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/

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.)

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.

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....

@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.

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