when you divide a number of meters by a number of seconds, you are expressing a ratio, an amount of meters for every second. you can add apples to oranges, but the result will be apples plus oranges. and it will mean number of apples together with number of oranges. if i do (3m)/(2s), because everything is being multiplied or divided and multiplication is commutative, i can rewrite that ad (3/2)m/s, or 1.5m/s. the m/s doesn't simplify further though. similarly, if a is apples and o is oranges, and i do 3a+2o, there's nothing i can do to rewrite that. it's just 3a+2o. it's NOT the same as adding 3+2, that is, 5(a+o), because that would be 5a+5o take away from this that units are multiplied and all the rules about multiplication with numbers are true for units as well.

ALSO: you could compare apples to oranges, and express that comparison with a ratio. SIMILARLY: you could not add meters to seconds.

i can write down 3m+2s if i want to. i think what you mean is they no longer have something underlying in common, like how if i have 3 apples and 2 oranges then i have 5 fruits. but imagine i'm doing a task where i need to move something 3 meters and then wait 2 seconds, or maybe i could wait 2 seconds and then move the object 3 meters, or maybe i could move it one meter, wait one second, etc. then 3m+2s is meaningful, as meaningful as 3 apples plus 2 oranges. the underlying unit could be "tasks", ie moving the object 1 meter is a task, and waiting 1 second is a different task.

This arbitrary (re-)definition of "task" does not make sense, however. If you move 0.3 meter and wait 0.6 second, how old is the captain?

I don't really agree with him but he's sort of trying to say you unify the summed units with an over arching unit. Like he would say it's fine to say 1 meter + 1 inch because it's 0.00_ miles. I personally think the issue is more that writing 1 meter plus 1 inch is just mathematically wrong on its own essentially because the operation + isn't defined without more information. For example you could consider 1 meter + 1 second to be somewhat sensible as distances in spacetime, but actually computing the distance you get from events separated by a meter and a second is the difference of the squares after adjustment by C, rather than just the sum after adjustment by C.

This sounds more like composition than addition, especially noting that it might not be commutative.. so tasks make a good metaphor actually

I don't think there's any interpretation where the expression 3m + 2s is meaningful. The example you provided with "tasks" saying that "the underlying unit would be tasks" isn't convincing at all, I've never seen an instance of that being used, and I guess if you needed that to make sense in that context you would just explicitly write a unit of "tasks" or leave it dimensionless, like Hertz being "things per second" but defined simply as 1/s, with no extra unit on top. Apple and oranges are both fruits, so their sum can indeed represent the total number of fruit, but "lenght" and "time" aren't things that you can sum to get a total number of some more general thing encompassing them

In mathematics, you can make any rules that you like, they do not need to be based in the real world. After all, we think of numbers as continuous even though everything in the universe is discrete, even space and time. We do not have to leave these concepts though, they do have an importance in the real world and even if they didn't, they are proof of our logical and creative abilities. Also, 3m + 2s could be useful if you are taking both of these things as resources. Like, if you had a space of 10m and 5s of time. You can represent that as 10m + 5s and then subtract 3m + 2s and see that you would be left with 7m + 3s. Of course, you can treat them seperately and say 10m - 3m would be 7m and 5s - 2s would be 3s. Its just a different way of representing it. We should not say that it is meaningless just because there is a better way to represent it. It is just as much a logically consistent way to express something.

>In mathematics, you can make any rules that you like, they do not need to be based in the real world That's true, I agree, but I still fail to understand what possible meaning cuold an expression like 3m + 2s have, not only in the real world, but even in abstract (without chaning the deifnitions of meter and second, since then you would just be adding some different kind of stuff that happens to be represented by the letters "m" and "s", without having any connection to meters and seconds) >After all, we think of numbers as continuous even though everything in the universe is discrete, even space and time This is not true: we don't know if space and time are discrete, and in fact I think most hep-th physicists don't believe that. (If you were thinking about "Planck's length" or "Planck's time" while writing this, read [this comment of mine](https://www.reddit.com/r/theydidthemath/comments/y6qmpj/comment/iyuvy24/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button) and the discussion that generated underneath, with links). But let's say they are discrete: using the continuum of the real line to describe numbers would still be a useful approximation, I agree. What is the usefulness of an expression like 3m + 2s though? I'll get to that in the next part of this comment >Also, 3m + 2s could be useful if you are taking both of these things as resources I don't understand how this makes any sense, what's the utility, or if it's even logically consistent as you claim. So, let's try to take this proposal at face value, let's play with it a bit and see where it takes us. Let's take this "resource" quantity (3m + 2s). What would be the units of this new "resource"? Can it be expressed as a single number? Would you say you have "5 resources", or 3m + 2s = 5r? But this depends on the choice of units of the first expression. What if you had written the first expression as 300cm + 2s? Would you now have 302 resources? Or maybe "centi-resources"? But only "centi" with respect to one of the two initial resources? This seems to break. So, you might say, don't express this "resource" quantity in terms of a single unit. Fine, what do we do with it then? How to we multiply two resources? Do we get that the product of two resources has partially units or area, partially units of absement? What does this quantity represent?

Let's say you're in a car with a bomb, rolling toward a cliff. The bomb is going to blow up in 10 seconds, and the car is 10 m from the cliff. Anything you do to fix this situation is going to take both time and distance, and the time and distance are independent of each other.

Ok, and? I don't see how this is relevant to what we're discussing here. How do you propose we use a quantity like (10m + 10s) to do anything here? How does this example help us understand the meaning of an expression in the form of [length] + [time]?

What's the utility of 3 + 4i?

You didn't answer the question, you just dodged with another question. Also, the point you're trying to make is silly: the immense utility of complex numbers all comes from the fact that i\^2 = -1, and all the multiplicative geometric structure that comes associated with that simple idenitity, which gives us all the nice things we care about complex numbers: algebraic closure, analytic functions, representations of scaling and rotations... etc Literally none of that usefulness happens with something like 3m + 2s. There's no deep geometric identity hiding behind this expression

I think we should consider a few examples first. Lets take temperature. To convert Celcius to Fahrenheit, we have the formula: y = 9/5 x + 32, where x is the temperature in Celcius and y is the temperature in Fahrenheit. And to convert Celcius to Kelvin, we have the formula, y = x + 273.15, where x is the temperature in Celcius and y is the temperature in Kelvin. This would give us results like these. 0°C is the same temperature as 32°F And 0°C is the same temperature as 273.15 K. But we can face a problem when we say, 0°C = 32°F or 0°C = 273.15 K Even though there is nothing wrong with these statements, we cannot apply the normal rules of arithmetic to them. If we treat °C and °F as normal units, we would find that °F and °C are equal to 0 which is nonsensical. 0°C = 32°F actually represents that 9/5 (0) + 32 = 32. Similarly in the example with apples and oranges. If we just say that 3 apples + 2 oranges = 5 fruits and statements of this kind, we would face a similar problem which would make us conclude that apples = oranges. What we are really trying to say is that 3 apples are 3 fruits but 3 fruits may or may not be 3 apples. How can we state this in a way that doesn't violate the rules of arithmetic? We can just make a function for fruits. And say that f(3 apples) = 3 fruits. And it is obvious that the function f does not have an inverse considering the existance of oranges. Now, I think the case of apples and oranges best fits the hypothetical situation I provided. There could be a function r for resources and we can say that r(1m) = 1 resource and r(1s) = 1 resource and there wouldnt be any issue concerning the units since r(100cm) would be the same as r(1m). Now, addressing your other concern about the utility. My previous comment might have been unclear. I just present this as another way of presenting the same data. I do not expect anyone would find any utility in this but that does not mean we should reject this as an idea. But neither do many of the things in modern mathematical research. It is an interesting idea and leads to discussions about what mathematics is actually about and I think that is enough a reason for it to be considered. Also, great point about the space and time, but my point still stands in other regards.

