how do you interpret multiplied units?

[StringJunky]

1 hour ago, jerricks54 said:

@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.

S² is seconds per second. 

[Genady]

1 hour ago, jerricks54 said:

I can see a m^2 in my mind's eye easily, and I can draw one.

No, you cannot draw m^2. You can draw a geometric shape, but m^2 is not a geometric shape. You can draw a square. It has various properties such as area, perimeter, sum of angles, etc. These are numbers. You do not draw them.

[jerricks54]

1 hour ago, StringJunky said:

S² is seconds per second. 

Yes, it is that, too, with meters involved! (m/s)/s = m/s * 1/s = m/s^2. Meters per second per second is synonymous with meters per s^2. 

In a real-world problem where we have to solve for time given acceleration and distance, we would need to find the square root of s^2, which would equal s. So, it's mathematically a squaring thing, not merely a notational "seconds per seconds" thing.

However...this just dawned on me...I miswrote at least one thing in my earlier post! It's not n*n as values like "1*1" or "24*24" or whatever other number we choose to use! It's "unit*unit," like "seconds*seconds." I need to rethink this and write a new post....

Edited March 26, 2022 by jerricks54

[swansont]

1 hour ago, StringJunky said:

S² is seconds per second. 

No, it’s not.

m/s^2 is “meters per second, per second” because acceleration is a rate of change of speed.

s^2 is in the denominator, which is crucial. 

Seconds per second is s/s, which would cancel

 

[studiot]

1 hour ago, StringJunky said:

S² is seconds per second. 

Definitely not.

 

Seconds squared although some may have got the idea from acceleration which is metres per second squared or (metres per second) per second.

 

4 minutes ago, jerricks54 said:

However...this just dawned on me...I miswrote at least one thing in my earlier post! It's not n*n as values like "1*1" or "24*24" or whatever other number we choose to use! It's "unit*unit," like "seconds*seconds." I need to rethink this and write a new post....

There is more in the Philosophy of Mathematics about this because we have to think of 'repeated operations' and also whether the output domain of the repeated operation is suitable for the next repeat.

I am rather busy at the moment but will post more detail in due course.

[jerricks54]

1 hour ago, Genady said:

No, you cannot draw m^2. You can draw a geometric shape, but m^2 is not a geometric shape. You can draw a square. It has various properties such as area, perimeter, sum of angles, etc. These are numbers. You do not draw them.

I don't understand how one cannot draw 1 m^2. Why is it not possible to draw m^2? I'm struggling with this concept : )

For example, let's say I have some strawberries sitting in front of me. The goal is to measure how many individual strawberries there are, to count the quantity, to count the frequency of "individual strawberries sitting in front me." I do this, and I come up with a value of 4. Now, if someone, with no context given to me, asked me to draw the quantity "4," how do I do that? I cannot. All I can do is write the character/number "4." So, in this sense, I see what you are saying. I cannot draw a numerical value. But, if I say, "4 what? I'm drawing the quantity of 4 what?" and they give me the crucial context of "strawberries," then I can draw out 4 individual strawberries. The count is 4, the unit is strawberry, and I just drew 4 strawberries on the paper. This is the part that I'm struggling to understand in relation to what you posted.

It seems like there are some units that lend themselves to visualization better than others. I can draw a visual representation of the unit "strawberry" that is similar, if not identical, to what everyone on earth sees in their imagination or mind's eye when they hear their language's word for the fruit we call in English "strawberry." I can do the same thing with the shape in English we call a "square." But, there are other units, like kg*m^2/s^3, that I have absolutely no idea how to draw a visualization of, and I seriously doubt there is any way to draw a visualization of it.

Edited March 26, 2022 by jerricks54

[Genady]

5 minutes ago, jerricks54 said:

I don't understand how one cannot draw 1 m^2. Why is it not possible to draw m^2? I'm struggling with this concept : )

For example, let's say I have some strawberries, sitting in front of me. The goal is to measure how many individual strawberries there are, to count the quantity, to count the frequency of "individual strawberries sitting in front me." I do this, and I come up with a value of 4. Now, if someone, with no context given to me, asked me to draw the quantity "4," how do I do that? I cannot. All I can do is write the character/number "4." So, in this sense, I see what you are saying. I cannot draw a numerical value. 

But, if I say, "4 what? I'm drawing the quantity of 4 what?" and they say "strawberries," then I can draw out 4 individual strawberries. The count is 4, the unit is strawberry, and I just drew 4 strawberries on the paper. This is the part that I'm struggling to understand in relation to what you posted. It seems like there are some units that lend themselves to visualization better than others.

If you are asked to draw an equilateral triangle with the area 1 m^2, will you draw a square?

