how do you interpret multiplied units?

[gib65]

When you have to multiply two quantities with different units together, how do you interpret the meaning of the combined units?

 

Take the formula for momentum, p=mv, for example - you have (say) kg times m/s. I know how to interpret m/s - for every second that passes by, so many meters are traversed. But what does kgxm mean? For every second, there are so many kilogram-meters. But what is a "kilogram-meter"?

[swansont]

Some unit combinations have special names and/or physical interpretation and others do not; kg-m/s are units of momentum. Others might depend on the context of how terms are defined or grouped in a formula. If you took gravitational potential energy, for example, PE = mgh, and arranged it to be PE/g = mh, you'd get units of kg-m, which would make sense if you were figuring PE for a given height on different planets (though why you'd be doing that is another question altogether)

 

Also note that a kg-m^2/s^2 is equivalent to a unit of energy or of torque, and they are conceptually different things. Units don't tell the whole story.

[the tree]

But what is a "kilogram-meter[-per-second]"?

It's the might amount of momentum to move a kilogram one meter in a second. I think.

[gib65]

Some unit combinations have special names and/or physical interpretation and others do not; kg-m/s are units of momentum. Others might depend on the context of how terms are defined or grouped in a formula. If you took gravitational potential energy, for example, PE = mgh, and arranged it to be PE/g = mh, you'd get units of kg-m, which would make sense if you were figuring PE for a given height on different planets (though why you'd be doing that is another question altogether)

 

Also note that a kg-m^2/s^2 is equivalent to a unit of energy or of torque, and they are conceptually different things. Units don't tell the whole story.

 

So, are you saying that some composite units have no meaning at all - other than how they were derived mathematically? That's a little dissappointing to me. I'd like to think that if a group of composite units are derived from a legitemate formula (i.e. it actually represents something real in the nature world), then any which way you slice it up also represents something real. For example, we know the formula for calculating kinetic energy is E=1/2mv^2. From that, we can extract mv which is the formula for momentum. Taking the units for each of those formuli, we can say that if kgxm^2/s^2 is a composite set of units for energy, then kgxm/s is a composite set of units for momentum. How this is is the question I'm asking myself. It tells us that somehow momentum is involved in the phenomenon of energy. But the only way to explain this is through the math. I'd like to derive a methodology for explaining it conceptually. I'd like to be able to explain how to make the connection between kgxm/s and momentum, and also show how that relates to the connection between kgxm^2/s^2 and kinetic energy, which itself is something that I'd like to be able to show through some kind of methodology.

 

I don't know if the above is clear. All I'm saying is that we've got tons of experts who can understand the math, and we also have experts, most of which are from the latter group, who understand what the individual variables (mass, distance, time, etc.) mean conceptually as well as what the formuli mean as a whole (momentum, kinetic energy, etc.), but I don't know of any experts or methodologies that can attest to the conceptual meaning any of the intermediate components of these formuli, such as kgxm or even m^2 or s^2, such that it can be shown exactly how the elementary phenomena (mass, distance, time, etc.) contribute to the composite phenomena (momentum, kinetic energy, etc.) from a conceptual perspective. It's almost as though we understand that there is a firm correspondence between the individual variables and a set of phenomena we can conceptualize (mass, distance, time, etc.), and there is an equally firm correspondence between the overall formuli and the a set of phenomena we can conceptualize for them (momentum, kinetic energy, etc.), but the road from the former to the latter relies solely on the rules of algebra - any conceptual understanding along that road has to be posponed until we get to the end.

 

I feel it's possible to invent a methodology to take any step in the mathematical derivation from individual variables to the overall formuli such that a conceptual model of the physical phenomena which that step represents can be uncovered. This methodology should work independently of the path one takes from the individual variables to the overal formuli - for example, from the derivation of mass, distance, and time to kinetic energy, one can either derive velocity first, square it, and then multiply by 1/2 the mass - or - square distance independently of time (which also gets squared) and then devide the former by the latter to get something other than velocity (velocity squared - which drives my point home - what on Earth is velocity squared???). I feel that if such a methodology can be put into practice, it would advance science, mathematics, and the understanding of the average layman by quantum leaps. Average people would find it so much easier to understand mathematical concepts. All too often, after inquiring into certain scientific topic - which they might genuinely be interested in - they get discourage by the load of mathematical jargon. They get lost in the numbers and symbols, and soon give up on their persuit of scientific knowledge. I feel that if such a methodology can be introduced into the school systems - probably at the university level - the reach of science would span so much more of the general population, and that would be a good thing.

