What really are units? Why can we ignore them, like in class?

[chingel]

The operations are all the same if you write out all the units or not, as long as it is consistent, ie you don't use meters one time and millimeters another time, then the numbers would come out different.

For example, you measure the length of a lap and the time to go around it. You get 100 m and 20 s. You want to find the average velocity, you divide 100m by 20s and you get 5 m/s. But you can also not write out the units and just keep in mind you had meters and seconds, and you would still get 5. The answer or the operations do not change if you keep in mind that the units are actually there and don't write them out. I also don't understand the problem, it would be very helpful if you would pick out a specific equation and ask specifically, how in this part here and here we can not write out the units and what you think would happen differently, if we wrote them out. It should be clear that no math operation somehow changes its nature if you don't write the units out and just keep them in mind.

The units are like a regular variable that multiplies something. If everywhere all the variables are multiplied by something that always go along with them, it doesn't change anything fundamentally. Like in the previous example, instead of 20 we have 20s everywhere, 20 is just multiplied by s and instead of dividing by 20 we divide by 20s, the s goes along with the 20 everywhere and it doesn't change the nature of operations.

 

[gufiguer]

Look on page 2

http://www.ime.usp.br/~tausk/texts/MathPhysics.pdf

 

[ShlomoBenAmar]

haruspex:

I don't have any good examples at the moment. What you said about me confusing "disregarding units with not writing them out" is I think quite accurate. A great part of my question asks why it's valid not to write them out. Like, why does it work? Why can we treat (in reference to your specific example) any variable, like q2, as a number, and make use of the familiar rules that exist for numbers? This is also the case for all physical (dimensionful) equations, like for instance, the ideal gas law $pV=nRT$. I can treat any variable in that equation as a number to produce, e.g., $p=\frac{nRT}{V}$ among other things. Do you have any ideas as to why this is valid apart from "units behave just like numbers"?

Not that I consider that reasoning to be incomplete, it's just that I already know about that and would like to see another point of view...

Thanks.

 

[haruspex]

Why can we treat (in reference to your specific example) any variable, like q2, as a number, with the familiar rules that exist for numbers?

This is a new question, yes? This is not about units, but about why algebraic operations, which we all first learned as operations on numbers, apply also to physical entities that have dimension.
It might help to have a more mundane example. I share 10 oranges equally among five students. What does each student get? Answer, 2 oranges. Your question becomes, why does the arithmetical rule 10/2 = 5 turn into (10 oranges)/5 = 2 oranges, or maybe (10 oranges)/(5 persons) = 2 oranges/person?
A bit too philosophical for me.

 

[ShlomoBenAmar]

haruspex:

Yes, I'm troubled as to why these things I learned early on by applying them on numbers, also work with dimensional quantities. Here "dimensional quantity" implies that "oranges" is some kind of mathematical object that is also manipulated around in the equation.

There are lengthy manipulations in physics (or whatever else, really) which carry out lots of algebraic steps. It bothers me that I can't really explain why, after all that lengthy procedure, both sides of the equation have the same units, and not only that, but the equation was also numerically (that is, referring to only the numbers involved, not the units) a valid manipulation.

I've already been told that "units behave just like numbers" so it's as if I were just manipulating x's and y's all over the place. I don't particularly like this POV though.

Any opinions?

I've been told to "let things which do not matter slide". This thing is seriously not letting me proceed with my regular studies. Do you think it could be a good thing to stop worrying about it and just proceed with math in a mechanical way (even though I'd hate to do such a thing)?

Thanks.

 

[haruspex]

these things I learned early on by applying them on numbers, also work with dimensional quantities

Yes, but they don't work always. You can multiply a speed by a time and get a distance, but is it the distance you wanted?
When we write a physical law, the claim is that if you match up the entities in the equation to the right entities in the real world then the equation, interpreted as arithmetic, will produce (near enough) the right answer. So your question becomes why do certain real world entities have relationships that correspond to mathematical operations.
This is referred to as 'the unreasonable effectiveness of mathematics'. But I feel it's not such a mystery.
First, we never can know how accurately the real world corresponds to the maths. Newtonian mechanics looked pretty good for a while.
Secondly, we defined numbers and standard operations on them as we have because they have such use. Group Theory covers all sorts of real world things that don't behave much like numbers.
Thirdly, the Anthropic Principle: maths works at all in the real world because the real world has a degree of predictability. In a universe without that, intelligent beings would not evolve, since they'd have no advantage.

 

[Drakkith]

Do you think it could be a good thing to stop worrying about it and just proceed with math in a mechanical way (even though I'd hate to do such a thing)?

Yes. While trying to fully grasp the underlying concepts of something can be good, if it's stopping you from learning the math you need to succeed in the real world then I'd put your question on the back burner for now. You can always come back to it later.

 

[Svein]

Here "dimensional quantity" implies that "oranges" is some kind of mathematical object that is also manipulated around in the equation.

No, Physical objects. Mathematics does not care. It just solves the problem you give it,whether it makes any sense or not.