[jerricks54]

1 hour ago, Genady said:

If you are asked to draw an equilateral triangle with the area 1 m^2, will you draw a square?

I'm beginning to see what you are saying! No, I would not draw a square. I see that language gives critical context to the math problem, language is so important. "Equilateral triangle" drastically changes what I sketch on my paper when I'm drawing the shape that has the area of 1 m^2.

Now, if someone said "Draw a shape with an area of 1 m^2, and that shape is the same one that the m^2 is based off of and finds its genesis in," I would then draw a square whose sides n have a length of 1 meter each.

But even if I do draw the "shape that m^2 is based off of" with an area of 1 m^2, it's not that I'm drawing 1 m^2 (unless I shaded in the area of the 1m x 1m square? Would that be equivalent "drawing"?), it's that I'm drawing the shape upon which the unit m^2, and the ^2 nomenclature, are based. 

It seems that the confusion for me is still the word "square" as a name for math operations in which either two identical values are multiplied, or two identical units (regardless of numerical value) are multiplied. In the case of two identical units being multiplied, using a geometric shape name "square" for "^2" connotes "shape" to the units being multiplied. Like a s², s³, or kg*m², or any other nomenclature/symbolism used to represent multiplying identical units (even if they have different numerical values). I first learned ^2 and ^3 with shapes on the chalkboard in front of me, the teacher saying things like "square" and "cube," so it got stuck in my head as "shapes are involved." In preschool, we even did math with little cubes and squares made up of the little cubes. It is hard for me to disentangle what was so firmly implanted in the beginning as "shape." I see s³, and automatically, without any effort or conscious thought, I feel and think "time has shape." Ugh! : )

...

Okay, I rethought through my original post about ^2 and "squaring," and I edited it to read as follows:

(Idk why, but it only lets me copy and paste as an image. It won't paste the plain text.)
(Also, it automatically merged my replies. It wouldn't let me do a separate one for each, one to Genady, and then one independently.)

Edited March 26, 2022 by jerricks54

[jerricks54]

Another piece of this whole discussion is what the values and the units represent in the real, observable, experienceable world. Some units, like the watt (a unit for power, which equals kg*m²/s³), don't easily pop into my mind or emotions as an experience. I can't relate. So, I have to have experience with that unit a lot of times to get an intuitive feel for it, to build a connection with what happens in the real world when that unit is used or at work. The best way I know to do this is by working lots of real-world practice math problems. Given real-world scenarios, I work lots of different practice problems. 

For the watt, this might be converting between different units of power, like converting watts to Horsepower. Or, it might be solving for joules given time. Etc. The more examples, the more values are changed, the more different variables are solved for, the more of a "feel" I get for what this unit is doing in the real world. It's even better if there are real-live events happening in front of me, like calculating electric bill based off of my washing machine churning away feet from me. The more real-world practice, and the more examples, the more varied, the more I can "visualize" what this weird unit "is" or "does."

I can't remember where I read this, but someone posted somewhere that really weird units with strange exponents and combinations is all about relationships between various values and units. The purpose of those units is not perfectly map onto some visualization we come up with, they are to show relationships, proportions, ratios, etc. 

I found these posts of use while writing this: 

https://physics.stackexchange.com/questions/32096/what-exactly-is-a-kilogram-meter/32103#32103

https://physics.stackexchange.com/questions/221847/visualizing-physical-units-in-phyiscs?rq=1

https://easierwithpractice.com/what-unit-is-kgm2-s3/

Edited March 26, 2022 by jerricks54

[Genady]

2 hours ago, jerricks54 said:

It seems that the confusion for me is still the word "square" as a name for math operations

Math and physics are full of instances when common words from everyday language are used in a new sense. I don't think it is productive to be attached to any one meaning. Natural languages also have such instances, called homonyms. It doesn't generally confuse people using them.

[studiot]

2 hours ago, jerricks54 said:

It seems that the confusion for me is still the word "square" as a name for math operations in which either two identical values are multiplied, or two identical units (regardless of numerical value) are multiplied. In the case of two identical units being multiplied, using a geometric shape name "square" for "^2" connotes "shape" to the units being multiplied. Like a s², s³, or kg*m², or any other nomenclature/symbolism used to represent multiplying identical units (even if they have different numerical values). I first learned ^2 and ^3 with shapes on the chalkboard in front of me, the teacher saying things like "square" and "cube," so it got stuck in my head as "shapes are involved." In preschool, we even did math with little cubes and squares made up of the little cubes. It is hard for me to disentangle what was so firmly implanted in the beginning as "shape." I see s³, and automatically, without any effort or conscious thought, I feel and think "time has shape." Ugh! : )

You and you teacher may have associated squaring with shapes but I will lay a bet that before your teacher mention s², she said something like

James has 5 apples. If Sally makes this up to five times as many, how many apples does James now have?