 

Sorry - got overly passionate there... but I meant what I said .

[swansont]

As I said before, there's more to the story than units. Using the example of energy and torque, it's possible to exert a torque and transfer varying amounts of energy — it all depends on how much of a rotation is involved, and angles are unitless.

 

Dimensional analysis can tell you if an answer is wrong (the units don't work out) but can't guarantee that the answer is right.

[Bignose]

So, are you saying that some composite units have no meaning at all - other than how they were derived mathematically? That's a little dissappointing to me. I'd like to think that if a group of composite units are derived from a legitemate formula (i.e. it actually represents something real in the nature world), then any which way you slice it up also represents something real. For example, we know the formula for calculating kinetic energy is E=1/2mv^2. From that, we can extract mv which is the formula for momentum. Taking the units for each of those formuli, we can say that if kgxm^2/s^2 is a composite set of units for energy, then kgxm/s is a composite set of units for momentum. How this is is the question I'm asking myself. It tells us that somehow momentum is involved in the phenomenon of energy. But the only way to explain this is through the math. I'd like to derive a methodology for explaining it conceptually. I'd like to be able to explain how to make the connection between kgxm/s and momentum, and also show how that relates to the connection between kgxm^2/s^2 and kinetic energy, which itself is something that I'd like to be able to show through some kind of methodology.

 

I don't know if the above is clear. All I'm saying is that we've got tons of experts who can understand the math, and we also have experts, most of which are from the latter group, who understand what the individual variables (mass, distance, time, etc.) mean conceptually as well as what the formuli mean as a whole (momentum, kinetic energy, etc.), but I don't know of any experts or methodologies that can attest to the conceptual meaning any of the intermediate components of these formuli, such as kgxm or even m^2 or s^2, such that it can be shown exactly how the elementary phenomena (mass, distance, time, etc.) contribute to the composite phenomena (momentum, kinetic energy, etc.) from a conceptual perspective. It's almost as though we understand that there is a firm correspondence between the individual variables and a set of phenomena we can conceptualize (mass, distance, time, etc.), and there is an equally firm correspondence between the overall formuli and the a set of phenomena we can conceptualize for them (momentum, kinetic energy, etc.), but the road from the former to the latter relies solely on the rules of algebra - any conceptual understanding along that road has to be posponed until we get to the end.

 

I feel it's possible to invent a methodology to take any step in the mathematical derivation from individual variables to the overall formuli such that a conceptual model of the physical phenomena which that step represents can be uncovered. This methodology should work independently of the path one takes from the individual variables to the overal formuli - for example, from the derivation of mass, distance, and time to kinetic energy, one can either derive velocity first, square it, and then multiply by 1/2 the mass - or - square distance independently of time (which also gets squared) and then devide the former by the latter to get something other than velocity (velocity squared - which drives my point home - what on Earth is velocity squared???). I feel that if such a methodology can be put into practice, it would advance science, mathematics, and the understanding of the average layman by quantum leaps. Average people would find it so much easier to understand mathematical concepts. All too often, after inquiring into certain scientific topic - which they might genuinely be interested in - they get discourage by the load of mathematical jargon. They get lost in the numbers and symbols, and soon give up on their persuit of scientific knowledge. I feel that if such a methodology can be introduced into the school systems - probably at the university level - the reach of science would span so much more of the general population, and that would be a good thing.

 

Sorry - got overly passionate there... but I meant what I said .

 

Just a few quick notes in reply here. 1) RE bolded part in your quote above: Energy and momentum are not necessarily linked. Potential energy doesn't have momentum in it, chemical energy stored in gasoline or ATP doesn't have momentum in it, the electrical energy stored in a battery doesn't have momentum in it, and yet all of these still use the same units.

