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

• ShlomoBenAmar
In summary, the conversation discusses the use of units in mathematical and physical equations. The speaker questions why units are often disregarded in equations with dimensional homogeneity, and wonders about their place in formal mathematics. However, it is explained that units are important for accurately representing physical quantities and that they are designed to obey algebra. While units may not be necessary in mathematical proofs, they are crucial for real-world applications.
ShlomoBenAmar
All my life the approach has been as follows:

In math class I learn the rules and almost always deal with purely numerical problems.

In physics class I apply the things learned in math class but this time our quantities have units. Now, once the equation is well put and has dimensional homogeneity the problem is magically reduced to a purely numerical problem like the ones dealt with in math class. You might say that units do not actually vanish and it's actually just for practical reasons that teachers choose not to write them down, but I would like to see an explanation as to why this is OK? (Ignoring the units).

For instance, in my mathematical physics class I learn about vector calculus and vector analysis. First I learn it abstractly in a mathematically rigorous manner mostly dealing with no numbers because we derive the equations and theorems via the manipulation of letters. Now, I'm supposed to apply the exact same theorems and the same reasoning for problems that are dimensionful and this tears me apart for some unknown reason.

Right now I'm studying EM waves and I'm just solving Maxwell's Eqs to see certain things that must follow for an EM wave. All of this is done with letters like ##\epsilon_o## and standard arithmetical rules. Never do we worry about units. Man, I don't even know what the units for ##\epsilon_o## are. Nobody seems to care! Of course they are important, though… But this isn't my point.

I need a profound reason or something that could prove to me that units shouldn't bother me at all. I kind of see dimensionless and dimensional math as somehow separate, so I need a good argument that tells me this shouldn't be my way of seeing things and that whatever applies to dimensionless equations applies to dimensionful ones.

I don't know if my question even makes sense, so thank you if you've read this far.

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PFuser1232
What does 6 meters + 4 volts equal?

symbolipoint
Yeah, it doesn't make sense to add those. But I don't think that's my question. Thanks.

ShlomoBenAmar said:
Yeah, it doesn't make sense to add those. But I don't think that's my question. Thanks.
Vanadium's point is that if you get in the habit of working equations in physics without the dimensions, there WILL come a time when you apply an element incorrectly or use the wrong equation or whatever and since you have no units, your equations will not reflect reality and neither will your answer but you will have no way of knowing that because your answer will just be a number, not reflective of a physical quantity.

ShlomoBenAmar said:
I need a profound reason or something that could prove to me that units shouldn't bother me at all. I kind of see dimensionless and dimensional math as somehow separate, so I need a good argument that tells me this shouldn't be my way of seeing things and that whatever applies to dimensionless equations applies to dimensionful ones.

That's just it, and what I think Vanadium 50 was trying to illustrate. Units should bother you. Units weren't just made up to annoy physics students or whoever. Units are what we use to classify certain physical attributes. Describing everything using numbers alone leaves out certain useful information and leads to ambiguous results.

I know units should bother me but there's something characteristic about equations with dimensional homogeneity that allows teachers and students alike to completely disregard them. And in the end, no contradiction is produced when disregarding them in dimensionally homogeneous equations.

I remember perfectly people shouting to the teacher "Prof! What were the units again?". Operating inside an equation with dimensional homogeneity is exactly like handling a purely numerical problem. Somehow your trapped inside a little universe of logic that won't pop out contradictions. Are units really just like numbers and thus they obey the philosophy of "having the same on both sides"?

I know the intention that wants us to have units. That is, to have meaningful quantities. But I want to know what their place in formal mathematics is. What are they? Why do they obey algebra?

I bet a large portion of physics students can't tell you immediately the SI units for some really important constants in physics. Thats because we seem to only care about mathematical soundness after there is dimensional homogeneity. But maybe this is irrelevant to my main question. Thanks.

It's not that units are ignored in equations which have dimensional consistency (rather than homogeneity), it's just that once the units on both sides of the equation agree, the numbers can take care of themselves, i.e., after this point, it's just doing the arithmetic.

Units obey algebra because they are designed to do so. There's nothing mystical about that.

In terms of the place of units in formal mathematics, whatever that is, units are units, and numbers are numbers. It's like asking what is the place of dollars and cents, for example, in formal mathematics.

ShlomoBenAmar
I think what you are running into is a Theorem does NOT need units, it applies to all measurement systems. However when you apply the theorem - applying REAL data, then you should ( always) use units, and keep them in the formula - when you are done the units should match the type of variable you expect.

