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If you add two magnitudes with same units you get an answer which is in the same units, but adding different units "does not make sense". But this is not true with multiplication, because you can divide meters by seconds to get meters per each second passed.

But what makes this true? What are the rules we have to follow to manipulate magnitudes with units? This question implicitly asks why cannot be units be added.

Addendum: I didnt know if this is a physics or math question, as it has aspects from both branches.

Here is all the info I found about this, and it almost tells nothing.

Garmekain
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  • You might be interested in "dimensional analysis": https://en.wikipedia.org/wiki/Dimensional_analysis – awkward Nov 17 '17 at 17:43
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    See https://math.stackexchange.com/questions/792291/why-cant-you-add-apples-and-oranges-but-you-can-multiply-and-divide-them -- do the answers to that question help you? If not, can you be more specific about what your concerns are? – David K Nov 18 '17 at 02:32

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If you cut a piece of tape as a measure of the width of your computer monitor, then to tell some one else remote from you this width you either have to physically send them the piece of tape or you have to compare the length of the piece of tape with a standard of length that you can both agree on e.g. the metre. Once you have both agreed on a common standard you can communicate a dimensioned number to tell them the screen width. The act of measurement is basically a ratio comparison of two quantities of the same type e.g. two lengths. If one of them is a standard length or some known proportion of a standard length, then the measured/calculated ratio has the dimension of that standard of length.

For convenience you will normally use a tape pre-marked out in standard units so the width of the monitor can be measured simply and directly, without thinking deeply about the underlying measurement process.

If you calculate the ratio of two lengths measured using the same standard then this ratio becomes dimensionless.

If you calculate the ratio of two lengths measured using different standards then this ratio becomes a conversion factor between the two different standards.

There are some oddities. e.g. energy and torque have the same units in the metric system, i.e. Nm, but they are physically different types of measured quantity.

James Arathoon
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