 

2) Following your methodology there in your last paragraph, say I come up with a quantity that has units of mass*(length squared)/(time squared), however I got there, again how do I know if that is a unit of energy or a unit of torque? Just as was said above, two very, very different quantities, but the same unit.

 

Lastly, I am not too sure what is really wrong with the current methodology. Take F=ma, force is defined as mass times acceleration, so force is defined to have units of mass*length/(time squared). Or Work dW=F*dl, Work=force times a distance, so work has to be mass*(length squared)/(time squared). Etc., etc. The units come from the definitions of the terms, and are what they are. Again, from your way, it looks to me like putting the horse before the cart, because just because a set of units come out to a certain form, doesn't mean that that combination is meaningful.

[gib65]

Just a few quick notes in reply here. 1) RE bolded part in your quote above: Energy and momentum are not necessarily linked. Potential energy doesn't have momentum in it, chemical energy stored in gasoline or ATP doesn't have momentum in it, the electrical energy stored in a battery doesn't have momentum in it, and yet all of these still use the same units.

 

2) Following your methodology there in your last paragraph, say I come up with a quantity that has units of mass*(length squared)/(time squared), however I got there, again how do I know if that is a unit of energy or a unit of torque? Just as was said above, two very, very different quantities, but the same unit.

 

Lastly, I am not too sure what is really wrong with the current methodology. Take F=ma, force is defined as mass times acceleration, so force is defined to have units of mass*length/(time squared). Or Work dW=F*dl, Work=force times a distance, so work has to be mass*(length squared)/(time squared). Etc., etc. The units come from the definitions of the terms, and are what they are. Again, from your way, it looks to me like putting the horse before the cart, because just because a set of units come out to a certain form, doesn't mean that that combination is meaningful.

 

Hmmm... all good points, but something inside me tells me that this represents a philosophical problem - not an answer. I mean, what you say makes good sense, but at the same time, all our formuli in science are said to map directly onto physical phenomena and serve as a formal expression of them. If this is true, then the same should be true for the operation each variable performs on the others, which should map onto the relations each unit bares to the others. For example, when it comes to velocity, we could say that each second is being allocated equal portions of the total distance. That is, in a *very crude* sense, time is dividing up distance among itself, and that's what "produces" the phenomenon of velocity.

 

This is all subject to interpretation of course, and there's actually a very good chance I'm out to lunch on this, but it's just one of those things that I can't justify yet I have a very strong sense that there has to be a meaningful relation between, not just the individual variables and the overall formuli, but each component of the formuli anyway you chop it up. Therefore, the same should be true of the units.

 

Obviously, this is not a scientific fact - and it's scarcely philosophy since I can't prove it... there's just that feeling.

[swansont]

There's not a unique mapping, though. Like some physics calculations (especially in thermo), answers can be path dependent — it's not just a matter of where you end up, it matters how you got there. I think units is also "path dependent" and you're focusing on the endpoint and ignoring the path, as it were. The endpoint does not contain the information about how you got there.

[gib65]

There's not a unique mapping, though. Like some physics calculations (especially in thermo), answers can be path dependent — it's not just a matter of where you end up, it matters how you got there. I think units is also "path dependent" and you're focusing on the endpoint and ignoring the path, as it were. The endpoint does not contain the information about how you got there.

 

Yeah, maybe you're right. Still, it would be nice to come up with a way of inventing abstract interpretations of the intermediate steps - as long as we always kept in mind that we were simply making something up in order to have something conceptual to hold onto. This might be a weak point, though, and I don't know if there'd be much interest in "making up ideas" by more serious practitioners - although I would still maintain that it could help novices or laymen to ease into science and mathematics more smoothly.

 

I don't know - might just be a pipe dream.

[finkaruniski]

This thread is pretty much dead and cold but I think it's relevant so I'm gonna try a little CPR.

 

I think that Gib65 is making an interesting point here that and I'm not sure if the rest of the forum is quite on the same page.

 

Why is it that we can so easily conceptualize the idea of division in units but not multiplication? We can understand the idea of "feet per second" (ft/sec) - it's a certain number of feet for every second elapsed. We can understand the idea of "miles per gallon" (mil/g)- it's a number of miles traversed for every gallon used.