Ok, now I feel like this way of formalizing it is better: I agree that it's a potentially interesting idea (regardless of utility), but wouldn't you then use this to write something like r(3m) + r(2s) rather than 3m + 2s?

You are right. I just used that notation because that was being used in the original comment.

Well my entire point is that that original notation was wrong. But apart from that, i think we agree. Thanks, have a nice day!

> even though everything in the universe is discrete, even space and time **\[citation needed\]** All of our fundamental, well-tested, and accepted theories of physics, special and general relativity; and quantum mechanics and quantum field theory, fundamentally have both space and time (or their combination) as continuous. You basically can't construct a regular grid that respects Lorentz symmetries, even approximately. Heck, even the more controversial ones like string theory are fundamentally continuous. You have to go to something even more outrageous like loop quantum gravity to get a discrete space-time.

Sorry, my knowledge in this regard was just based on hearsay, but wouldnt you agree that there are some physical quantities that are discrete?

Sure, some (particle number, energy levels in bound systems, dimensions of spin states). But not time or space.

The broader abstraction here is still useful though. In order for unlike units to be added, they must be converted into the same unit. In the case of apples and oranges to fruits the conversion just so happens to occur at a constant one-to-one. However "distance + time" likely wouldn't be in most situations. (Taxi cabs, for example, charge for distance and time and so must convert each to dollars before they can be meaningfully added together.)

> In order for unlike units to be added, they must be converted into the same unit. Not if you treat those different units as different dimensions, like adding vectors.

literally complex numbers

The point you're trying to make is silly: the immense utility of complex numbers all comes from the fact that i\^2 = -1, and all the multiplicative geometric structure that comes associated with that simple idenitity, which gives us all the nice things we care about: algebraic closure, analytic functions, representations of scaling and rotations... etc. Literally none of that usefulness happens with something like 3m + 2s. There's no deep geometric identity hiding behind this expression, nor a clear definition. What would the product of such "numbers" even look like?

not the point, a + bi uses notationally the fact that you cannot simplify it into a single number, which effectively gives you a vector space without the tediousness of writing it in vector form. Effectively, this is all just category theory. You're trying to elevate two numbers in R into a single element in C with some additional structure on top.

I mean, I agree with everything you're saying about complex numbers, but I fail to make the jump/analogy with summing quantities with different physical dimensions. Once I accept that complex numbers make sense as an extension of the addition structure on real numbers, how does 3m + 2s make sense? Mind you, "m" and "s" need to preserve their original meaning as "meter" and "second", otherwise if we just make them generic letter variables with no physical meaning, then yeah ok sure, but the point is making sense of 3m + 2s while still maintaining the original meaning of what a meter and a second are and how they are defined in SI

I get what you’re saying and I don’t disagree but it’s never a bad idea to think of other contexts where something might be true. In the same way you could describe a composition of rigid motions (although we wouldn’t use a plus sign), maybe we’re describing a set of instructions or some series of actions.

Well you can represent them in 2d space with each axis representing a quantity. Then you have (3,0) + (0,2) is (3,2). And this can be used in the taxi example given by u/kblaney in response to this comment. You can think of taxis having a "fare matrix" which when multiplied by the 2d vector gives you a single dollar amount.

This is a different thing than adding quantities with different units, though. Notice that in this new vector representation you proposed, when you perform a vector sum you still compute it by constructing the new vector adding component by component: and the components you're summing still share the same units. You're never writing 3m + 2s, you're writing (3m, 2s) + (2m, 4s) = (3m + 2m, 2s + 4s) = (5m, 6s) Also, as an aside, I don't quite get the "fare matrix" that gets you a dollar amount when multiplied by a vector. Isn't matrix-vector multiplication defined to output another vector? Not a single scalar amount. Whatever this new operation does, it probably shouldn't be called matrix-vector multiplication

The notations we use for vectors do allow us to write 3m+2s. You just need to let m be the unit vector along one axis and s be the unit vector along the other axis.  About the fare matrix, it would be a 1x2 matrix, where the columns are named distance and time, and the row is named currency. It transforms a 2D (distance, time) column vector to a 1D (currency) vector. I agree things feel a bit far-fetched by looking at it like this. But I added my comment only because I feel it's still an entirely legitimate way of looking at things. With this view, the notion of multiplying things of different units can be viewed as a tensor multiplication of 1D vectors. That's why it is so comfortable for us, because even the result is 1D and is reduced to a single number. It does seem to break if you try to divide vectors that have more than one dimension, but as far as addition is concerned it captures the concept quite neatly.

I think I disagree with almost everything you wrote, sorry. Let me go point by point > The notations we use for vectors do allow us to write 3m+2s. You just need to let m be the unit vector along one axis and s be the unit vector along the other axis While yes, this works if m = (1,0) and s = (0,1) (because then 3m + 2s = (3,2)), this is a new "+" operation in this vector space, and not the same "+" operation we use on real numbers, which is the one that you still need to keep using when you sum the individual components of these vectors. This is what I feel people mean when talking about "summing quantities with different units": they're talking about addition of real numbers, not vector addition in a new vector space where each component represents a different unit. While perfectly legitimate, it sounds misleading to frame the problem this way > About the fare matrix, it would be a 1x2 matrix, where the columns are named distance and time, and the row is named currency. It transforms a 2D (distance, time) column vector to a 1D (currency) vector Calling a row vector a "matrix", while _technically true_, is very misleading. The word "matrix" in this context refers to an (n x n) object representing linear transformations, not (1 x n) objects representing linear functionals. You just call it a functional, or a dual vector, or a 1-form. And the result of the product is a scalar, in fact, it already has a name and is called "scalar product", or inner product, or dot product. > With this view, the notion of multiplying things of different units can be viewed as a tensor multiplication of 1D vectors This looks like it's contradicting what you've been saying up until now: you said that we should think of the units of m and s as unit vectors in the direction of some axes. Then, m = (1,0) and s = (0,1). These are 2D vecotrs, not scalar (or "1D" vectors as you (confusignly) call them). So, their tensor product is a (2 x 2) matrix, not a scalar This doesn't look consistent, legitimate, or useful at all

Okay, here's my viewpoint: The discussion was whether we can add objects of different types to each other, not their quantities. If you want to limit yourself to real numbers to represent everything, it won't work. But objects can be represented as more than just real numbers and symbolically representing them, or representing them as vectors, is a valid attempt.  About the matrix, we were transforming an object in 2D space to an object in 1D space. That's why it happened to be a row vector. But even if in this specific case you want to call it a functional, it's still an operation on this object which is a sum of things of different units. I don't think the discussion of what to call the operation is relevant to the purpose of the example.  Regarding the tensoring, I did a bad job of communication. What I meant to say was that we are comfortable multiplying two 1D objects together (by which I mean they live along an axis) because they result in 1D objects. When written in this vector space, 5 m times 3 s would be (5,0) tensored with (0,3) which would give you (0,15,0,0) which is representable by a single number 15 and a unit that really just says "co-ordinate 2", or m s in the index of the tensor space.

