# Why choose symbol $\sum$ to be the summation operator in maths?

AFAIK there's rare evidence to show that $\sum$ has the meaning of "sum" in Greek, so why mathematician chose $\sum$ to be the summation operator?

• The comment at Math.SE answers your question. – Wojowu May 8 '16 at 8:42
• I guess that "sum" starts with s and ∑ is sort of an s. – Billy Rubina May 8 '16 at 15:32

See Forian Cajori, A History of Mathematical Notations (1928, also Dover reprint), Vol.II, page 61:

438. The sign $$\Sigma$$ for summation is due to Leonhard Euler, Institutiones calculi differentialis (St.Petersburg, 1755), Cap.I,§26, page 27, who says:

summam indicabimus signo $$\Sigma$$.

This symbol was used by Lagrange but otherwise received little attention during the eighteenth century. [...] The $$\Sigma$$ to express "sum" occurs in 1829 in Fourier's Theory of Heat, published in 1822, and in C.G.J. Jacobi's Fundamenta nova theoriae functionum ellipticarum (1829).

Note We have to take into account that the latin sum was already the source for Leibniz's symbol $$\int$$ for integral:

Utile erit scribi $$\int$$ pro omnia, ut $$\int l = \text {omn.} l$$, id est summa ipsorum $$l$$.

The exact mathematical Greek word for "sum" is written "Αθροισμα". But the "everyday" word for "sum" is written "Σύνολο".

The interesting thing is that, as a mathematical term, in Greek the word "Σύνολο" is used to mean "set".

• In Classical Greek σύνολος means "all together" and is not used to mean a mathematical "sum". This latter is modern Greek. – fdb May 10 '16 at 22:47
• @fdb ...which is exactly what I have written in my post. – Alecos Papadopoulos May 11 '16 at 0:45

This is a typical example of mixing Greek with Latin in terminology and notation. Greek letter which is equivalent to S stands for a Latin word summa which begins with S. Such things are very common. Some purists object this kind of mixture for aesthetic and logical reasons, but this kind of confusion is so common that it is impossible to "correct" it. The reason is that scientists who introduce notation usually do not know much of Greek. (And modern ones also of Latin). I don't know whether Euler had a working knowledge of Greek.

Here is another example: from the Greek noun "cone", one can make an adjective "conic" using the Greek suffix -ic. In Latin the same role is played by the suffix -al. So the modern English adjective "conical" has two suffixes from different languages which mean exactly the same. From the point of a linguist this is a nonsense. But scientists do not mind.

• I think your explanation is right, except for the knowledge part. Basically, using the Greek or Roman alphabets never had much to do with Greek or Latin themselves, and using Greek to create symbol, and Latin as a language for science, does not denote a "confusion". – VicAche May 8 '16 at 13:23
• As I said some purists consider this a confusion: mixing symbols or terms from two languages. – Alexandre Eremenko May 8 '16 at 17:19
• This is not purism. The latin alphabet derives from a way to spell Greek ( en.wikipedia.org/wiki/Archaic_Greek_alphabets#Euboean ), defending such views (notably by using the word "confusion") and calling for a correction seems out of place here. – VicAche May 8 '16 at 21:16
• "From the point of a linguist this is a nonsense" would require a reference and it'll be very hard for you to find one because the general opinion among linguists should be precisely that transcription is a core aspect of modern (as in in modern use) alphabets. – VicAche May 8 '16 at 21:18