Why do we square 'things' ?

amilton542

Joined Nov 13, 2010
496
I've realised we have a habit of squaring things.

For example, the linear distance between two charges associated with Coulomb's Law is squared. The speed of light celeritas associated with Einstein's famous equation is squared.

I concluded, anything relative to displacement, acceleration or velocity is always, more often than not, squared.

If someone could elaborate on this it would be much appreciated, thanks.

1chance

Joined Nov 26, 2011
42
In a nutshell, we square to keep negative numbers from reeking chaos. Since a negative can mean a direction rather than a value, that is left vs right or down vs up, it is useful to think in terms of going continuously from one point to another without "negatives" canceling out the distance. This is my 2 cents answer. I'll let someone else do the elaboration.

K7GUH

Joined Jan 28, 2011
191
If one studies the basic phenomena of physics, one finds such things as the law of inverse squares, wherein the illumination from a point source is inversely proportional to the square of the distance from the source. The law was derived from studying the physical phenomena, so far as I know. If it has anything to do with negative numbers, that is news to me, but then my high school physics course ended in 1951.

MrChips

Joined Oct 2, 2009
19,154
There is a simple reason why we square things. It is because a x b represents an area.

Suppose a can of paint can cover a wall 10' x 10'.
Now we make the wall twice as long and twice as high, i.e. 20' x 20'
How many cans of paint would we need? The answer is 2 squared = 4 cans of paint.

If I shine a light on a wall 10' away, the wall will be a certain brightness.
If I move away so that I am 20' away from the wall, would the brightness on the wall be 1/2. No, it will be 1/2 squared = 1/4.

Another example is the area of a circle = $\pi r^2$

Last edited:

steveb

Joined Jul 3, 2008
2,431
There are many ways to begin to answer this, and I suspect that there may be more than one thing at work here. The above answers from other AAC members tend to confirm this. I think many of the square laws can be tied back to the principle of least action and the form of the Lagrangian for various laws of physics. Basically, energy tends to be a square of a basic variable (length, velocity, field amplitude etc.). It turns out that energy and the Lagrangian need to have particular properties that can only be met by particular functions (the square being one of them). I'll try to track down some good descriptions of the details of this.

I like 1chance's answer though and I think this ties in to a basic symmetry property. I can't do a better job of elaborating myself, but I like the description given by one of the famous books on mechanics, as follows.

"We can now draw some immediate inferences concerning the form of the Lagrangian of a particle, moving freely, in an inertial frame of reference. The homogeneity of space and time implies that the Lagrangian cannot contain explicitly either the radius vector r of the particle or the time t, i.e. L must be a function of the velocity v only. Since space is isotropic, the Lagrangian must also be independent of the direction of v, and is therefore a function only of its magnitude, i.e. of v^2 = v^2:"

from: Landau and Lifshitz, "Course of Theoretical Physics : Mechanics" (Kindle Locations 570-574). Butterworth-Heinemann. Kindle Edition.
In a nutshell this is saying that our notion of kinetic energy being the square of the velocity is directly traceable to the homogeneity and isotropic nature of space and time, in any inertial frame. An inertial frame is just a frame of reference that is not accelerating.

In a sense any answer we give is in some ways begging the question because one can then ask a more fundamental question. Why is it that all of our well established physical laws are expressible in terms of Lagrangians and the principle of least action, and why is our universe governed by laws of symmetry, and not chaos. These are very difficult questions to answer and physics really does not provide the tools to answer them fully.

atferrari

Joined Jan 6, 2004
3,380
In a nutshell, we square to keep negative numbers from reeking chaos. Since a negative can mean a direction rather than a value, that is left vs right or down vs up, it is useful to think in terms of going continuously from one point to another without "negatives" canceling out the distance. This is my 2 cents answer. I'll let someone else do the elaboration.
I understand that we square things "because" a law of some kind just express such a relationship between two or more entities and not "to" do this or that.

I dare to say that the original question is as valid as asking "why do we add?"

