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I would interpret $\sum_{i=1}^2 x_i + y$ as $x_1 + x_2 + y$, but I would interpret $\sum_{i=1}^2 x_i + y_i$ as $x_1 + y_1 + x_2 + y_2$. I realize this is a little inconsistent. Should the latter be written as $\sum_{i=1}^2 (x_i + y_i)$?

Or, in other words, does the sum operator have precedence over + and - ?

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1 Answer 1

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It should be written with parentheses to avoid ambiguity, yes. If you think about the "sum" symbol as a function, it makes sense:

$$ \sum(\cdot) $$

This is a function which takes a list $\{x_1,x_2,\ldots\}$ of numbers (or other mathematical objects which you might want to add), and adds them in order. This list could be finite or infinite: the "sum" function figures out how long the list is, and adjusts its indexing accordingly (i.e. if there are 10 things in the list, your index will go from $1$ to $10$).

If you want to add a sequence which is itself the addition of two sequences, like your example of $\{x_1+y_1,x_2+y_2\}$, you'll need to drop the whole sequence into the function:

$$ \sum(\{x_1+y_1,x_2+y_2\})=\sum_{i=1}^2(x_i+y_i) $$

For finite sums of numbers, we always have the property that

$$ \sum(\{x_1,x_2,\ldots\}+\{y_1,y_2,\ldots\})=\sum(\{x_1,x_2,\ldots\})+\sum(\{y_1,y_2,\ldots\}) $$ however the following would have a different interpretation:

$$ \sum(\{x_1,x_2,\ldots\})+\{y_1,y_2,\ldots\} $$

Hence the parentheses!

Interesting, yet more advanced side note: "breaking up" a sum doesn't always work if the sequences are infinite!

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    $\begingroup$ Thanks! Does this mean that if I wrote $\sum_{i=1}^2 x_i + y_i$, you would read it as $x_1 + x_2 + y_i$? Or would you have understood that I meant $x_1 + y_1 + x_2 + y_2$? Or would you just have thought "Geez, what sloppy notation!" :) $\endgroup$
    – Nagel
    Commented Apr 16, 2013 at 15:41
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    $\begingroup$ Most people would understand what is meant, but yes we would also think it was a bit sloppy =) I see $\int f+g$ all the time as well, similar confusion. $\endgroup$
    – icurays1
    Commented Apr 16, 2013 at 16:14
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    $\begingroup$ Correct, but sometimes people are lazy and don't put the 'dx'. $\endgroup$
    – icurays1
    Commented Apr 16, 2013 at 17:06
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    $\begingroup$ However, $\Sigma^2_{i=1} x_i + y$ would be $x_1 + x_2 + 2y$. If that is your intention, it would be preferable to write $\Sigma^2_{i=1} (x_i + y)$. Otherwise, one would write $y + \Sigma^2_{i=1} x_i$, which would be unambiguously taken to mean $x_1 + x_2 + y$ . $\endgroup$ Commented Apr 16, 2013 at 17:19
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    $\begingroup$ Agreed. So $\sum_{i=1}^2 x_i + y$ would be considered somewhat ambiguous, or would most people take it to mean $y+\sum_{i=1}^2 x_i$ without hesitation? $\endgroup$
    – Nagel
    Commented Apr 16, 2013 at 18:09

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