When are square and curved brackets interchangeable?

Is it ever acceptable to interchange square and curved brackets? E.g. are the following both acceptable (and identical)? $$x = t(a + [b + c])$$ $$x = t(a+(b+c))$$

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They are absolutely identical. The addition of square brackets in algebra, or I have even seen $()$, $[\,]$, and $\{ \}$ in a deeper expression, is solely for purposes of enhanced clarity. You may interchange them at will as they have no other meaning than additional clarification, all things being in context.

Note that I do not like seeing $\{ \}$ in texts as a third parentheses style, but I have seen it in published text.

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There's no mathematical meaning to square brackets used just for grouping; they are simply to make a visual distinction between the groups, particularly when there's a lot of nesting.

On the other hand, there is this rather awful notation $[x]$ for the "greatest integer" function now given in modern notation by $\lfloor x\rfloor$. Don't use parentheses for this!

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Yes, they are both acceptable and identical (Unless there is some weird use explicitly expressed by the author).

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By commutative and distributive properties, the first one ends up becoming $x=(t*a + t*b + t*c)$ and so does the second one. Brackets and parentheses generally exist to organize terms together.

I believe that {} is less specific than brackets [ ] and those in turn are less specific than parentheses ( ). Meaning that in an expression that is complex you would go $x*${$t[a+(b+c)]$}. This is how it is often used. That does not mean that you must necessarily write it a certain way.

So by convention many people would prefer to write it the second way rather than the first way, as well as in the way $x=t[a+(b+c)]$

Either way, both ways you have written are acceptable and interchangeable since they do not mathematically change the answer.

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