# Why can we multiply different units but not add them? [duplicate]

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?

• I don't see division as "repeated subtraction". Can you explain that part? – Ertxiem - reinstate Monica Apr 30 '19 at 14:49
• I'm not quite sure what you mean? Could you maybe give some examples of what you're talking about. – Ollie113 Apr 30 '19 at 14:50
• 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. – Ollie113 Apr 30 '19 at 14:52
• Possible duplicates: physics.stackexchange.com/q/337092/2451 and links therein. – Qmechanic Apr 30 '19 at 14:57
• Possible duplicate of What justifies dimensional analysis? – Kyle Kanos Apr 30 '19 at 14:58

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".

• @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. – jacob1729 Apr 30 '19 at 15:02
• @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. – jacob1729 Apr 30 '19 at 15:11