For example

F=m*a -- Force = Mass * Acceleration.

Technically you can take ANY unit of mass(drams) and ANY units of acceleration( say furlongs/ fortnight^2) (sorry this was a standard format we would turn in in college when the question was ambiguously defined(;-) --- technically this is valid, yet the resultant force of (d*f)/F^2 -- is not helpful to anyone because it is not a standard unit.

So in the real world - a real application... we use standard units... in short 1 N = 1 kg⋅m/s^2... Newton, kg, meters, seconds... same math, with standard units.

Soooo... if you are doing a proof... a mathematical derivation this is typically dimensionless, but to solve a problem you have units.

ShlomoBenAmar said:
I bet a large portion of physics students can't tell you immediately the SI units for some really important constants in physics. Thats because we seem to only care about mathematical soundness after there is dimensional homogeneity. But maybe this is irrelevant to my main question. Thanks.
Well, that's because they are careless or poor students. If you look at the problems submitted in the Intro Physics HW forums on this site, you will see that all too common, many posters omit units, or they don't understand why units are necessary (even students who have been taught only SI), etc. And it shows when they obtain a ridiculous result from their calculations, and they are unable to understand immediately that something has gone wrong.

For example you can think that every variable in a physical equation has a numeric part times its unit. You can think of the unit as a regular variable, which multiplies the number. Now, if at the beginning the units on both sides of the equation were the same, then all the operations you usually do with equations will preserve the equality of the units. If you divide both sides of the equation by some variable, both sides get divided by the same unit. If you gather some terms on one side, the units will still fit because you move a term with the same units to the other side. If you try to isolate the electric field E from an equation, you are only multiplying, dividing, factorizing, gathering terms etc and these all will preserve the units. Even differentiation and integration preserve the equality of units, so does the quadratic formula for example.

So everything you usually do with equations preserves the equality of units. Using this, you can check if you made a mistake. Because if the units don't match, you did something wrong.

ShlomoBenAmar
Why do we need units in physics? Because all our data is based on measuring or counting. And without units the measurements are useless. Without a clue to what you are measuring, how do you know to measure it? If you think about voltage and length as without units, what instrument would you use to measure a voltage? A measuring tape?

An example where the unit really tell you how to measure:

"One recommended British unit of thermal conductivity - useful for calculating the heat transmission through walls - is: BThU/hour/sq ft/cm/°F "
- from Random Walks in Science

symbolipoint
ShlomoBenAmar said:
But I want to know what their place in formal mathematics is. What are they? Why do they obey algebra?

The study of manipulations with units is called "dimensional analysis". You can find axiomatic treatments of the topic. ( I don't find the axiomatics of the subject mathematically exciting. It makes such strong assumptions that little is left to be proven.)

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symbolipoint and ShlomoBenAmar
Stephen: Hey, are you absolutely sure about that axiomatic approach? A quick survey has yielded me nothing good. Could you maybe tell me what to look for? Thanks!

Technically you can take ANY unit of mass(drams) and ANY units of acceleration( say furlongs/ fortnight^2) (sorry this was a standard format we would turn in in college when the question was ambiguously defined(;-) --- technically this is valid, yet the resultant force of (d*f)/F^2 -- is not helpful to anyone because it is not a standard unit.

For those interested, Google gives the conversion 1 furlong/fortnight^2 = 1.375 * 10^-10 m/s^2 (so I highly doubt we'll be adopting this system of measurements).

ShlomoBenAmar
ShlomoBenAmar said:
Stephen: Hey, are you absolutely sure about that axiomatic approach? A quick survey has yielded me nothing good. Could you maybe tell me what to look for? Thanks!

Any decent applied mathematics book should start out with a chapter or two about dimensional analysis and scaling. Try Logan or Holmes, both good books. And thanks to Stephen for that link.

Dimensional Analysis took no more than about 15 minutes of instruction time during the course of Physics 1 and the importance was very clear with very little explanation.

symbolipoint said:
Dimensional Analysis took no more than about 15 minutes of instruction time during the course of Physics 1 and the importance was very clear with very little explanation.

Did it cover the Buckingham Pi theorem? http://en.wikipedia.org/wiki/Buckingham_π_theorem

Stephen Tashi said:
Did it cover the Buckingham Pi theorem? http://en.wikipedia.org/wiki/Buckingham_π_theorem
The instruction was just from classroom instruction, not requiring more than maybe 20 to 25 minutes. The instruction was not from a book. We were expected to account for all dimensions of MASS, TIME, LENGTH. The focus was not on the units, but just on the physical dimensions.Your question: NO.
That is why our instruction took no more than about 20 minutes.