> The discussion was whether we can add objects of different types to each other, not their quantities What do you define as "the quantity" of an object? > I don't think the discussion of what to call the operation is relevant to the purpose of the example I agree with this, it's not super relevant In the end I guess that your construction is consistent (I don't actually know this, but it seems so), now the problem is, how would we ever use something like that? Why is it useful?

In any game or sport where both length and time are desirable, your score or rating could just be length + time. For example, a game show where you have 10 minutes to construct a marble run out of lego and your score is based on the number of meters the marble rolls + the number of seconds it rolls for.

But this isn't an example of summing length and time, you're just summing dimensionless quantities: the formula you described to compute the score S would be: S = L/m + T/s Where L is the length of the track, T the rolling time, m is 1 meter, and s is 1 second. Notice that (L/m) and (T/s) are both dimensionless numbers, they don't have units of \[length\] and \[time\]

It is correct in a strictly formulaic sense, that the calculation isconverting both to dimensionless quantities and then adding them to get a score of arbitrary "points". However, it's still an an example of a context where adding "metres" to "seconds" has a practical meaning - when summarizing the rules to contestants it could even be expressed as simply as "your score is metres plus seconds". > > > I don't think there's any interpretation where the expression 3m + 2s is meaningful. Within that game, the expression is meaningful. If the points are based on twice the time in seconds plus thrice the length in metres, this could be meaningfully expressed as p = 3m + 2s. The TV show "Taskmaster" often has timed tasks where a distance is also measured and/or some context-specific penalties and/or bonuses are applied, that leads to these sorts of situations. > Choose a length for this pole then guide it through the course. The person who completes the course with the longest pole wins. Every time your pole touches something other than your hands, ten centimetres will be taken from its length. You have a maximum of ten minutes. Your time starts now. > Get the duck into the lake. You must not touch the beak. If the duck leaves the course, it must re-enter at the point it left the course. If your duck touches the boundary or a flamingo or a pineapple, one minute will be added to your time. Fastest wins. Your time starts now.

I think I understand now where the confusion lies, and maybe I can try to explain what's wrong. There are two different things you seem to be conflating: a length L, and a length L in meters. There is a subtle difference here, these quantities are not the same. In particular: L is a distance. It has dimensions of [length], and whichever units of [length] you want to express it in. "L in meters" is NOT a distance. It's a pure dimensionless number, has no units, and is equal to the amount of meters that make up the same length as L. To write the number "L in meters", take L (measured in any units, not necessarily meters) and do L/m Repeat the same distinction with the quantity T for time, and the quantity "T in seconds". Everytime in your examples you think you're summing meters and seconds, you're actually summing "number of meters" and "number of seconds" I hope this clarified something

“Bob walked for 3 meters, then an additional 2 seconds” is absolutely a meaningful description of the manner in which Bob is walking. The issue arises by not having enough information to determine either how much time this took him, or how far he went. But the statement still has meaning. What’s interesting is that if we know Bob is walking with a constant speed (represented as m/s) or a constant pace (represented as s/m), and we know what that speed (or pace) is, then we can convert 3m + 2s into an expression that we can simplify and find out both how far he went and how long it took him. Which again says that 3m + 2s is a meaningful statement. It’s just that getting at what that meaning is can be difficult, or even impossible.

> “Bob walked for 3 meters, then an additional 2 seconds” is absolutely a meaningful description of the manner in which Bob is walking. I completely agree with this, however, what does this have to do with writing down something like 3m + 2s? Your statement isn't equivalent to the expression "3m + 2s". In fact, the latter isn't even a statement, it's just a single (undefined) quantity. The way to correctly write down you statement about Bob is this: let x(t) represent distance Bob walked at time t. Let t=0 be the time Bob starts walking, and T the time at which Bob stop walking. Your statement then becomes: x(T-2s) = 3m Nowhere in the above the quantity "3m + 2s" appears, as it's meaningless and doesn't represent anything measurable

You’re incorrect. Both 3m and 2s represent *either* a distance Bob walked or a time Bob walked for. Meaning both expressions can represent both ideas. So 3m + 2s can represent both as well. If Bob is walking 6 m/s, then 2s can convert to 12 m and 3m + 2s becomes 15 m. Or 3m becomes 0.5s and we get that 3m + 2s represents 2.5 s. Either way 3m + 2s certainly represents something and has meaning. To say otherwise is quite strange actually. Here’s another example: Bob has 3 bars of gold and $500 in cash. Since gold has value, and dollars have value, we could say that the value of assets in Bob’s current possession is 3g + 500c. This certainly is a meaningful way to represent the value in Bob’s hands. But we can’t *combine* them into a single number until we know what he can either sell his gold bars for, or how many gold bars he can buy with $500. But either way it’s still a perfectly reasonable representation of how much value Bob has.

I’m not incorrect, and you seem to have a fundamental misunderstanding of what units are or, more in general, what mathematical expressions mean and how to use them. Here are the correct ways to write the quantities you suggested: If Bob is walking at a constant speed of v = 6 m/s and he has already covered L = 3m of distance, and we’re interested in the total distance D he will have covered if he keeps walking for another t = 2s, we write that as D = L + vt = 3m + (6 m/s)\*2s = 15m. We *don’t* write it as D = 3m + 2s. If, in the same scenario as above, we're instead interested in how much total time T Bob has been walking for, we write it as T = t + L/v = 2s + 3m/(6 m/s) = 2.5s. We *don't* write it as T = 3m + 2s. If Bob has w = 3g of gold (I'm assuming one standard gold bar weights 1 gram) and c = 500$ of cash, and k = 83 $/g is the value of 1 gram of gold, and we're interested in how much total value M that's worth, we write it as M = c + wk = 500$ + 3g\*(83 $/g) = 749 $. We *don’t* write it as M = 500$ + 3g I don’t think this will convince you, since you seem to be quite convinced of your mistake, and online discussions always end up in debates where people never want to admit when they’re wrong. So, if you still want to argue that you’re correct, could you link me a single academic/technical/reputable source where anyone ever writes the sum of two quantities with different physical dimensions, such as 3m + 2s? An introductory textbook maybe, or some lecture notes, a physics paper, a video extract of a lecture with some writing on a blackboard, really anything will work. Good luck with your search!

