Reason why multiplication gives fraction of a number

[omagdon7]

Division and multiplication and all things mathematical are created and defined by man. Trees are not concerned with 1/2 of their apples, trees in fact do not think. Cheetahs do not calculate a minimum velocity required to catch their prey. Rocks do not count. There is no other possible way for division to work and I am at as much of a loss as everyone else as to what sort of obfuscation is blinding you.

 

[mtanti]

Guess I am just trying to understand fractions and their relationship to operators. Does it work that way because it is defined that way that a/b means multiply by a and divide by b or is there more to it?

 

[selfAdjoint]

Guess I am just trying to understand fractions and their relationship to operators. Does it work that way because it is defined that way that a/b means multiply by a and divide by b or is there more to it?

There are a few surprises. For example what happens when b is itself a (proper) fraction? What happens when b = 1/2?

 

[mtanti]

Well mathematically you could multiply the numerator and denomenator by 2 so that it becomes 2a/1 which is then easier to understand. But in practise (assuming mathematical axioms represent reality) you are saying divide a quantity into 1/2 equal pieces and take a of them. So how can you take 1/2 equal pieces?

 

[mtanti]

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.

 

[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.