Dimensions can help out in real life too. Example: If a sprinter can do the 100m in 10s, he obviously has an average speed of 100m/10s =10m/s. But how does this speed compare to say the speed of a car (where the speedometer reads in miles/hour)?

Dimensional calculus: 1mile = 1609m. 1 hour = 60 minutes. 1 minute = 60 seconds. Thus 1 hour = 60*60s =3600s. From there on it is simple arithmetic:
$$1m =\frac{1mile}{1609}, 1s =\frac{1hour}{3600} \therefore \frac{10m}{s}=\frac{\frac{10miles}{1609}}{\frac{1hour}{3600}} =\frac{10\cdot3600}{1609}\frac{miles}{hour}=22.4\frac{miles}{hour}$$

I actually started out wondering if you were simply a troll but I'm guessing not.

The answer is that, without units, the math is simply academic. Now, you may choose not to show units as you solve a problem you've set out on, but that's only because, if I might be frank, you are lazy; this makes you prone to errors.

Your instructor is wrong in leaving out the units.

Vanadium asks what 6 meters plus 4 Volts comes to. To a European (it's across the water to the East of you) the answer would be six different approximations to 4 Volts, unless you could afford quite excellent meters ! How can anyone live without differentiating between measuring devices and units of measurement Oh, I see; a 'measuring unit' ! seriously the answer is, of course: "probably, but not on a Saturday."

ShlomoBenAmar said:
In physics class I apply the things learned in math class but this time our quantities have units. Now, once the equation is well put and has dimensional homogeneity the problem is magically reduced to a purely numerical problem like the ones dealt with in math class. You might say that units do not actually vanish and it's actually just for practical reasons that teachers choose not to write them down, but I would like to see an explanation as to why this is OK? (Ignoring the units).

For instance, in my mathematical physics class I learn about vector calculus and vector analysis. First I learn it abstractly in a mathematically rigorous manner mostly dealing with no numbers because we derive the equations and theorems via the manipulation of letters. Now, I'm supposed to apply the exact same theorems and the same reasoning for problems that are dimensionful and this tears me apart for some unknown reason.

Right now I'm studying EM waves and I'm just solving Maxwell's Eqs to see certain things that must follow for an EM wave. All of this is done with letters like ##\epsilon_o## and standard arithmetical rules. Never do we worry about units. Man, I don't even know what the units for ##\epsilon_o## are. Nobody seems to care! Of course they are important, though… But this isn't my point.

I need a profound reason or something that could prove to me that units shouldn't bother me at all. I kind of see dimensionless and dimensional math as somehow separate, so I need a good argument that tells me this shouldn't be my way of seeing things and that whatever applies to dimensionless equations applies to dimensionful ones.
I'm not sure that any of the responses so far have fully apprehended the question. It seems not to be about dimensionality. But I'm not sure what it is about because it appears to switch between arithmetic and algebra.
In a given expression there may be a mix of numeric values, and symbols representing values. In each case, the values may be dimensionless or have some dimension.
As thoroughly commented elsewhere in this thread, dimensions must be consistent. They do not need to be written out expressly unless performing dimensional analysis. (Indeed, I'm not aware of that ever being done.) Units are another matter.
Any number that has dimension should be accompanied by the units that go with it. Without the units, the numeric value is uninterpretable.
Symbolic values are usually considered to represent the given entity regardless of units. Thus, g represents 9.8m/s2 (say) or 32.2 f/s2; they're the same thing. It is not necessary (indeed, wrong) to write g m/s2. This explains your ##\epsilon_o## example. One could choose to define h to be the numeric value of g when it is expressed in m/s2, so one would write h m/s2, but that would be most unusual.
There is no need for units to be consistent through the equation provided they are all specified. Thus if a car goes 120 miles at 60 kph then the time taken is 120 miles/60 kph = 2 miles/kph = 2 hour-miles/km = 2h(1.6) = 3.2h.

ShlomoBenAmar
I of course know what the purpose of units are. Yeah, 2V and 5apples don't make 7 volt-apples. That's not my question. I'm bothered as to what are units in mathematics? Terry Tao has written a blog post on this, but it's well beyond me and I don't really get it. (See: https://terrytao.wordpress.com/2012/12/29/a-mathematical-formalisation-of-dimensional-analysis/ )

In a more indirect way, I want an argument that proves that if you have dimensional consistency and all that, then you could completely disregard units (Something which is done! Even the Boas mathematical method for physicists book does this) and then do valid mathematical operations and your answer will come out as sensible. In fact, if you do the unit analysis afterwards you will even find out that this dimensional consistency has been preserved!

