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Like units can be added together or, subtracted from one another. However, multiplication and division of units does not have such boundations. multiplication is just repeated addition, similarly division is repeated subtraction. How in the world we don't have same conditions as that on addition of like units?

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  • $\begingroup$ I don't see division as "repeated subtraction". Can you explain that part? $\endgroup$ Commented Apr 30, 2019 at 14:49
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    $\begingroup$ I'm not quite sure what you mean? Could you maybe give some examples of what you're talking about. $\endgroup$ Commented Apr 30, 2019 at 14:50
  • $\begingroup$ You make the point that multiplication is repeated addition, and this is true if you are multiplying by a number, e.g. 3kg = 1kg + 1kg + 1kg. When you multiply by a quantity with a unit, however, you are doing something else, e.g. length times length gives units of length squared. $\endgroup$ Commented Apr 30, 2019 at 14:52
  • $\begingroup$ Possible duplicates: physics.stackexchange.com/q/337092/2451 and links therein. $\endgroup$ Commented Apr 30, 2019 at 14:57
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    $\begingroup$ Possible duplicate of What justifies dimensional analysis? $\endgroup$ Commented Apr 30, 2019 at 14:58

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If you mix (add) $5$ oranges and $2$ cars, you still get $5$ oranges and $2$ cars.

However, if you mix (add) $5$ oranges and $2$ oranges, you can compute the sum and we say that we get $7$ oranges.

The point is: to add and subtract, you need to have the same type of "things".

These examples use integer arithmetic, since it is a concept that we can visualise. However it would be easy to expand to continuous measures like:

Mixing (adding) $0.5 \ kg$ of sugar with $0.2 \ kg$ of sugar gives $0.7 \ kg$ of sugar; while mixing $0.5 \ kg$ of sugar with a ruler $0.2 \ m$ long gives the $0.5 \ kg$ of sugar and a ruler $0.2 \ m$ long.

With respect to multiplication, the multiplication can be thought as having $2$ boxes with $5$ oranges each, which results in:

$2$ box(es) $\times 5$ oranges/box = $10$ oranges. Note that oranges/box can be read as "oranges per box".

I tried to give a simple answer to your question. A more complex answer could lead us to dimensional analysis.

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  • $\begingroup$ To be honest you are making two different ad hoc examples to justify the claim: "multiplication can be thought of as.. no, multiplication is by definition repeated addition and that's it. $\endgroup$ Commented Apr 30, 2019 at 15:00
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    $\begingroup$ @gented multiplication is not by definition repeated addition. It might be for the integers, but for $R$ we abstract beyond that. IMO this is part of the OPs confusion. $\endgroup$ Commented Apr 30, 2019 at 15:02
  • $\begingroup$ @jacob1729 Well, yes and no. For rational numbers multiplication essentially follows the same lines as for integers, whereas for real numbers it can be defined based on its properties. This said, the original question is still valid even restricting oneself to the integers only and the answer above still fails to provide a reason. $\endgroup$ Commented Apr 30, 2019 at 15:09
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    $\begingroup$ @gented you can multiply dimensionful numbers by integers. Eg $7 \times 3m = 21m$. And that can be viewed as repeated addition, yes. But if you do $7s \times 3m$ there's no way of adding $3m$ to itself $7s$ times. So its the same case as in the reals - you can't inherit all the way from the integer definition and you need to give up and abstract. $\endgroup$ Commented Apr 30, 2019 at 15:11
  • $\begingroup$ @jacob1729 You're right, but the reason for this is hidden in the initial definition of "unit" and "number with a unit" - it has nothing to do with real numbers, rational numbers or integers. The answer to the original question is, as you in fact mention, hidden in the fact that dimensionful numbers don't necessarily posses the same ring properties as dimensionless numbers (by definition, somehow). $\endgroup$ Commented Apr 30, 2019 at 15:15

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