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

  1. Jan 18, 2015 #1
    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.
     
    Last edited: Jan 18, 2015
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  3. Jan 18, 2015 #2

    Vanadium 50

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    What does 6 meters + 4 volts equal?
     
  4. Jan 18, 2015 #3
    Yeah, it doesn't make sense to add those. But I don't think that's my question. Thanks.
     
  5. Jan 18, 2015 #4

    phinds

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    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.
     
  6. Jan 18, 2015 #5

    SteamKing

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    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.
     
  7. Jan 18, 2015 #6
    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 wont 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?
     
  8. Jan 18, 2015 #7
    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.
     
  9. Jan 18, 2015 #8

    SteamKing

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    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.
     
  10. Jan 18, 2015 #9
    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 any one 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.
     
  11. Jan 18, 2015 #10

    SteamKing

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    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.
     
  12. Jan 19, 2015 #11
    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.
     
  13. Jan 20, 2015 #12

    Svein

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    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
     
  14. Jan 20, 2015 #13

    Stephen Tashi

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    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.)
     
    Last edited: Jan 20, 2015
  15. Jan 20, 2015 #14
    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!
     
  16. Jan 20, 2015 #15
    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).
     
  17. Jan 20, 2015 #16

    Stephen Tashi

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  18. Jan 20, 2015 #17
    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.
     
  19. Jan 20, 2015 #18

    symbolipoint

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    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.
     
  20. Jan 21, 2015 #19

    Stephen Tashi

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    Did it cover the Buckingham Pi theorem? http://en.wikipedia.org/wiki/Buckingham_π_theorem
     
  21. Jan 21, 2015 #20

    symbolipoint

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    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.
     
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