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Here's a quick question I can't find an answer to anywhere: is a sum of zero real numbers defined as equal to $0$, or is the quantity undefined?

$$\sum_{x\in\emptyset} x \stackrel{?}{=} 0$$

I can think of contexts in which either answer might make sense so my intuition says that this may be an issue of an ill-formed question, but I just wanted to see if there is a generally-accepted answer one way or another.

Next-Door Tech
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  • I think it’s common to define it as $0$, just as an empty product is defined as $1$. – Pranay Feb 08 '25 at 02:52
  • A sum with no terms is equal to $0$. $$\sum_{x\in\varnothing}x=\log\prod_{x\in\varnothing}e^x=\log1=0$$ – user14111 Feb 08 '25 at 02:55
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    "Here's a quick question I can't find an answer to anywhere" - There is an entire Wikipedia article on this topic: https://en.m.wikipedia.org/wiki/Empty_sum – Filippo Feb 08 '25 at 03:37
  • Can't find an answer anywhere? How about https://math.stackexchange.com/questions/2310758/empty-set-and-empty-sum or https://math.stackexchange.com/questions/3247615/how-to-find-p-2-here/3247630#3247630 or https://math.stackexchange.com/questions/2038317/vector-subspaces-of-zero-dimension/2038332#2038332 or https://math.stackexchange.com/questions/1017441/why-is-empty-product-defined-to-be-1/1017473#1017473 or https://math.stackexchange.com/questions/1877144/bounds-of-summation-what-happens-when-the-upper-bound-is-below-the-lower-bound/1877816#1877816 – Gerry Myerson Feb 08 '25 at 04:28
  • or https://math.stackexchange.com/questions/771133/sumn-i-n1-1-is-0-or-underfined-or-identity-element-for-the-operation/771137#771137 or https://math.stackexchange.com/questions/4130814/is-xn-yn-x-y-sum-limits-i-0n-1xn-1-iyi-for-integer-n-wh/4130819#4130819 or https://math.stackexchange.com/questions/2195922/interpretation-of-sums-using-cdots/2201664#2201664 or https://math.stackexchange.com/questions/4470269/product-of-n-terms-of-sequence-where-the-nth-term-is-of-the-form-xan/4470433#4470433 or about 50 other places on this very website? – Gerry Myerson Feb 08 '25 at 04:31
  • No need to be rude people... has anyone considered that perhaps I couldn't find the answer because I didn't know to call it the "empty sum" due to being unfamiliar with the topic? (Hence, the reason for asking the question...) Try searching anything along the lines of "is a sum with zero terms equal to zero" and you get a bunch of questions talking about the sum of all reals/integers equaling zero for some reason, nothing on this topic. – Next-Door Tech Feb 08 '25 at 08:33
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    I typed sum with no terms into Google, and the first link that came back was https://en.wikipedia.org/wiki/Empty_sum I apologize, if giving you nine places to find an answer to your question came across as rude. – Gerry Myerson Feb 08 '25 at 23:01

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It is and it must be defined as zero. Put $L = \sum_{x\in\emptyset}x$. To see this, let $X$ be some set of numbers. Since $X = X \cup \emptyset$ with the union being disjoint, we have $\sum_{x \in X}x = \sum_{x \in X}x + \sum_{x \in \emptyset}x$, forcing $L = 0$.

Gary
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温泽海
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