Consider the following fraction:
$$\frac{\frac{\frac{a}{b}}{c}}{a}$$
How to explain that:
$$\frac{\frac{\frac{a}{b}}{c}}{a} \ne \frac{\frac{a^2}{b}}{c}$$
Obviously, you get different results. But a-priori, one might think that $a:b:c:d$ is associative. Is there any good way proving/explaining the difference, other then showing the different results?
Thanks.
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$\begingroup$That example is a bit more complex that you need to show that associativity fails. You only need one counterexample: $$ 2 \div (1\div 2) = 4 \ne 1 = (2\div 1)\div 2 $$
More intuitively (?), division is not associative, because if you look at one of the operands to a nest of divisions, the result will vary either in proportion to it or inversely to it depending on whether it's to the right of an odd or even number of divisions. And the basic associative law changes the number of operations the rightmost operand is to the right of by 1.
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