How one should teach using brackets in summation?
For example, why is it correct to write $\sum_i a_ib_i$ but $\sum_i a_i+b_i$ should be written as $\sum_i (a_i+b_i)$ But $\sum_i\frac{a_i+b_i}{2}$ is, in turn, correct case not to use brackets?
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$\begingroup$ I would teach it the other way around: conceptually, there are brackets everywhere; but we have conventions that tell the reader how to add missing brackets in their head when brackets are missing. For instance, a+bc means a+(bc) by convention. So if you want to write a+(bc), you can afford to omit the parentheses in writing, and the reader will put the parentheses correctly in their head. If you want to write (a+b)c, the convention won't help you, so you can't afford to omit the parentheses. The $\Sigma$ symbol works no differently. By convention, juxtaposition takes precedence over + symbol. $\endgroup$– StefFeb 4 at 14:35
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$\begingroup$ Aeryk's answer is incorrect. See this. Juxtaposition is conventionally treated differently from explicit multiplication symbols. $\endgroup$– user21820Mar 10 at 6:41
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3 Answers
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I would say that in terms of order of operations, the summation symbol is between multiplication/division and addition/subtraction. So when you write $\sum_i a_i+b_i$, the implication is to do $\sum_i a_i$ first, and so you need parentheses. However in $\sum_i a_ib_i$ or $\sum_i \frac{a_i+b_i}{2}$ the multiplication and division by 2 come before the summation.
Note though that $\sum_i a_i+b_i$ implicitly means $\sum_i (a_i+b_i)$ because the dummy index on the $b_i$ doesn't make sense outside the context of the sum. The only real confusion comes in when you have something like $\sum_i a_i+3$.
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11$\begingroup$ A good answer, +1, but I think an important aspect needs to be emphasised: The purpose of adding brackets is to avoid ambiguity, and from that understanding springs all the uses of brackets, IMO. I feel, In teaching we too often concentrate on the 'how' and 'what', but leave out the 'why'; understanding why something is the way it is, equips the student to think for themself, which is what we hope to achieve. $\endgroup$ Feb 3 at 9:12
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2$\begingroup$ I would very strongly argue that $\sum_i a_i + b_i$ means "someone made a mistake". In my experience it's more likely that the mistake is "there are two distinct variables with the same name $i$" rather than "some parentheses were forgotten". For instance, maybe there was already a variable called $i$, and someone erroneously wrote $\sum_i a_i + b_i$ when they really meant $\sum_j a_j + b_i$, but not $\sum_i (a_i + b_i)$. The notation is wrong and you have to guess what the mistake was; there isn't one correct implicit meaning. $\endgroup$– StefFeb 4 at 14:27
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1$\begingroup$ My reading of $\sum_i a_i + b_i$ lies between @Stef's and Aeryk's: I'd argue that it implicitly actually means $\sum_p a_p + b_i,$ where $p$ is dummy (bound by that summation) and $i$ may or may not appear elsewhere in the context, however in practice I believe that its intended meaning is probably $\sum_i (a_i+b_i).$ $\endgroup$– ryangFeb 7 at 22:37
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Teacher: What do you want to add ?
Student: $a_i+b_i$
Teacher: Put it between brackets.
Student: $(a_i+b_i)$
Teacher: Sum it!
Student: $\sum_i (a_i + b_i)$
Same for the fraction:
Teacher: What do you want to add all together?
Student: $\frac{a_i+b_i}{2}$
Teacher: Put it between brackets.
Student: $(\frac{a_i+b_i}{2})$
Teacher: Sum it!
Student: $\sum_i (\frac{a_i+b_i}{2})$
=> this provides students with a fixed procedure, which is very clear, and does not generate any misunderstandings.
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7$\begingroup$ But you don't really need the brackets for the second one, because putting the sum above the division line already groups it together. Few people would add them since they're redundant. $\endgroup$– BarmarFeb 3 at 15:11
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3$\begingroup$ @Barmar That's actually a good way to think about it. Always start by adding the brackets (at least in your head), then remove the brackets that are redundant and keep the ones that are not redundant. $\endgroup$– StefFeb 4 at 14:45
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There are actually two questions that you're asking: how to teach "the summation symbol" and the followup question that asks for concrete explanations for examples.
I think the followup questions are non-questions if the summation symbol is carefully introduced and defined, I'd do something like
- $\sum$ is a shorthand for a (probably infinite) summation.
- Following the order of operations, the terms that are summed up stretch to the next (in terms of order of execution) addition or subtraction step in the term, if any.
- It then becomes clear that $\sum ab$ is a sum of products and that $\sum a + b$ is a sum of many $a$s and one $b$.
- To sum multiple sums, you have to use parentheses: $\sum (a+b)$ is a sum of sums.
