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We all come across these two words in literature: summation, addition.

I personally do not know any difference between them and I view both of them the same in all mathematical aspects.

Is there any case where I cannot use the two words interchangeably? If not, then why do these two words evolve?

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  • $\begingroup$ You could ask the same question about the two words product and multiplication. $\endgroup$
    – KCd
    Nov 1 at 10:19
  • $\begingroup$ @KCd The answers to this question might give a clue! $\endgroup$
    – hanugm
    Nov 1 at 12:30
  • $\begingroup$ "Your salary was \$25000 per year and then they added \$5000 to it". Or "Bob robbed three banks: for \$30000, \$27500 and \$45000. Adding these numbers together the Judge put him in jail for 10.5 years". $\endgroup$
    – markvs
    Nov 1 at 22:29
  • $\begingroup$ This is the distinction between '$\sum$' and '$+$', and, directing attention to the comment by @KCd is similar to the distinction between '$\prod$' and '$\times$' (or '$\cdot$', implicit binary multiplication, etc.) $\endgroup$
    – Alexis
    Nov 30 at 17:56
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I do not know whether this distinction is universal, but to my knowledge addition primarily applies to the adding together of two entities, the augend and the addend, while summation applies to the adding of any number of operands, usually more than two, the summands.

All these words are derived from Latin. The verb addere means to attach, the verb augere means to enlarge. The -nd ending indicates the gerundium, so the augend is "that which is to be enlarged" and the addend is "that which is to be attached". Summation goes back to French somme and that in turn to Latin summa, total amount (often of money). This is turn is from Latin summus, highest point or top, which is also the root of the English word summit.

According to Klein's Comprehensive Etymological Dictionary of the English Language (Elsevier 1971), the Romans did not sum quantities top to bottom, with the sum total below all summands. Instead, they summed from the bottom up and wrote the sum total at the top.

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