No, you *are* incorrect. The way I was writing it is a *reasonable* way to write it, that, yes, still conveys *meaning*. You’re trying to force the situation into official equation types. But the beauty of math is that we don’t *have* to. The misunderstanding is that you think there’s only one way to use plus signs. It’s not *addition* here, it’s appendation, which is more general form of addition. I’m not creating a linear equation. The task was simply to write down an expression that conveys meaning about a situation. I mean if you’re saying the expression Assets = 1 house + 2 cars + $70,000 Lacks meaning then I don’t know what to say. Not every situation has to be drilled down into official equations. We can form more general expressions than that, to convey *ideas*. It seems what you really lack here is imagination. Oh well, I hope you figure it out someday. So long. Edit: this isn’t a response to whatever you wrote below. I haven’t read it. No, it dawned on me that this whole time you’ve actually tacitly agreed with me. If the whole point was that the expressions still convey meaning, then you admitted as much by understanding there are ways to convert them into proper equations. The only reason you’re able to do that is you understand what my expressions *mean*, i.e. they *have* meaning. The reason you can’t tell you’re actually in agreement with me is that you think all of these ideas *must* be rewritten in a proper format, which just isn’t true. It is perfectly reasonable to use more general expressions to get points across. And in fact, this is usually how we represent things when doing mental math. I’m betting that my very first example is a lot easier for non math folks to understand than the linear equation you wrote. And it has the added benefit of using no variables and describing exactly what took place. But I highly doubt any of this sways you. You’ve shown yourself to be dogmatic in your approach. But like I said, whether you admit it or not, you’ve actually accepted all along that my expressions have meaning and can be interpreted. You wouldn’t have been able to respond as you did if they were meaningless. So I’m glad we agree on that point.

>The misunderstanding is that you think there’s only one way to use plus signs. It’s not addition here, it’s appendation Well, I guess this whole thing ends here right after you wrote this sentence. I don't know if you are being dishonest or if this was a genuine misunderstanding, but we were OBVIOUSLY talking about addition when writing the symbol "+", as every normal person would assume in the context of someone asking why they can't ADD units of different dimensions. We were not talking about appending, about a new logical operator meaning "and", about imagining new meanings of our notation: we were talking about addition. When one asks "why can I multiply different units, but not add them?" They are talking about the arithmetic operations, they're not talking about using the symbols "+" and "×" to assign them a new meaning to use outside of "official" equations (whatever that means) Nice way of coping out of your claim once you realised you were writing nonsense. Good luck!

"Formal sums"

meters and seconds are both units. if i have one meter and one second, then i have 2 units. if i have 100 centimeters and 1 second, then i have 101 units. this is probably not as useful as counting fruit, but nothing is mathematically wrong here. if you object on the basis that this implies 2=101, that is wrong, because by changing both meters and centimeters to "units," i have forgotten how a centimeter is related to a meter, and only remembered that they are units. so indeed 2 does not equal 101. if you object on the basis that that's useless, we're no longer talking about math, because formally this is exactly the same as counting fruit.

... what? Help me follow what you're proposing, since it makes absolutely no sense to me. Would you say that the following chain of equalities is valid, according to what you wrote above? 101 units = 100 units + 1 unit = 100 cm + 1 s = 1 m + 1 s = 1 unit + 1 unit = 2 units

editing this to hopefully clarify: you should take my comment more as a reason why we don't add things with unlike units, because that kind of nonsense happens when we do. you could replace units in my comments with "things" or "blorglesnaps" or whatever you prefer. plenty of other comments on this thread, including my previous comment, have given examples where it makes sense to bookkeep meters with seconds. if you're converting them to something they have in common (like apples and oranges are both fruits), then of course you have to keep track of that conversion. maybe i'm playing a game with two moves, move or wait. i can move in units of 1 meter, or i can wait in units of 1 second. say i have 3 of the former to use and 2 of the latter. then i have 5 possible moves. i don't suddenly have 302 possible moves because a meter is 100 centimeters, because moving 1cm is not a valid rule of the game.

I've read it, and it's borderline word-salad to me, so I'm trying to understand better what you're proposing by asking a clarifying question with an example. Could you answer my question? Would the chain of equalities I wrote above be valid/true according to you? If not, could you point out the specific step where the mistake lies?

edited

You kind of can add meters and seconds because of relativity, but it’s often assumed that your meter is perpendicular to your second. If you use natural units, the speed of light is 1 (unitless). Edit: your answer explains what OP was asking, though their question is flawed, since it used a bad example.

This involves tweaking your perspective so that you are actually measuring time and space using the same unit though.  It's saying something very deep and special about the relationship between space and time.  It's very different to trying to combine two arbitrary units.

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It’s an interesting idea, but I doubt that Einstein just randomly had the idea that mass and energy are related. It is much more likely that he developed that concept through years of rigorous study at the forefront of theoretical physics. That is to say, I don’t want to discourage inquisition and curiosity, but I’ve heard a lot of garbage pseudoscience come out of that. Like the quantum mysticism crowd ugh

This sounds like bad physics on top of bad mathematics. You cannot add meters and seconds. Even when you use natural units - whatever the hell they are.

Technically he’s not wrong, but as someone said: it requires a change in perspective that invalidates the actual brilliance of the equation. Just because there is a relation between two things does not mean that they are the same thing.

As a practical matter, this type of dimensional analysis is very useful for sanity checking your work, and even deriving formulas in physics.

I tell students sure you can add apples and oranges, but you have to figure out how they are similar first (working with elementary, this helps scaffold common denominators) So 3 apples plus 2 oranges is 5 fruit (or objects/items, whatever) But we remind them the unit measure has to change to accommodate both objects.

I'm not sure how it is expressed in math, but in comsci adding 2 apples to 3 oranges you would get 5 fruits. Maybe set theory? Someone educate me here.

This means you added 2 fruits to 3 fruits.

Object oriented programming. Apple and Orange are derived classes that inherit from the fruit base class. But the fruit class doesn't have a +operator, so it still doesn't make sense to add apples to oranges. That's why you create a fruit_basket class that stores fruits.

This *feels* like an abstract algebra thing

The free vector space on {apples, oranges} simply declares apples and oranges to basis elements. so you could represent 2 apples + 3 oranges as (2,3)

What is your explanation for multiplication then? Like in division you are expressing ratios an amount of one thing for every unit of other thing , what is it that you are doing in multiplication?

Just like division is inverse multiplication, multiplication is inverse division. Mathematically, it's just reversing which of the two is more "fundamental." Trying to come up with a physical interpretation of what apple times orange as a unit means is left as an exercise to the reader.

In this scenario, Apples are matryoshka dolls that each contain a fixed number of Oranges 

If you _really_ want to you can cast it as a second-per-foobar (where a foobar is a m^(-1) ).

Second third fourth this

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Is there any context where you would actually use that?

You can add apples to oranges. 5 apples plus 4 oranges is (5 apples and 4 oranges). The problem is that it won't reduce. I can subtract from this (4 apples and 2 oranges) to get (1 apple and 2 oranges). It's not reducing into a single unit because we're not introducing any relations between apples and oranges. But keeping the unit separate and doing arithmetic this way is just an example of tensor algebra. Specifically since we only deal with apple and orange, we are dealing with rank ≤ 2 tensors. Once some relation between apples and oranges is established, like apples = oranges = fruits, the tensors may collapse and for example the problem above becomes 5 fruits + 4 fruits = 9 fruits, subtract (4 fruit + 2 fruit) = 6 fruit and get 1 fruit + 2 fruit = 3 fruit. Or maybe the price of 1 apple is half of 1 orange, and so for example 1 orange = 2 apple = 1 dollar, so now there's no conversion from fruits to dollars, but there is a different way to collapse the apples and oranges tensors. Now 5 apples + 4 oranges = 6.5 dollars, etc. On the other hand, another type of relation is saying for example that 1 apple + 1 orange represents the unit 1 "of each". Now 2 apple + 2 orange is 2×(1 apple + 1 orange) and therefore 2 "of each". And (4 apple + 6 orange) = 2 orange + 4 "of each".

