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Why multiplication produces new uniits but not addition or subtraction

  1. Jul 16, 2013 #1
    In physics new units are usually multiplication or division of some physical quantities. Why is there no unit which is a result of addiing or subtracting other units?

    I'm sensing this is a rather stupid question but still I'm going to ask anyway. Thank you!
  2. jcsd
  3. Jul 16, 2013 #2


    Staff: Mentor

    This isn't a stupid question but one steeped the lore of life experience and mathematics so any answer to your question will be more philosophical.

    Addition came out of the need to count things. Someone would be asked how many oranges are in the baskets and initially they would be counted one by one until a mathematician came along and invented a new operation called addition with tables and rules to do the counting faster. Counting implies a common unit of measure and hence addition requires it.

    Subtraction complements addition and so the same need for a common unit of measure.
  4. Jul 16, 2013 #3
    When you have e.g. 3m and 5m, what you really have there is 3*1m + 5*1m since the basic unit of length is actually 1m and not just m.
    That of course is equal to (3+5)*1m
    In algebra this is called "distributivity".
  5. Jul 16, 2013 #4
    I might be missing something but I really can't get anything out of your posts.

    Here is my doubt explained a bit more clearly: -
    Force x Distance is "something" called work. But Force + Distance is nothing. Ofourse this is because Force + Distance does not make sense. But across all physics there are no examples where addition or subtraction leads to a new unit. Only multiplication and division. Any reason?
  6. Jul 16, 2013 #5


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    Staff: Mentor

    Multiplication does NOT produce "new" units, but rather combines the units and maintains their relationship. Is, newton-meters means newtons times meters. But sometimes if those combinations of units are useful, they are shortened and given names.

    Similarly, addition of different units would invite invention of new units if it makes sense to do such addition: five "pears" plus four "apples" equals nine "pieces of fruit".
  7. Jul 16, 2013 #6
    Your statement clears my doubt too. So addition can also create new units? But we have no physical thing that does, right?
    Last edited: Jul 16, 2013
  8. Jul 16, 2013 #7
    you should atleast have knowledge of vector algebra or geometry!!!
  9. Jul 16, 2013 #8
    Dr. Zoidberg's response is about as concrete as it gets.
    adding same units makes sense.
    ##3m\cdot 2m=6(m\cdot m)=6m^2##
    multiplying same units makes sense and gives a new, combined unit of ##m^2##
    there is not algebraic way to combine ##m## and ##s## into one new unit
    ##3m\cdot 2s=6m\cdot s##
    This particular unit doesn't mean anything, but it is mathematically acceptable

    You could as why the rules of elementary algebra are the way they are, but the answer to that would be that the early mathematicians were trying to model the real world and this is the way the world works... so, in some ways there is no answer. "It wouldn't make sense" is the best, but that still draws on the fact that this is the way the universe works.
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