But multiplication is much harder to understand. What is a "foot-pound", a "meter-second", a "kilogram-watt"? I don't mean "what are they?" literally. I understand what they are and how to use them mathematically. BUT, if 1 meter/sec is a meter for every second elapsed then what is 1 meter x sec? What does that mean? Why can't we conceptualize it like we can conceptualize division based units?

I think the answer has to do with the innate limitations that we (as humans) have in understanding the concept of time. Meters/sec makes the best example.

 

As self-proclamed geeks we all understand the idea that time can mathematically be manipulated like any other component of an equation. Mathematically time can go forward, backwards, breakdance, whatever. Of course in reality time only flows in one direction, at least for us. By doing this it's very nature is to distribute events across itself, essentially, to divide. If you traverse 30 meters in 2 seconds, you have moved 30m at 15 m/s. We can understand this easily because we are used to manipulating time in this way. It is, in fact, the only way that we are used to manipulating it.

 

Now at this point you may say, "Yes but time doesn't have to be an element of the unit for it to be conceptualized. Take for instance miles/gal. 'A certain amount of miles traversed for every gallon used', no time units at all!"

Yes, this is true BUT miles/gal INFERS that time is moving in a fashion that we are used to dealing with, where as meters x sec does not.

 

When you really think about it, it's rather bizarre that we can conceptualize any mathematically manipulated unit. Miles/hr is a totally intuitive concept to us but why? All it really means is the distance traveled in miles, divided by the time it took to traverse it in hours. It is entirely a mathematical concept and it makes no sense that it should be more or less intuitive than a "foot-pound".

 

At this point you may think that this is just one of those situations in which you have to throw up your hands and say, "whatever, it just IS". But I find that there are SOME multiplied units that are easier to conceptualize than others. Take, for instance, a Newton, or, for the purposes of our discussion let's call it a "kilogram-meter per second squared" (kg x m/s^2). This multiplied (and divided) unit is, of course, derived from the formula F=ma, just as m/s is derived for the formula V=d/t.

 

Think about it, you CAN conceptualize this unit. If someone punches you in the face it is easy to understand that the amount of pain you experience is a product of the mass of their fist and the amount that said fist is accelerating. Small fist x big accel.- not so bad. Big fist x small accel.- tolerable. Small fist x small accel.- barely noticable. Big fist x big accel.- expensive dental bill. The differences in how your jaw feels afterwards IS a conceptualization of a "kilogram-meter per second squared".

 

Remember, things like Force and Kinetic Energy are fundamentally different than things like Velocity and Frequency, so conceptualizing their units requires a different sort of thinking. If you expect them to feel the same, you will be disappointed.

 

I hope I have contributed in some positive way here, and I hope that SOMEONE that posted here still checks their account, considering that this thread has been left alone for 5 years. This is just something I have been thinking a lot about recently so I felt the need to share. Happy to be a part of the forum

 

Finkaruniski

[John Cuthber]

If you are going to put the word "INFERS" in capital letters you should find out what it means.

An abstract concept like miles per gallon isn't in a position to infer anything. You probably meant implies

 

"Why is it that we can so easily conceptualize the idea of division in units but not multiplication?"

Do you have a problem understanding square metres or a cubic foot?

What about the Joule- roughly the gravitational energy released by an apple falling off a table (1 Newton metre)? It seems you are OK with that.

We can handle multiplication of units, as long as we understand what the product is.

You ask about the " kilogram watt" and I grant you that it's obtuse.

But why would you ever calculate it?

[finkaruniski]

I DID mean "implies", actually. Thank you for the correction. And thank you also for the condescending tone that accompanied it.

 

John, I understand what you are getting at but your point is kind of parallel to the idea that Gib65 and I are discussing. I think the confusion here is a semantic one involving the word "conceptualize".

 

As I said in my post, I can UNDERSTAND multiplied units, I can grasp their meaning and manipulate them accordingly. Nonetheless there is a difference in the level of intuitive understanding between a multiplied unit and a divided unit.

