Example: 5 x 18 = (5 x 10) + (5 x 8) instead of 5 x 10 + 5 x 8?
What is the justification to teach the (redundant) use of parentheses in multiplications?
Early in their education (even well into learning algebra), students don't naturally see the structure within algebraic expressions. It takes a bit of mental overhead to see that 5 x 10 + 5 x 8 is the sum of two products.
Presenting the expression as (5 x 10) + (5 x 8) saves students the mental overhead, letting them focus on the message that's trying to be communicated.
I recently found myself adding unnecessary parentheses to first-order logical formulas in order to make them easier to read, and thus more effective at quickly communicating the idea I was trying to get across.
3$\begingroup$ That's a great analogy! Anyone who's been through a college-level math course immediately recognizes PEMDAS, so much so that we don't even remember the days when it wasn't as intuitive. But logical connectives trip lots of people up, even experienced mathematicians. Despite it being fairly standard, lots of people don't know this precedence ordering. $\endgroup$ Jan 28 at 16:17
7$\begingroup$ Reducing that mental overhead is also really helpful in any kind of computer programming involving math; saves you time while giving you more confidence that the code is doing what you think it's doing $\endgroup$– anjamaJan 28 at 17:39
$\begingroup$ @Silvio Mayolo: I also like the logical connectives comparison. In the little bit of logic I've worked with, I've found the conventions for understood parentheses like learning a 2nd language, a language totally unnecessary for most anyone NOT specializing in mathematical logic. Indeed, this isn't even what the vast majority of people in mathematical logic need to know all that well anyway. Despite this unimportance (except in certain formal meta-logic proofs), many textbooks seem to expect people to learn this language very well, (continued) $\endgroup$ Jan 28 at 19:08
$\begingroup$ or have machine-like reading ability for complicated logical expressions. In my own writing I've mostly ignored each approach, and freely use different bracketing symbols for better comprehension and short-cut terms (e.g. "(line 3)" rather than writing out what "line 3" is). For example, see the style used in this MSE answer. I think it would be better to spend less time testing the evaluation of "artificial interpretation problems" like $5 \times 10 + 5 \times 8$ (continued) $\endgroup$ Jan 28 at 19:08
2$\begingroup$ @anjama Especially when dealing with operators which aren’t handled uniformly across languages. For example exponentiation is left-associative in some languages (Python for example) and right-associative in others (Wolfram Alpha for example), so
4^3^2is ambiguous without parenthesis, and possibly misunderstood without a proper knowledge of the language in the absence of parenthesis. $\endgroup$ Jan 29 at 2:55
5 x 10 + 5 x 8then very first thing I need to do is think about order of precedence and mentally map that to
(5 x 10) + (5 x 8)anyway. So certainly presenting it as such up front saves me a step. What would be the justification for not doing it? You think that the extra step is worthwhile in itself? Or just that you personally find doing it so intuitive you think everyone else will too? $\endgroup$