I doesn't look like a tensor algebra to me... If it were, then all your maps (collapsing, price, etc) should be bilinear, although they are linear ; and multiplying (1 apple + 1 orange) by 2 would give (2 apple + 1 orange) = (1 apple + 2 oranges). Aren't we simply computing in the vector space Apples ⊕ Oranges, and all your maps are linear ?

>Aren't we simply computing in the vector space Apples ⊕ Oranges, and all your maps are linear ? I'm pretty sure this is true. I guess another way of saying it is apples and oranges are just two orthogonal real valued axes, so we're just looking at R^2 here. If I were to guess what op means with tensors, maybe it's that further combinations of apples and oranges can become tensors? Like transformations ((apples/apple, oranges/apple), (apples/orange, oranges/orange)) are tensors I guess? Though the statement rank <= 2 doesn't make sense in that context. There I think rank is arbitrary, but each individual dimension is <= 2 To add to the discussion with a more relevant analogy / example: there is a really common case where we combine two different types to objects - space and time. We first extend to space time vectors with a direct sum. We then consider many different higher rank objects, but the individual dimensions are always bounded by the direct sum. There is also nothing stopping us from adding more space and time dimensions analogous to more types of fruit, increasing the maximum size for each dimension of relevant objects while keeping the rank the same (usually)

I want to caution that sometimes you can do things mathematically after the conversion that don’t make sense before, and whenever you are thinking about real life situations you need to disallow things that don’t make sense or else find alternate interpretations. For example, if you have 3 apples and 4 oranges, you have 7 fruits. If you want to eat 5 oranges, that’s 5 fruits, and 7 fruits - 5 fruits = 2 fruits. But you can’t actually eat 5 oranges because there were only 4 oranges. Perhaps the issue is that we don’t really have equality between apples and oranges. They both can be viewed as fruits, but they cannot be converted freely between apples and oranges. The bigger lesson here is that there are mathematical models of the world, but not everything that makes mathematical sense in the model makes realistic sense, and so care is needed to decide which operations are valid and which aren’t.

I might misunderstand, but where is the tensor product in your example, specifically where are the rank 2 tensors? For me all that's going on is that we're in a vector space (or free module, or even just commutative monoid) generated by the basis {apple, orange}.

Yeah it's a little weird. I'm thinking probably of the free algebra on {apple, orange} aka the tensor algebra on the vector space on that basis, to define counting them in a way that can actually measure how the formal sum apple + orange has two pieces that can't really be brought into one without quotienting by more relations. Over the field of real numbers we could model it as the diagonal embedding taking e.g apple +orange to apple(x)apple+ orange(x)orange or really as apple(x)y+z(x)orange for some basis y,z of the same 2 dimensional space. Now this is a rank 2 tensor to actually capture that there are two parts of this "unit." Basically it's just another way of capturing the fixed ordering of a fixed basis to the define the support of a vector in terms of that basis, i.e. the nonzero entries.  So now it's meaningful to say that two units are counting the same thing like fruits by quotienting them to the same side so that it's a rank 1 tensor. 

But you can do the quotienting also just fine on just a free module? I still don't see the use for this rank 2 tensor. Maybe some sort of product becomes necessary if one wants to actually multiply quantities with units - but then it would make more sense to take a polynomials/symmetric algebra instead of the tensor algebra, I believe.

You're right, there are isomorphisms that don't make it super meaningful to call these things tensors. All I'm saying is there should be some structure beyond a vector space that captures the idea that apples and oranges are two units that don't mix together, so looking at an expression 5 apple + 4 orange is more than just one vector, it has two pieces. My mind went to how not all tensors are simple, i.e. rank 1, and filled in the details ex post facto. I didn't really anticipate having to hold it up to any scrutiny.

but what if we had two linear equations for our apples and oranges? Then we could find unique solutions I believe

Well first of all two linear (affine, not necessarily homogeneous) equations have either 0, 1, or infinitely many intersections. But the generic case is that there's a unique solution. I gave two equations already as examples orange = apple = fruit, and orange = 2 apple = 1 dollar. Calling them fruits or dollars is irrelevant, we have the equations x = y and x = 2y and therefore x = y = 0. So now orange = apple = dollar = fruit = 0. I said before that in this situation "there's no conversion from fruits to dollars". Perhaps the correct abstraction is that the exchange rate fruit / dollar becomes 0 / 0 and is therefore undefined. So you're right, imposing different relations simultaneously gives a system of equations. But because these relations are supposed to be representative of something real/physically determined about apples and oranges, we should tread lightly when assuming two or more relations hold at once. It might not make sense physically if both relations are true at the same time.

Hmm interesting. What is representation theory? I'm a freshman undergrad. Is that like abstract algebra, because I took a course on that last semester.

i dont get the tensor analogy sorry. ok so units are tensor spaces, addition is direct sum so unit multiplication is what product of tensor spaces? and what is unit division , taking a quotient on those tensor spaces ? (according to which relationship)

You can add apples to oranges if you are counting the total number of fruits.

i like that but how about m+s

You can add any two disparate things as long as you define what the result of that operation means.

I found a [stack exchange post](https://math.stackexchange.com/questions/792291/why-cant-you-add-apples-and-oranges-but-you-can-multiply-and-divide-them) with an (eerily?) similar question which prompted a lot of interesting discussion.

It's not that eerie. Apples and oranges are the go to example of comparing two dissimilar objects and this is a common problem when first encountering units.

i assure you that i was too lazy to google it before asking it here in a drunken haze

The only possible answer

You must be thinking about it in a scientific context? It is because scientists will use math to explain what they are seeing. There really is no utility to having some quantity X of apples,oranges. But there is utility in examining the ratio of 2 things (this allows us to identify relationships between two quantities). Knowing how meters are related to seconds is key in figuring out your velocity.

meters minus seconds makes no sense... meters per second gives a rate.. a velocity in this case to elaborate, it is the same with addition and multiplication. kilograms plus meters makes no sense, but 1Kg\*m a kilogram-metre is one kilogram moved one metre. (i spelled metre/meter both ways) edit again i was wrong about that see below

I mean, nothing stops you from taking the direct sum of the additive groups or whatever (similarly to the complex numbers identified with copies of R), but we don't do that because it wouldn't necessarily be useful.

oh yeah thank you ! total amount of apples AND oranges is a totally valid example , as would apples per orange as an exchange rate . (i-hat)apples+(j-hat)oranges =\]

Zonkadoodles per booglesnaps makes no sense... apples plus oranges gives a meaningful grouping of items... perhaps a fruit basket in this case My point is that just dismissing the question by saying adding units "makes no sense" isn't very meaningful or useful.

thank you i was hoping to give an "intuitive" answer but forgot something crucial. i earned my D- on this =/

Making a fruit basket is different from adding different units, though

Multiplying apples with oranges is easy, you just express the produce with a new unit, the apple-orange. But unless you have a conversion factor for apples to oranges, the sum can’t be expressed in any other way than n oranges + m apples.  It’s like polynomial addition. You can only add like terms. 

