How come units are only meaningful if they are multiplied or divided, not added?

Question

Why is a meter/second more meaningful than a (meter + second) unit?

etc.

That doesn't answer my question, it just basically restates what I said in answer form. WHY?

But multiplication and division are essentially shorthand of addition, no?

Answer 1

A meter+second is impossible. Think about it physically, how would you add something that is 1 meter long to something that is 1 second long...you can't.

A meter/second however can be seen as a measure of speed. You move at a rate of meters per second.

Answer 2

I find this a very interesting question. Not even all unit combinations using multiplication and division have physical interpretations. For example, what would a meter times a second represent?

I was reading an old book called Dimensional Analysis (sorry--I forget the author) that claimed you could actually redefine units so everything was measured in just one unit! Certainly not a very useful concept and highly confusing, but it made the point that there's more to choice of units than first meets the eye.

One thing that struck me after reading that book was that newton-meters (forces in newtons times distance in meters) have two entirely different interpretations. One is energy in joules but the other is torque in newton-meters. These quantities seem to have the same units but they are not at all the same thing.

As far as constructing something like a meter+second unit, my instinct bristles at the idea, but right now I don't have a clear idea of exactly why. I'll devote some more thought to it...

Answer 3

The answer to "Why?" is inherent in the definition of addition. To add two things mathematically, they must have the same meaning in terms of what the numbers are describing. Adding 2 elephants to 2 crocodiles will get you 4 animals, but that no longer is meaningful in terms of mathematical addition, because the rigorous nature of mathematics is such that the answer in addition is defined as being equivalent in nature to both items being added. So you can add 2 (big) animals to 2 (scaley) animals and still get 4 animals. It is really just a question of the basic meaning of addition.

By the way, a similar discipline applies in multiplication and division. You can divide four elephants by two waterholes and you will get two elephants per waterhole. But you won't get two elephants period. - and you won't get two waterholes. You cannot adjust the meaning of the units or the operations without violating the discipline of mathematics.

ADDED: Sure - multiplication and division are shorthand - but the discipline of the basic meaning of operations and the use of units remains constant. That is very important - otherwise, people would have different interpretations for the same thing. Mathematics is like a language that has no words with two meanings - (Steve H points out an interesting exception below - but the math is still sound - its the physical application that is fuzzy). That's why mathematics works for all people in all cultures - everyone knows what you mean.

Answer 4

Er... you need to think about what a "metre + second" would MEAN. What would you measure with such a unit (after all... measurement is what units are all about, right?). If you had something that was "5 metres+seconds"... what type of physical quantity would it be?

The answer is nothing. There is not physical phenomenon which has dimensions of units ADDED together. You're trying to add entities with different dimensionalities - that can't be done. You can only add entities with the SAME dimensionalities.

The mathematical equivalent is when you have algebraic expressions with x^2 and x's and constants. You can only add together entities with the same powers of x.

The issue is so fundamental that it can't be explained any further.

Later: Multiplication is related to addition ONLY in the very simplistic "counting" sense... 5 x apples = apple+apple+apple+apple+apple. Fundamentally, multiplication of dimentional quantities results in a product quantity which is different from either of the two original quantities (eg. length x length = area). There is no way to ADD lengths to give an area - all you get is a bigger length. So forget it about it... addition of different units just doesn't make sense.

Answer 5

Building off of what Steve said.

Take momentum, we will build a unit for it using our mental idea of it. One thing we know is that momentum increases with mass. It also increases with speed. these are the only two things that affect momentum, so we change it into SI units like so: kg m/s, kilogram metres per second.

Looking at the units for electric charge, the ampere second (the coulomb). You can figure out by looking at this that electric charge varies proportionally to electric current as well as time. but that's all you know. Two things that have the same units simply vary proportionally to the same things.

I'm being very incoherent about all this, but the point is that a newton second measures something that varies proportionally to force and time. On the other hand, a newton plus a second would

I don't have the slightest idea, except that it is meaningless. This is a really good question.

Answer 6

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