Reason why multiplication gives fraction of a number

[selfAdjoint]

OK forget about the assuming axioms represent reality, think of numbers instead of quantities. And the last sentence was how can you divide into 1/2 equal pieces.

Well "divide into equal pieces" is only one model of numerical division. Another way to model a/b is to imagine a ruler and measure a length "a" units along it and ask how many lengths "b" units long fit into it. So if b = 1/2 you are thinking of lengths a half a unit long. How many times does half a meter fit into four meters?

 

[Sagredo]

I think the source of the puzzlement here is that mtanti is not posing a problem purely about numbers, but about the multiplication of a physical unit by a number. The problem is, why does (to quote his earlier example) 2 apples times 1/2 equal 1/2 of 2 apples or 1 apple? This is not a purely arithmetical expression, but one involving a physical unit, in this case, of apples. So the problem is, why does 2 apples times 1/2 equal 1 apple?

And in fact sometimes 1/2 of 2 apples is not 1 apple! It could in fact be 2 half apples.

So if one thinks in terms of pieces, or units, or magnitudes, or quantities which are not arbitrarily divisible, then indeed it is not always necessary that a quantity times a fraction equals the same fraction of that quantity (when we understand the last expression purely mathematically). If one wishes to see it this way, then one must forget about the original units in the answer.