You can only simplify or "reduce" like terms. You can absolutely add unlike terms in a formal sum.

You can't add unlike UNITS. apples and oranges aren't variables here. You could maybe treat apple orange as basis vectors of a vector space and then you'd be able to add them together.

You said I can't and then just provided a way of thinking about it that does let one. https://en.wikipedia.org/wiki/Formal_sum Or consider geometric algebra, where objects of different grades (and hence units) are allowed to be added together, and provide useful abilities.

Dude, nobody is saying you can't add different objects together, but you can't add unlike UNITS. If you ask what is 2km plus 5m you don't get 7 (km + m) that's simply not how maths works. You can say that 2km + 5m is ... 2km + 5m, but that isn't adding units together. The units remain completely separate.

Horrible analogy using something that converts to 5x10^(-3) km lmao. You can easily add these together. It is 2.005km

If you convert the units to be the same then you aren't adding unlike units...

You can't convert miles into gallons like you can convert km into m. It's a bad analogy. "Unlike units"implies the units are not proportional to eachother and can never be converted

You can add apples and oranges in the free abelian group generated by apples and oranges.

you can't add meters to seconds either...

Exactly. So odd that this isn't higher.

I mean if you ate 3 apples per orange then you’re eating at a rate of three apples/oranges

You CAN add apples to oranges if what you're concerned with is how many fruit you have You CAN'T if what you're concerned with is how many apples you have You divide distance by time to express a speed, not distance OR time Because speed is by definition the amount of time it takes to cover a distance

You can think of quantities as elements in a one-dimensional vector space (in this sense, choosing a unit amounts to choosing a basis). Now, just abstractly (without choosing a basis), you can construct a new one-dimensional vector space whose elements are products of one element of the first and one of the second vector space - the tensor product (this construction also works for vector spaces of other dimensions). If you think of your first vector space as representing velocities and the second as representing time, you can think of this tensor product as representing distance. With a bit more trickery, you can build a one-dimensional vector space whose elements are quotients of an element of the first vector space by a non-zero element of the second vector space (this construction does rely on the second vector space being one-dimensional but it does not need a basis). This way, you can make sense of velocities as ratios of distances and times. Further, for both of these constructions, a choice of basis (unit) for both vector spaces gives you a basis for the new vector space. On the other hand, you cannot abstractly construct a one-dimensional vector space from two one-dimensional ones whose elements are sums of elements from the first and the second. However, you can build a two-dimensional vector space (the direct sum) in which case you would think of 2 apples plus 1 orange as having 2 apples and 1 orange, and if you choose a basis for both vector spaces, you can build a one-dimensional space (which does, however, depend on the bases/units) and only gives you the useless interpretation that 2 apples plus 1 orange are 3 whatevers.

There is a field of math, often frowned at by anyone outside of physics, called dimensional analysis. Sometimes you can figure out a solution just by looking at the units attached to be inputs and outputs of a process, and derive an equation that would balance just the units. Fourier loved it, but it isn't very rigorous by modern standards of proof.

You can add apples to oranges! Units can be thought of as polynomials over some ring with a bunch of variables (with the names of the units). Let's think of the real numbers as our ring here. (We also need some extra structure to divide between them.) Adding x Apples and y Oranges gives you `x Apples + y Oranges`. What is usually meant by that phrase is that you can't use the addition from the real numbers. You'd need to consider some extra structure that lets you join the units to get e.g: x Apples + y Oranges = (x + y) (Apples + Oranges) The sum x+y on the right is what people usually mean you don't have.

When you add an apple to an orange, you have 2 fruits (because both are fruits) or 1 unit of an apple-orange set. But if the number of apples and oranges isn't the same, there is no counting possible in apple-orange sets directly. Instead, you will have some apple-orange sets and some apples/oranges. Dividing meters by seconds tells you how many meters in one second. If I travel x meters in y seconds, then on average I traveled x/y meters in 1 second. It goes like this: x/y in 1s 2x/y in 2s 3x/y in 3s . . . yx/y = x in y seconds Of course, saying I traveled x/y meters in one second (or meters per second, meters/second) assumes that I am travelling at a constant speed of x/y. Not necessarily true. That's why dx/dy would be more accurate. But the units remain the same. Now I would say something like: I traveled f(t) meters in t seconds. Here, you get f(t) by integrating v(t) w.r.t. t, where v(t) is the function that models your speed at a given time t.

Just make apple and orange juice and then its easy to add them in terms of liters of juice 😁

You can add apples to oranges.

You can add apples and oranges. It just depends on the context: 2 apples +3 oranges equal 5 pieces of fruit.

This is technically not a math question. It belongs more in the realm of physics and engineering. The short answer is you can do whatever you want with arithmetic operations. Arithmetic operations are based on abstractions from counting some generic objects. It does not care what the objects are. For everyday use and physics, we attach "meanings" to these generic objects at which point we have to further define what operations make logical sense for our calculations to have a meaning. So for example, 4 apples + 3 oranges = 7 fruits would be completely logical and fine. E: somewhat related video: https://www.youtube.com/watch?v=GZegwJVC_Pc

You *can* add apples to oranges. You just then have one measurement of apples and another of oranges. 3 apples + 4 oranges gives me 3 apples and 4 oranges

because multiplication is different from addition. you can't add m to s either, and you CAN multiply/divide apples and oranges. think of it as like how 2+3i doesn't simplify but 2\*3i does. multiplication combines different kinds and exponentiates like kinds, addition only combines like kinds.

but we can still add x=2 and y=3i even if the final expression (which we would write down as 2+3i) doesn't simplify any further

a+bi is equivalent to \[a,b\] on a space made of a real axis and an imaginary axis. with this framework, you can imagine "adding" different units to give coordinates in a space made of those measurements, like an "oranges axis" and an "apples axis". addition in this sense is perfectly fine, but it is not as "total" as addition between like kinds.

Indeed, we can identify the complex numbers with the direct sum of two copies of R. On the other hand, nothing stops us from doing the same with the space of apples and the space of oranges.  I assume here by total you sort of mean always-producing-a-scalar (total in the sense of being-defined-on-all-inputs would be strange here, as complex number addition certainly is). If we limit our understanding of addition with units of measure to scalars / same-unit-valued vectors, then sure, adding apples to oranges doesn't really make sense. A practical example where we don't limit ourselves to that would be automatic differentiation using dual numbers — the first component might represent meters, yet the second one would represent m/s. In that case, adding x m to y m/s would correspond to an object being at position x with velocity y — very neat!

yeah, the whole "you can't add apples and oranges" thing is just about the basic scalar-result addition, not addition interpreted as creating a series of coordinates.

You could reason about a ratio of apples to oranges, but not add meters and seconds together. Multiplication and division create new units, addition and subtraction need the same units to make sense. To put it simplistically.

Great ELI5 answer!

When you operate between two different units youre essentially creating a new unit as a result -- for example: meters divided by seconds results in meters/second. I suppose you could even say adding apples and oranges results in the unit "apples and oranges".