What is a m/sec? Well, it's a "meter for every second". That is conceptually intuitive.

What is a square meter? It's a "meter times every meter". What does that mean? It is not conceptually intuitive.

 

What we are exploring here is "why?", why is there a difference between one and the other.

 

This is a rather abstract concept and, being that, it is important to TRY to understand it, to make a mental leap to grasp the idea that we are trying to connote. This is especially true for a limited format such as an online forum.

Many abstract concepts are like this, they cannot be understood unless you TRY to understand them. If you instead set out with the intention of proving your superior intelligence by TRYING to refute them they will slip through your fingers.

Remember that just because a concept is abstract doesn't mean that it is irrelevant. Many important ideas started off as completely abstract flights of fancy. (What Einstein used to refer to as "a thought experiment")

 

I invite corrections, and criticism, but please try to be constructive. Try to promote the growth of ideas.

 

finkaruniski

[michel123456]

Very interesting question.

See one of my preferred:

The multiplication of bananas by umbrellas.

[finkaruniski]

Fascinating!

Now that's what I call a contribution!

[JEntler]

First, you can properly interpret a kg*m/s^2 by thinking of it like this: one newton is the amount of "force" it takes to accelerate one kilogram one meter per second squared. Further, you can interpret momentum as this: one kg*m/s or one "momentum" is the amount of momentum one kilogram going one meter per second has. Momentum is just a qualitative idea that we thought would be useful to measure and so we assign it to things as something quantitative. The only reason we combine these "things", increasing/decreasing things by certain factors, is to provide a certain insight into the phenomena itself or insight into another phenomena.

 

In terms of interpreting these "units", don't forget the key principle behind them; multiplication and division, which are two forms of the same thing! Division is a form of multiplication rather, being the multiplication of the inverse. So, it is fair to say that you can think of these multiplied quantities just as you've been thinking of these "divided" quantities.

 

When you say you understand a "meter per second" m/s, you are saying you understand a "meter per one-over-a-second" or a "meter times one-over-a-second" m*(1/s). It just so happens, in terms of context, you understand it's principle. Now when we think of m/s^2 things seem to get a little trickier. But they don't. All this is saying is "a meter per second per second". Now you don't find it strange that we have those two 'seconds' multiplied together in the denominator, because we know what it means; we are "accelerating" one meter-per-second per every second that goes by.

 

All units can be thought of this way. It takes a bit of imagination, you just have to think about it! A meter squared is a meter for every meter, a meter-in-the-x-direction per every meter-in-the-y-direction. Per just means "for each".

 

Energy is an abstract quantity with a peculiar quality; it just so happens to be conserved in many different forms. Thus, it has many different units that all represent the same thing (this is where the physicists are guilty of being a little nutty in a sense that they can't get the units straightened out!). As a result, you may think of energy any way you want. Convert joules to calories if you'd like! Or even Celsius - if it really makes it simpler for you to understand. Or you can think about it in context; a joule is the amount of energy it takes to bring a newton of weight up one meter from an arbitrarily defined surface - which now represents the amount of energy that that newton-weighted object has in terms of "potential energy" - which is said to be "stored".

 

I hope this is of any use to you so many years after...

 

Joe

[fancypants]

What an excellent discussion. As a layman trying to grasp physics equations (e.g. F=ma), I often get tripped up by what is actually happening to the units being manipulated mathematically, and what they actually "mean" in the real world. The analogy of multiplying bananas by umbrellas to get a grandfather clock (mentioned in the article cited by michel123456) especially resonated with me.

 

Thanks to everyone who has contributed; I hope this bump gets more people to chime in.

Edited January 31, 2017 by fancypants

[wtf]

The discussion so far has been about the physics; but since this is the math section I'd like to say a word about the math.

 

As a math-trained person when you tell me that 2 times 6 is 12, I believe that. I could drill it down to the Peano axioms. And it tracks a highly obvious and familiar fact of nature, namely that two rows of six are the same as six rows of two and there are twelve of them altogether. I can see the living proof of this in the world every time I buy a carton of eggs.

 

However if you ask me what 2 feet times 6 pounds is, I know that's 12 foot-pounds and I can conceptualize it physically.

