Where does the word “totient” come from?

Question

In math we learn about the "totient function". It rhymes with "quotient" when math teachers pronounce it.

But I cannot find the definition or etymology of this word in any dictionary, nor on any web-site.

Anyone have a clue?

Answer 1

The word "totient" comes from Latin.

tot: "that many, so many"

From the University of Notre Dame Latin Word Lookup:

tot indecl. [so many].

totidem indecl. [just as many].

toties (-iens) [so often , so many times].

totus -a -um genit. totius , dat. toti; [whole, complete, entire; wholehearted, absorbed].

N. as subst. totum -i, [the whole]; 'ex toto, in toto', [on the whole].

While the suffix of iens apparently goes back to Sanskrit.

Euler created his totient function to answer the following question:

For a given positive integer n, how many smaller positive integers, relative to it, are prime?

Source: The Words of Mathematics, by Steven Schwartzman

Link to the page on Google books

Special Thanks to Mr.Disappointment for finding some of this info. We decided to combine it to make it more readable for the community

Answer 2

The totient function, a particular number theory function, was discovered by Euler, but he was not the one to give it that name.

The word 'totient' was introduced by Sylvester in "On Certain Ternary Cubic-Form Equations", Amer. J. Math 2 (1879) 280-285, 357-393, in Sylvester's Collected Mathematical Papers vol. III p. 321.

So for a neologism, the etymology is really what the coiner says it is, because they just made it up out of thin air. Surely Sylvester was thinking of the Latin root 'tot-' but as with most mathematical coinages, there is little that is meaningful in the metaphor that corresponds to 'relative primality'. Sylvester was famous for coining mathematical terms including discriminant and matrix.