2 apples + 3 oranges = 5 fruit More specifically, if a ∈ A and b ∈ B, with A ⊆ C and B ⊆ C, then a + b is valid and is an element of C. For example, 𝜋 + i* *∈ ℂ.

ok now explain m/s and m*s using sets

You still can’t add metres to seconds.

that is exactly my question

In what context would you *want* to add them? What are you trying to do that the "don't add meters to seconds" rule is getting in your way of?

Your question is "why can't I add apples and oranges?" and then you mention meters and seconds as if meters and seconds are different in this regard. But meters and seconds work exactly the same as apples and oranges. You can't add them, but you can multiply or divide them.

Second one isn't useful, first one could be potentially nonsense too.

You can add apples and oranges. 5apple + 2oranges = 7fruits. You can as long as it's meaningful.

The way I think of it is that apples and oranges are two different objects. Often in math we deal with just one object called a number. Just a object called a number in it's most basic sense. By that I mean a number that is not complex. We can add a number to a number but we can't add a number to something that is not a number unless there is some kind of relation. Adding an apple to an orange is just that by definition. They are two separate objects with no defined relation. With multiplication and division a relation is implied. It is multiplication by a scalar. Again apples times oranges will be it's own thing unless there is a relation. I view it as multiplication and division is more relational but addition and subtraction can't be done without a relation. Except for counting totals of the objects and the relations that come with that.

The way i think about it is - the point of a “unit of measurement” for some quantity is that everything can be expressed as some multiple of the unit. So, if you’re measuring length, every length can be expressed as a multiple of a metre. Dividing metres by seconds to get m/s preserves this property - every speed can be expressed as some multiple of 1 m/s In contrast, adding apples to oranges doesn’t preserve this property - you can’t express (2 apples + 3 oranges) as a multiple of say (1 apple + 1 orange). It kinda comes down to the fact that R^2 is 2-dimensional, it can’t be written as the span of a single vector.

multiplying/dividing different units gives you a rate

You could say "one apple plus two oranges" but then the result is as useful as making a set instead of sum. Of coulre you can still have unit conversions like 3 oranges + 2 apples = 3000 mili oranges + 2 apples, but this is not very useful as is essentialyl equivalent to {3000 miliorganges, 2 apples }. And "big nono" is simply due to the fact that in all physics equations, geometry equations things that are added together are of the same units. So if at some point you end up with adding oranges to apples, you have a mistake in your calculations.

Note that you can also divide apples with oranges! You would calculate a ratio then, just like velocity is a ratio of distance to time.

Because if you change units you will get a different answer. Imagine computing 1 meter per 2 seconds = 0.5 (meter per second). If we compute it instead using millimeters and seconds, we get 1000 millimeters per 2 seconds = 500 (millimeters per second) and we can get back to meters per second by dividing by 1000 again, because both of them measure the same thing, and the difference of units is only a different encoding of the same information. And of course, the conversion from (meter per second) and (millimeter per second) is always to multiply by 1000, regardless of what the original numbers were. This makes sense. Let's try addition now: Imagine adding 1000 millimiters + 2 seconds = 1002 (of something). But if we compute this with meters instead, we would get 1 meter + 2 seconds = 3 (of something). But these two numbers are completely different. How do you translate (change units) from the other? If you say something like "add 999 to convert back", you will see that this doesn't work for other values. If we started with 2 meters instead, it wouldn't have worked out the same. So, adding meters and seconds gives different information than adding millimeters and seconds. This mans that "dividing" has meaning regardless of what units you're using, but adding can only have meaning when adding with specific units.

Yes you can add Apples and Orange if you want to know sum of fruits in your buckets. It depends of your goal

You can also multiply or divide apples by oranges. You'll end up with a quantity that has "apple-oranges" or "apples per orange" as its unit. In fact, the latter even makes intuitive sense. If you have 20 apples and 4 oranges, then you have 5 apples per orange.

2 Oranges + 3 Apples = 5 Fruits

Apples and Oranges can be added. Your result will however not be only Apples or only Oranges.

Well, this going to be a weird take but, if you cant add apples to oranges its because their axis are perpendicular, you know, like a vector. If you can multiply and divide meters with seconds its either because you know they are on the same axis, you maybe you dont know anything but you preserve their relationship say meter per second or.... Meters squared per second cubed, you know that sort of things. Maybe there's something wrong here in the details but the most important is you seeing where Im going with this thing.

It is perfectly acceptable to multiply/divide quantities with different units. Have you ever heard of "man-hours" needed for a task? That quantity is exactly a number of people multiplied by a number of hours. You can similarly divide apples by oranges: 5 apples divided by 4 oranges is 1.25 \*apples per orange\*. And in fact we add quantities in different units all the time: "To bake this cake, I need 2 eggs plus 100g flour plus 50g sugar plus..." As another answer said, the only issue is that these don't reduce; you just have to keep track of the units, because you cannot substitute 100g flour for some other ingredient and get the same cake at the end. Dividing different quantities is also fine: The SI unit of speed is "metres per second": it is a length unit divided by a time unit. When everything is in the \_same\_ units, there's nothing to keep track of so we often omit it (which might be what is confusing you?) and we end up with a \_dimensionless\_ quantity: 3 meters divided by 2 meters is 1.5 \*meters per meter\* -- Writing out the units like this sounds daft, so we don't do it in practice. Because it is dimensionless, we will get the same answer no matter what units we started with: If we convert "3 meters" and "2 meters" to yards and then divide, we will still get 1.5.

You are never holistically combining the two in *either* process. If you divide a unit of length by a unit of time, the resulting object is a ratio of length to time. It did not really become simplified, but a combination. For the obvious example in math, if you add a real number to an imaginary number, you do not get a simplified object, but a combination of the two, which for convenience we've *labelled* that specific combination "complex". A complex number is some real number added to some imaginary number. You could label a combination of apples and oranges an 'Orpple', for example, but it would still represent their combination, not a fundamentally simplified new representation. Or you could divide apples by oranges to get a ratio of apples to oranges. The unique representation of that unit is still dependent on its two distinct constituent parts.

You can actually add apples to oranges if you tell both are fruits. 3 apples + 2 oranges = 5 fruits But that is not what you asked for. Let's give an example. If I have 5 apples. You can't take me one orange from the 5 apples. And if you give me one orange, it won't make 6 apples, or 6 oranges, it will make 5 apples and one orange. But you can divide apples by oranges. For example if I give you 1 apple for each 10 oranges that you bring, we have a 0,1 apples/oranges ratio. Let's say you bring me 50 oranges, that means I will give you 50 oranges x 0,1 apples/oranges =5 apples

Is called dimensional analisis. And your example is flawed in a sense as you cannot add meters and seconds for example But you can multiply them and divide them. If you divide 4m by 2s you get 2m/s the two dimensions of these magnitudes merge and gain a new meaning. Longitude divided by time gives you speed You could also teoreticaly also divide apples by oranges. If you somehow could merge an apple by an orange to create a new thing which would be a ratio of apples to oranges. I can divide 3 apple by 2 orange. I would get 1.5 apple/orange. Which is a ratio. I could divide 6 apples by 4 oranges and get the same 1.5 apple/orange So it carries meaning!

You can divide apples by oranges (for eg ratios)

It's important to note that you can't add meters and seconds.

