or any other multiple of c) equal c?
Maybe, you believe due to bad formatting and wrong interpreting of it, that the formula is ##e= mc *2##, but it means ##e=mc^2=m*c*c##.

Staff Emeritus
I’m obviously looking at this the wrong way but…..with reference to the formula e = mc2 isn’t “c” (the speed of light) a constant? So if that is true doesn’t c2 (or any other multiple of c) equal c?

It depends on your unit system. If you use geometric units where c=1 and c is dimensionless, then you could say c^2=c.

However, if you are using the standard SI units (meters, kilograms, seconds), c neither has a numerical value of 1, nor is it dimensionless.

1 foot does not equal 1 foot squared, they are different things. The same with c, in the usual unit system where it has dimensions.

2 does not equal 2^2, the only case where dimensionless numbers satisfy x^2=x is when x=1 or when x=0.

The square of a dimensionful quantity is never equal to the unsquared value, it is a units error.

• vanhees71