Some people say, "well, units are just like numbers; if you trust algebra then no contradiction will be found". I guess that's ok, but in my opinion is leaving too much to some kind of faith in the mathematics. I like my mathematics to be clear in every step, I don't have any special commitment for algebra or anything. Plus, if they're just like numbers then how do you make sense of drawing some vector in a space by considering only its component numbers? For example, take a position vector. How do you know how to draw it if you don't know what [m] is? in fact, it could be [-1] and you've drawn the wrong picture. You might then say "well, we draw it them by convention that way, because it makes sense!" well then units are not like numbers at all!

Maybe you consider this is trivial, but I want to know what units are as mathematical objects. I KNOW about dimensional analysis and all that stuff. This isn't my question. In fact, Terry Tao says this is an interesting question because dimensional analysis can be useful to pure mathematicians studying, perhaps, PDE's.

If you think my question is stupid or useless, tell my why so, so that I can continue studying other things as this it taking much of my time.

edit: As another example of where units are disregarded remember your differential equations classes. You thought about them numerically, always. Everybody in class did. Nobody was breaking their necks about correct units. Suddenly you learn that the solutions to these equations are the solutions to dimensional ones too. Where's the connection? How does it come about? Somebody here told me that units were designed to work. I think that begs the question.

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ShlomoBenAmar said:
I of course know what the purpose of units are. Yeah, 2V and 5apples don't make 7 volt-apples.
Again, a confusion between dimensions and units. You can't add 2V and 5 apples because the dimensions are different. You can add 7V and 5mV because the dimensions are the same even though the units differ.
ShlomoBenAmar said:
if you have dimensional consistency and all that, then you could completely disregard units
I see you liked my post, but don't appear to have understood it.
If in the equation there are numbers (as numbers) that have dimension you cannot disregard their units.
Where a symbol represents an entity with dimension you can ignore units because the symbol does not represent a number - it represents an entity of that dimension. G represents the universal gravitation constant. You can choose to to express its value in various different units, with correspondingly different numbers, but G itself does not 'have' units, only dimension.

If you feel this still does not answer your question, please post specific examples of what bothers you.

I think I disagree with you. You can disregard the units, do consistent operations, and the solution to the equation will come out as correct. Do you write out all the units when solving differential equations? Do you write the units for every single thing you do with mathematics? I definitely don't many smart people I've come across in my life don't. In fact, I've never seen anyone do it, to be honest. It's such a hassle.

In my previous post I've made my questions really explicit.

You're welcome to criticize my way of thinking right now, maybe it could be beneficial ..
Thanks.

ShlomoBenAmar said:
You can disregard the units, do consistent operations, and the solution to the equation will come out as correct. Do you write out all the units when solving differential equations?
I feel you are confusing disregard of units with not writing them out. Here's an example I just happen to have seen on another thread:
F = 9*109 q1 q2/r2.
That is not strictly correct. It should be F = 9*109 q1 q2/r2 Nm2/C2.
If you know you will be plugging in the two charges and their separation (as pure numbers) in the corresponding units, and you intend to interpret the resulting number as a force in Ns, then of course it will all work out. This is not disregarding the units since you knew what was really meant, you just didn't spell it out.
But it does not make the expression correct. In the thread in question, the charges were plugged in as a number of microCoulombs. That would have been ok if the units had been included in the original equation, and the values plugged in along with their units.
ShlomoBenAmar said:
In my previous post I've made my questions really explicit.
Sorry, but it's still not crystal clear to me what your issue is. Please humour me by supplying a specific example.

ShlomoBenAmar
Think about what units represent. For example, one Newton. The units for one Newton are 1kg × 1m/s^2. What does this mean? A force of 1N is the same thing as saying 1kg of mass accelerating at a constant rate of 1 metre per second, every second. Units can help us visualize what exactly the numbers we're finding mean so they aren't just arbitrary numbers found.

ShlomoBenAmar said:
I think I disagree with you. You can disregard the units, do consistent operations, and the solution to the equation will come out as correct.
That is because you are doing mathematics, not physics. Mathematics are for solving abstract problems. Physics is for translating real-world problems into mathematical problems and translating the answer back into real-world answers. The translations are much easier if you keep track of the units.

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.

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.

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ShlomoBenAmar said:
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.

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.

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