You also can't add meters with seconds, thats the one thing your not allowed to do. Also you can multiply/divide apples with oranges: 3 red apples (a) and 5 yellow oranges (b); 3a*5b = 15ab 3a/5b = 0.6 a per 1 b.

You can. 7 Apples + 8 oranges = 15 snacks

You can do both but in most cases the former is a lot less useful than the latter.

I often think about units as a variable. 3x+5y cannot be simplified, and the same would hold for 3 Apples + 5. However 3x / 5y can be written as ⅗ x/y. In the same way we can take 3m / 5s and get ⅗ m/s. Sometimes we can simplify the unit part further by using other units (like kg * m / s² is the same as N)

Just change the units. 3 fruits + 2 fruits = 5 fruits. Doesn't matter what variety.

If you have 20 apples and 4 oranges, you have 5 apples per orange. If you go 20 miles in 4 hours, you are going 5 miles per hour. We could compare any two units in this way, however in general we cannot add two units of different categories

You can't add apples and oranges, but you also can't add meters and seconds. You can multiply or divide meters and seconds, but you can also multiply or divide apples and oranges. If I have 10 apples and 5 oranges then I have > 10 apples / 5 oranges = 2 apples per orange. That makes perfect sense.

You can. You can add two oranges to the two apples in your shopping cart. They just won't become 4 apple-oranges because they belong to two different dimensional systems. Likewise, you can divide them... you can trade two apples to two oranges and end up with a 1:1 apple/orange ratio. And you could also say 1 banana + 1 apple to 2 orange ratio. And you can combine lots of different unit systems together when you have extensive versus intensive properties. Extensive = property applies to a small amount and large amount equally. Intensive = property is dependent on an amount.

But you can actually add apples to oranges? As long as if you draw a clear relationship between the mathematical structure you use (Natural numbers…) and the physical phenomena, you can work with the properties of the structure. You are responsible of choosing the math structure you use, and there’s nothing wrong as long as if you are consistent with the rules of the structure and that all properties of the structure are physically meaningful.

You can take the union of a set of two apples and a set of three oranges

As a science teacher I'd say it's all about units. Meters divided by seconds=m/s You could add apples and oranges you'd just have to give it a different unit Apples+oranges=fruit

you can divide or multiply apples to oranges. I could say I have 1 apple per 1 orange. Adding is like having one more of the thing you're looking at, multiplying is like combining the things you're looking at. For example I could have a bus full of 25 people, or I could divide those 25 people up among 5 buses, 25/5, so that there are 5 people per bus. You can also say the opposite that I have 5 buses each with 5 people and if I combined it into one bus there would be 25 people. Adding a bus in this situation would add to the total number of buses but not the total number of people, likewise adding a person doesn't add to the number of buses. Essentially the answer to your question is because multiplication and division combines or distributes quantities, and additions or subtraction increases or reduces quantities.

Actually, you *can* add apples to oranges. They’re two linearly independent elements of the fruit space is the real vector space of all fruit. The fruit algebra, meanwhile, is the tensor algebra of the fruit space. In the fruit algebra, the product of 2 apples and 3 oranges is the elementary tensor of 6 apple-oranges. An apple-orange is an object which possesses the units of both apples *and* oranges.

From my perspective, it is about whether the units you get would make any sense. To add apples and oranges, you have to redefine both as "fruits", the result will tell how many fruits you have, BUT will no longer tell how many of each fruit. The same logic will allow you to add bananas and watermelons, and if you are a botanist instead of a chef, tomatoes and eggplants. Meters per second is a different MEANINGFUL unit than either meters or seconds. Meters are a measure of distance, and seconds a measure of time. Since distance per time is how we define speed, you are now measuring speed instead of distance or time. If you do vectors instead of scalars, you will get displacement instead of distance, and velocity instead of speed (i.e. the amount of movement will be directional, and the rate of travel can now become negative without being nonsensical). NOT all ratios are useful either. While grams per cubic centimeter would be the useful ratio of mass/volume=density, grams per centimeter would not be as useful. Without knowing the width or depth of the substance, we could say that so many centimeters length of it will weigh so many grams, but we could not identify the substance, tell how much space 50 grams would require, or whether it will float in water. You can relate any two quantities as a ratio, but that is not a compelling reason to do so. We could come up with a ratio like ice cream trucks per violent crime (using statistics for number of both in a given city), but that would not mean that selling ice cream from trucks somehow causes crime, nor vice versa.

You could add apples to oranges, but, much like when you divide meters by seconds you end up with a new unit. If I have 5 apples and 7 oranges I have 12 apples and oranges. Whether or not you can perform an operation with units mostly has to do with whether or not there is a conversion between the units (i.e. if the 2 units are feet and meters) and how useful the new unit would be (5 apples * 7 oranges would be 35 apples*oranges but that's a mostly meaningless statement).

5 apples + 7 oranges = 12 apples and oranges 8 apples + 4 oranges = 12 apples and oranges => 5 apples + 7 oranges = 8 apples + 4 oranges => 1 apples = 1 oranges Not useful :(

Just because you lose information (like the exact composition of apples and oranges) doesn't mean it's not useful. If you know you have 5 apples and 7 oranges and I ask how many apples and oranges you can say 12, because you can add them together. If you know we have 12 apples and oranges and I ask how many apples we have, you can't answer cause you're missing information. When you work with different units you have to be extra careful that your steps make sense, but that doesn't mean that it can never be done.

5 apples + 7 oranges is more likly to be 12 apples xor oranges. 12 apples and oranges makes me think of 12 oranges and 12 apples

The simplest way I can think of it is this: if s and m were variables instead of units you'd be able to factorize them like: 3 m / 2 s = (1.5) * (m/s). So you get a sort of 'meaning conservation law' where you can always 'factor out' the meaning for any combination of meters and seconds that you divide (or multiply for that matter). You can't do the same thing with addition: there would't be a unit for general sums of meters and seconds. What I mean is that 1s + 2m would have different 'units' than eg 1s + 3m. There's no way we can express, say, (x s + y m) = z (m+s) for example. So while you can do these linear combinations if you want, you'll hardly be able to compare different combinations to each other. Each combination would live in 'its own universe'. (That's not quite true, but if you plot each of these combinations as coordinates in a 2D plane if (x axis m, y axis s eg), each line through the origin would be 'its own universe'.) Because of the factorization, all divisions/multiplications 'live' in the same universe.

Sure you can add apples to oranges to create a new unit. Just like velocity = x \[meter/ second\], I will create a new unit called darth = x \[apple + orange\]. Now you can derive some operations on this new unit we created (what happens when we add 2 darths, how about divide 2 darths, muliply 2 darths, etc).

You can't add units together to make a new unit. That's not how units work. Imagine creating a new unit called the (km + m) which combines the kilometre and the meter. This would not be consistent at all. 2km + 5m = what? .... 7 (km + m)? So does that mean that 5km + 2m is also 7 (km +m)? You can't add units like this.

Ah yes I see, thanks! This makes sense.

This is a physics question, not a math question

It’s spelled metres in the civilised world. Meters are instruments for measurement. Like water meters.

that's rather imperial way of looking things. I don't really think an empire that thrived on the suffering of billions is a civilized one.

It’s spelled civilised in the civilised world. Civilized is a barbarism.