Dimensions of A and B are different, how to express?

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The discussion centers on the mathematical principles governing the combination of different dimensions, specifically addressing why multiplication is permissible while addition is not. It is established that adding quantities with different dimensions, such as pounds and miles, yields nonsensical results, whereas multiplying them produces a coherent unit, such as pound-miles, which can represent energy or torque. The conversation emphasizes the importance of understanding units as constants of proportionality that facilitate meaningful comparisons between measurements across different scales.

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Indranil
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If the dimension of A and B are different, then how to express the dimension of A and B together? how to write?
 
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Simple rule: multiplications are o.k., additions are not. Can you give an example what you mean, and especially what "together" means?
 
fresh_42 said:
Simple rule: multiplications are o.k., additions are not. Can you give an example what you mean, and especially what "together" means?
Could you explain why multiplications are ok but additions are not? It could be division like A/B or could be A-B. I am confused. Please get it clear.
 
Indranil said:
Could you explain why multiplications are ok but additions are not? It could be division like A/B or could be A-B. I am confused. Please get it clear.
If you add a measured number of pounds (force) to a measured number of miles, you get garbage. If you change one unit or the other, the result will change. But by no fixed proportion.

If you multiply a measured number of pounds (force) by a measured number miles, you get a quantity with units of pound(force)-miles. This is a unit of energy (or of torque). If you change one unit or the other, the resulting product will change in proportion to the ratio of the new and old units.

Units can be understood as a constant of proportionality that allow you to relate measurements made using one scale to measurements made using another.
 
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Indranil said:
Could you explain why multiplications are ok but additions are not? It could be division like A/B or could be A-B. I am confused. Please get it clear.
##A/B = A \cdot B^{-1}## and ##A-B= A + (-B)##, so from a mathematical point of view, there is no difference between multiplication and division, resp. addition and subtraction. Addition is obviously not allowed, because there is no common domain where it would make sense to add, e.g. length to time. By multiplication we define a new domain of the multiplied dimension, e.g. distance per time results in velocity which is a new dimension. One could probably formally construct domains with length plus time, but this has no useful real life correspondence. It will always remain a pair (length ; time) whereas length / time consists of all possible velocities.
 
Thank you very much, sir.
 

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