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$\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}$. If at first we divide only by b, and then their result by d, It is clear to me. I do not understand why if we divide directly by bd, it will give the same result? Is this a consequence of the axiom of associativity law (if I rewrite this as $a\cdot c\cdot b^{-1}\cdot d^{-1}$)?

I know how to operate with fractions, I'm interested in mathematical confirmation.

For example, I want to reduce the number 16 by 4 times, it will be 4. Then I want to reduce the number 4 by 4 times more. That is 16:4:4=1 or 16:16=1, they are identical.

It's the same with multiplication, if I take the number 1 and want to increase it 4 times more, I get 4. Then I increase the number 4 four times more, I get 16. That is 1*4*4=16 or 1*16=16, they are identical too.

I know that - no matter how much I try to calculate it, the result will be the same. But why does it work like this, it's just a fundamental property of multiplication - associativity? It's just a thing that doesn't have an explanation, it just works like that?

Sorry if this question is a duplicate, I didn't find the answer.

I am not an expert in mathematics, my level of knowledge is high school.

Thank for you answer.

Honza Prochazka
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    You may be interested in the formal construction of the rational numbers: https://en.wikipedia.org/wiki/Rational_number#Formal_construction. The product formula is really a consequence of this construction. This may be out of reach at the high school level - but you can keep it in mind as you continue to learn. – Fibonacci Cube K May 16 '20 at 03:49
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    Dupe of https://math.stackexchange.com/questions/1337036/understanding-the-multiplication-of-fractions Was wrongly gold-badge reopened by answerer – Bill Dubuque May 18 '20 at 15:21

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$\dfrac ac$ is the number $x$ such that $cx = a$. $\dfrac bd$ is the number $y$ such that $dy=b$. What is $xy$? Well, $(cd)(xy) = (cx)(dy) = ab$. So this is a consequence of associativity and commutativity of multiplication (you need both).

Matt Samuel
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