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

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Discussion Overview

The discussion revolves around how to express the dimensions of two quantities, A and B, when they are different. Participants explore the mathematical operations applicable to these dimensions, particularly focusing on the validity of addition versus multiplication in this context.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants assert that multiplication of different dimensions is acceptable, while addition is not, prompting a request for clarification on what "together" means.
  • One participant questions the rationale behind the rule that allows multiplication but not addition, suggesting that division or subtraction could also be considered.
  • Another participant illustrates that adding incompatible units, such as pounds and miles, results in nonsensical quantities, while multiplying them yields a meaningful unit of energy or torque.
  • A further argument is made that multiplication creates a new dimension, such as velocity from distance and time, while addition does not produce a useful real-life correspondence.
  • There is a mathematical perspective presented that equates division with multiplication by the inverse and addition with subtraction, arguing that addition is not permissible due to the lack of a common domain for different dimensions.

Areas of Agreement / Disagreement

Participants express differing views on the operations applicable to different dimensions, with no consensus reached on the validity of addition versus multiplication or the implications of these operations.

Contextual Notes

Participants highlight the limitations of combining dimensions, noting that while multiplication can create new dimensions, addition lacks a meaningful context when dimensions differ.

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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