# Whats the purpose of squaring?

1. Jul 2, 2004

### Dook

Whats the purpose of squaring? As in E=MC^2, what does squaring the speed of light do for the equation?

2. Jul 2, 2004

### Muzza

"c squared" is shorthand for c times c. So e = mc^2 is equivalent to e = m * c * c.

3. Jul 2, 2004

### Dook

Why multiply the speed of light times itself though?

4. Jul 2, 2004

### Muzza

I don't know. That's just how nature is, I suppose.

5. Jul 2, 2004

### robphy

For one thing, it makes the right-hand side have the units of energy.
(kg)*(m/s)^2 = (kg m/s^2)*m = Newton*m = Joule

6. Jul 2, 2004

### Dook

Hehe, that's how nature is or that's how our understanding of it is? Our understanding is not always so correct. But then, this is from Einstein.

If I was to do a calculation what would squaring it do for me? What's the purpose? Why did Einstein square the speed of light? Not because of anything he saw in the universe. I think his theories all came from math and not observation or experimentation so why multiply c times c?

Last edited: Jul 2, 2004
7. Jul 2, 2004

### Zurtex

I think your example is too advanced for your question. Maybe you should start on something more simple like:

Kinetic Energy = 1/2 * Mass * Velocity * Velocity

Then maybe someone can actually explain it (I'm not a physicist so I can't)

But there are many mathematical patterns in nature, it’s sometime quite amazing.

8. Jul 2, 2004

### jcsd

It comes from pre-relatvistic physics + the two postulates of relatvity.

To see exactly why it is c^2 you have to look at the derivation:

http://www.ams.org/bull/2000-37-01/S0273-0979-99-00805-8/S0273-0979-99-00805-8.pdf

9. Jul 3, 2004

### Dook

Okay so Energy = Mass times the speed of light times the speed of light. Now it seems to me the only time one would multiply something by itself is when they are trying to get the square area of it in two dimensions. I find this very confusing. The formula is telling us how much energy is hidden in mass (a great deal it seems) but mass has three dimensions so shouldn't the formula be E=MC*C*C?

10. Jul 3, 2004

### Antiproton

You can't think of it in terms of area. That doesn't mean anything. You have to, instead, accept that Einstein didn't invent E=mc^2 out of nothing. There was a natural progression of thought using math that already existed. It was derived, not invented. As an analogy, consider the example of Kinetic energy as stated previously. KE = 1/2mv^2. This wasn't invented either, but was derived (specifically, it is the integral of Force over a distance ds by the Work-Kinetic Energy theorem).

So, making a short story long, the c isn't squared because Einstein likes squares, nor does it have anything to do with 'area' or 'volume'. It's purely mathematical.

11. Jul 3, 2004

### Janitor

Yes indeed. Ultimately the square of c comes from the fact that we live in a universe where right triangles have a relationship between length of hypotenuse and lengths of sides that goes as the sum of the squares, a fact known back in the days of ancient Greek geometry, and probably known by some people even earlier than that.

Note to purists: Okay, given a curved spacetime, I should technically have said that in the limit of small right triangles there is a Pythagorean relationship...

12. Jul 5, 2004

### lvlastermind

E=mc^2 is equal to sqrtE=sqrtM*c

it was just easier to write E=mc^2

13. Jul 5, 2004

### uart

Dooky, perhaps you should consider an easier example in which you can clearly see and follow the derivation that results in squared terms that are unrelated to area.

How about this one. Consider a simple spring, the more that you wish to compress the spring then the harder you have to push it, easy enough to understand right. So mathematically this corresponds to F = k x, where k is the spring constant and x is the distance that you deflect the spring (from it's equalibrium position).

Now the stored energy E is equal to the integral of force F with respect to x. So,

E = integral (kx,dx)
= 1/2 k x^2

See that the x^2 is not directly related to area but appears in the formula just the same.

Last edited: Jul 5, 2004
14. Jul 5, 2004

### MathematicalPhysicist

arent area and volume mathematical (geometrical) definitions?! (rhetorical question).

15. Jul 5, 2004

### quartodeciman

Calculating a square area is NOT the only reason for taking the second (or higher) power of a quantity.

Here is an example.

Suppose you have 1000 cells of bacteria and the bacteria reproduce at the rate of +20% every minute, which covers both reproduction and death. After the first minute, there should be

1000*(1 + .20) cells of bacteria

. This amounts to 1200 cells of bacteria. For two minutes the actual amount of increase is larger, because there are 1200 cells after the first minute, not just 1000. So, the amount after two minutes is

1000*(1 + .20)*(1 + .20)

or

1000*(1 + .20)2

. This amounts to 1440 cells.

. I have in fact calculated the second power of the growth factor to get the amount after two minutes. But it has not a thing to do with geometric squares.

After three minutes, the amount would be calculated with a third power,

1000*(1 + .20)3

The amount is 1728 cells.

This has nothing to do with geometric cubes.

These calculations are just a contingency of handling net growth over time as a fraction of quantity.

This is just one example where powers of quantities have nothing to do with geometry.