BTW, we also deduct, take square root of things and....

amilton542

Joined Nov 13, 2010
496
Thank you for your positive replies. This should keep me busy until I find the link and conclude things for myself. Thank you

Last edited:

#12

Joined Nov 30, 2010
18,076
@steveb
The only reason I looked at this post is because I knew you'd be here. You get over my head a bit, but it's probably good for me. Thanks for spending your time on questions like this.

amilton542

Joined Nov 13, 2010
496
I've done some research and I started off with E = mc^2 and I came across the most intriguing video on YouTube.

The video is called 'Einstein's Big Idea' which is a breakdown of scientific breakthroughs leading upto his famous formula. It's more of a narrated movie than a documentary, which starts off with Faraday claiming light is an electromagnetic phenomenonen and finishes off with the nuclear bomb.

However, the chapter '2 is for squared' and the work Émilie du Châtelet did makes my problem more clear.

Steveb I would be grateful if you could view this ten minute chapter of the video and explain in more detail.

The chapter '2 is for squared' begins at 54:36 and goes on for approximately ten minutes.

K7GUH

Joined Jan 28, 2011
191
OK, guys, now that you're squared away, how do you explain the formula for the volume of a sphere?

steveb

Joined Jul 3, 2008
2,431
The video is called 'Einstein's Big Idea' which is a breakdown of scientific breakthroughs leading upto his famous formula ...

However, the chapter '2 is for squared' and the work Émilie du Châtelet did makes my problem more clear.

Steveb I would be grateful if you could view this ten minute chapter of the video and explain in more detail.

The chapter '2 is for squared' begins at 54:36 and goes on for approximately ten minutes.
Thanks for posting the video. I watched the whole thing and found it educational, interesting and entertaining. I especially enjoyed seeing my heros (Faraday, Maxwell and Einstein) come to life.

I'm not sure of what to say about that section of the video, but I can make an attempt as follows.

My best interpretation is that we need to realize that Newton and the scientists of his day did not have our perspective on Newton's laws of mechanics. There is no doubt that Newton fully understood his own laws, but how these laws interface with all other physics that we know today is something that he could not know about. In a sense, he invented the first reasonably complete theory of mechanics, and it was difficult to link this theory with other areas of science. He had no knowledge of theories of relativity, quantum mechanics and electrodynamics, nor did he live to see the Hamiltonian and Lagrangian formulations of his own laws. Hence, the fact that Newton and Leibniz might not see eye-to-eye on the importance of kinetic energy versus momentum is perhaps not too surprising.

We now readily think about and understand the concept of energy. We know that for the many experiments energy is conserved and mass is conserved independently. We also know that other experiments can show that mass and energy can be converted and are in a sense different forms of the same thing.

Apparently, Newton did not think from the same point of view, even though his laws of mechanics had the concept of energy embedded within them without doubt. Is seems Newton viewed momentum (mv) as the more important quantity of interest, while Leibniz recognized kinetic energy (mv^2/2 or mv^2 or anything proportional to v^2) as an important characteristic of a body. In some sense this comes down to semantics. Newton was not really saying that mv is energy, but most likely was just saying that mv is important to consider when looking at interactions. In reality both are equally important, so maybe the video is giving a biased view of what the debate is about. Fundamentally, momentum and energy are both conserved and are both important quantities to consider. Indeed, modern relativity theory links both of these into one 4-vector indicating that our perception of these quantities depends on our frame of reference.

amilton542

Joined Nov 13, 2010
496
@ Steveb

Thank you for your generous interpretation. Me, myself, also enjoyed the video and I now fancy the actor who played Émilie du Châtelet.

As a reward for my gratitude, you will recieve new bedsheets of your named 'heros' through the post in the next few days.

panic mode

Joined Oct 10, 2011
1,774
OK, guys, now that you're squared away, how do you explain the formula for the volume of a sphere?
it is actually very simple, in fact you get formula for any shape - using calculus.
the key idea is to divide object into many small pieces and 'count them up'.
the smaller the pieces (use limit), the more accurate sum. this is what integrals are - sums when 'pieces' are infinitely small.

area is square function, volume is cubic... cube of side a has volume a^3 etc. there may be some other scaling factors involved (like 4pi/3) but ultimately volume is cubic (in volume equation for sphere it is r^3).