 

But if I put on my formalist hat, I confess I have no idea what that means in math. I can't drill down foot-pounds to anything I know in set theory. I actually have no idea what it really is.

 

As someone noted in this Stackexchange thread, we tell kids you can't add apples to oranges, and then we tell them to multiply feet times pounds. What kind of sense does that make? http://physics.stackexchange.com/questions/98241/what-justifies-dimensional-analysis

 

No less a genius than professor Terrance Tao has blogged on exactly this subject. https://terrytao.wordpress.com/2012/12/29/a-mathematical-formalisation-of-dimensional-analysis/

 

I don't have time to read this today otherwise I'd summarize as much as I understood. Hopefully I'll get to that later. Meanwhile I wanted to toss these links out there because this really is a good question. What is a foot-pound, really?

Edited February 1, 2017 by wtf

[swansont]

 

As someone noted in this Stackexchange thread, we tell kids you can't add apples to oranges, and then we tell them to multiply feet times pounds. What kind of sense does that make?

 

 

We don't add feet to pounds, though, we multiply them. (You could have units of apple-oranges, too, under some bizarre set of circumstances where apple and oranges were a unit and not a description.) Adding different units doesn't happen.

[wtf]

We don't add feet to pounds, though, we multiply them. (You could have units of apple-oranges, too, under some bizarre set of circumstances where apple and oranges were a unit and not a description.) Adding different units doesn't happen.

(Wiseguy kid): But isn't multiplication just repeated addition? How can you have apple-many oranges?

[swansont]

(Wiseguy kid): But isn't multiplication just repeated addition? How can you have apple-many oranges?

 

 

No, it's not. Despite the fact that you can do that as an algorithm, it's not the same concept.

[studiot]

Multiplication and division can introduce some funny effects when applied to quantities with units.

 

for instance multiplication can loose all the information as in

 

frequency times period = 1

 

Division, on the other hand, can loose the units as in velocity ratio or mechanical advantage or many other dimensionless ratios.

 

It is also clear that whilst multiplication / division can be modeled as repeated addition / subtraction for integers, the model suffers when we introduce non integer quantities.

Physically the model is also suspect with say a hydraulic force multiplier which multiplies the input by a fixed scalar.

[tech_boi2019]

I believe it's like saying "either-or" for example  5kg * 2m / 2s = 5kg m/s where you could say force momentum to move 5kg 1 m every 1 second OR 1kg 5m every 1 second.

to get really BASIC it's just like saying 5*2 is five 2's OR two 5's.

Edited May 30, 2019 by tech_boi2019

[jerricks54]

I stumbled upon this thread by Googling. I have almost the exact same question as @OP, and what @wtf said resonates with me to some degree. 

I understand the math operations - I can do the equations and derive the units and do the dimensional analysis - but I am not able to "visualize" or "make concrete" or "imagine" or "intuit" units like kW x hour, or kg x m^2/s^2, or lb x ft. Other units, like miles/gallon, or meters/second, I can visualize.

For example, I can close my eyes and see a car go down a road for 30 miles, while the one gallon of gas in the tank slowly dwindles to nothing. I can see a sprinter dash on the track 100m, and then look down at my timer and see the seconds that elapsed during the sprint. But, when I try to visualize the energy powering my blender in the kitchen after I plug it into the wall, I can't imagine/see/visualize the kg x m^2/s^2...at all! Yet, I hear and see the blender working, and my smoothie ingredients are now a semi-tasty snack, so kg x m^2/s^2 is happening right before my eyes! Yet I cannot "interpret" or connect the logical dots that lead from the smoothie blending in front of me to kg x m^2/s^2. 

I found these, and they've helped me a little:

https://physics.stackexchange.com/questions/9052/how-can-i-understand-counterintuitive-units-like-texts2/9068#9068

https://physics.stackexchange.com/questions/252675/is-multiplication-in-physics-purely-mathematical-or-is-there-a-physical-explanat/252691#252691

These helped me a tiny bit:

See @Wildcard's answer here - https://math.stackexchange.com/questions/1980010/why-do-units-from-physics-behave-like-numbers/1983842#1983842

See OP's second question and responses to it - https://physics.stackexchange.com/questions/122358/force-and-newton-seeking-intuitive-explanation/122361#122361

[studiot]

1 hour ago, jerricks54 said:

I stumbled upon this thread by Googling. I have almost the exact same question as @OP, and what @wtf said resonates with me to some degree. 