You said as much yourself

2 hours ago, jerricks54 said:

For example, let's say I have some strawberries sitting in front of me. The goal is to measure how many individual strawberries there are, to count the quantity, to count the frequency of "individual strawberries sitting in front me." I do this, and I come up with a value of 4. Now, if someone, with no context given to me, asked me to draw the quantity "4," how do I do that? I cannot. All I can do is write the character/number "4." So, in this sense, I see what you are saying. I cannot draw a numerical value. But, if I say, "4 what? I'm drawing the quantity of 4 what?" and they give me the crucial context of "strawberries," then I can draw out 4 individual strawberries. The count is 4, the unit is strawberry, and I just drew 4 strawberries on the paper. This is the part that I'm struggling to understand in relation to what you posted.

So how many square apples does James have ?

 

Or is the 'unit' not square apples or (apples)² ?

 

Were you going to reply to my previous post ?

[StringJunky]

13 hours ago, studiot said:

Definitely not.

 

Seconds squared although some may have got the idea from acceleration which is metres per second squared or (metres per second) per second.

 

There is more in the Philosophy of Mathematics about this because we have to think of 'repeated operations' and also whether the output domain of the repeated operation is suitable for the next repeat.

I am rather busy at the moment but will post more detail in due course.

Yes.

[Reduct Blog]

On 9/7/2007 at 9:28 AM, gib65 said:

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"?

Great question, and one that I struggled with while trying to understand the meaning of momentum.

The best explanation I have found, for what the product of two magnitudes is, comes from Robert Knapp's "Mathematics is About the World"
 

Quote

Yet, as I have already argued, momentum is a specific property of a moving object. It is a property; indeed it is a magnitude, a magnitude that can distinguish one moving object from another...

In general, a product of magnitudes is a condensation of certain aspects of a physical situation. The product reflects the measurements of each factor, but it also omits certain characteristics of the physical context, retaining only the arithmetic product and the combination of physical units that relate this product to the physical context of the measurement. ..

In relation to these units, the product measures a physical characteristic of the physical situation to which it relates....

And this product is properly regarded, in its own right, as a magnitude.

In general, whenever two magnitudes are connected within a category of physical situations, their product can be taken as a third, derived magnitude pertaining to and measuring something about that type of physical situation.

The ontological status of momentum as a magnitude is not affected by the indirect means required for its specification and measurement.

Knapp, Robert. Mathematics is About the World: How Ayn Rand's Theory of Concepts Unlocks the False Alternatives Between Plato's Mathematical Universe and Hilbert's Game of Symbols (p. 147 - 149). Kindle Edition.

 

He also tackles the case of adding 4 apples to  3 oranges ( 7 "fruits"), and multiplying "umbrellas by bananas" (as per that hilarious article title that michel123456 linked to):

Quote

When one multiplies a number of units of one kind by a number of units of a second kind, the result is a number of units of yet a third kind.

Knapp, Robert. Mathematics is About the World: How Ayn Rand's Theory of Concepts Unlocks the False Alternatives Between Plato's Mathematical Universe and Hilbert's Game of Symbols (p. 156). Kindle Edition.

So, to answer your specific question as to what a "kilogram-meter", I would say that it is a type of unit, a compound unit that is composed of kilograms and meters (obviously enough).

And in fact, this unit (or rather "kilogram-kilometers") is used in the freight industry, as part of a rate like "CO₂ emitted per kilogram-kilometer".

Visually, this is hard to imagine I suppose, but I see a kilogram weight sliding along a meter stick, to represent a sort of dynamic unit, one which can be used to compare the movement of arbitrary kilograms of freight over whatever distance.

When you move a kilogram a meter, you see only the kilogram and the meter distance, but there is an indirect magnitude here, something resulting indirectly from that movement of the kilogram that distance.

Once we consider the amount of time taken for that kilogram to move that meter, then we have momentum.

It's like the kilowatt-hour, which results from letting a kilowatt device run for an hour.

The kilowatt-hour is exactly like the kilogram-meter, in that the first unit (kW or kg) is "run" over the second unit (hour or m), and we create a new, indirect, derived "third" unit (kw·h or kg·m) : in the case of the kilowatt-hour, this compound unit is a unit of energy.

(I don't know enough physics to know what a kilogram-meter measures, but it doesn't seem too far from energy to me -- although my concept of "energy" is rather vague)

I'll be tackling multipart rates and compound units over the coming months over at my blog, which is against the rules to advertise as part of a discussion thread, so feel free to check it out.

Edited May 24, 2023 by Phi for All
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