Welcome, I hope the various responses spread over a decade and a half helped but if you want to discuss this further please amplify and try to explain your exact difficulty as I for one couldn't catch it from your post.

[aside]

'responses per decade'  how's that for a unit ?

[/aside]

Please note two points about forum etiquette for this forum.

We like members to make their point here, not tell others to go to the middle of somewhere else.

Posting links is a great way to support something you have said here though.

Also note the anti spam rule limit new members to 5 posts in their first 24 hours.

[jerricks54]

@studiotThank you for the note on etiquette for this forum - this is my first time posting, just found y'all while Googling about this! 

I think I finally figured out what was confusing me! Here's my best stab at it:

For example, m/s^2. What on earth is a s^2? I can see a m^2 in my mind's eye easily, and I can draw one. But how do I draw a s^2? The confusing part is the word "square."

     When I first learned about squaring (which is: multiplying a number by itself once), I was shown a literal square. Geometry. I saw the shape, and the teacher called it a "square." They showed me how every side of the square is the same distance = 1 meter. They said if you multiply one side by another side, you get the area of the "square," which is 1. This area is in a unit called "square meters," written as m^2, so it's area is 1 m^2. The "m^2" is written that way to show that the two dimensions of the square - length and width - are measured in meters, are equal to one another, and were both multiplied together. In "m^2," "m" means the distance of one side of the square is measured in meters, and the value is equal for every side of the square (the value is 1). In "m^2," the "^2" means that the value "m" represents the 2 dimensions of the square (length and width) and that they the 2 of them are multiplied together "m*m." The resulting square with its area is now dubbed "1 m^2," or "1 square meter." Any shape can have an area in square meters, because it's how many 1 meter x 1 meter squares are inside of the 2-dimensional shape. "Yep, that triangle over there has about 2.7 of those 1x1 meter squares in it! We call the 1x1 meter squares 'square meters,' and the shorthand symbol for that is "m^2." So, we say the triangle has 2.7 m^2 in it, or that the area is 2.7 m^2."

     Now, in other instances of math besides geometry, there are times when a number "n" is multiplied by itself once, "n*n." A name had to be given for n*n. It seems someone said, "Hey, you know in Geometry with the square, you know, the shape with four sides that all have the same value? How we multiplied the length and width of the square together, and came up with 1 square meter, and wrote it as m^2. Well! Since that is an instance of a number "n" times itself, "n*n," why don't we call that same operation "n*n" in other parts of math and science "squaring," you know, since the same thing happens with the square?" Thus, the term used to describe other instances of "multiplying a number by itself once, n*n" came to be called "squaring." Hypothetically, n*n could have been called anything, but it was fun and made a good deal of sense to call it "squaring." 

     So, the image of the square is still associated with "multiplying a number by itself once." This is the confusing part. In other instances of math beyond geometry and measuring dimensions of squares, there are many equations in which n*n occurs that has nothing to do with shapes at all, much less square shapes. But because the word "square" implies a shape, it is tempting to automatically, intuitively feel and think, "square shape."

     Going back to s^2, we think or feel something like "Oh, does time now have shape of some kind? Is there a brand new dimension of time suggested by this math?" We can leave such theorizing about multiple time-dimensions to cutting-edge experts. For the rest of us, no. No, s^2 does not suggest time has new dimensions or shape. All that is suggested by ^2 is "a number multiplied by itself once." That is all. 

     It would be inconvenient, but for the purposes of disentangling shape from a mere multiplication operation, we could do away with the term "squaring" and call it "duplicating," "multiplicating," "copying," "burritoing," "sporking," "anythinging" - it is, to some degree - the geometry and square shapes being fun and logical foundations for "^2" aside - an arbitrary name for a mathematical operation. ^2 just means n*n.

Edited March 26, 2022 by jerricks54