Why is it Euler's 'Totient' Function?

Question

The function $\phi(n)$ calculates the number of positive integers $k \leqslant n \space , \gcd(k,n)=1$. It was found by mathematician Leonhard Euler. In 1879, mathematician J.J.Sylvester coined the term 'totient' function. What is the meaning of the word 'totient' in the context? Why was the name coined for the function?

I have received replies that 'tot' refers to 'that many, so many' in Latin. What about the suffix 'ient'? It can be seen in words such as 'quotient' etc. Finally isn't there any reference to 'relatively prime numbers' ?

Answer 1

The Latin tot is correct as an origin for the root, but the suffix '-iens' doesn't originate with Sylvester either who was undoubtedly thinking of the already fully-formed word totiens when he coined 'totient.' Compare this to how quotiens enters into English as 'quotient.'

Sylvester knew Latin well enough that he would have been aware of the parallel between totiens and quotiens, which is actually a very manifest parallel since they function together as correlative conjunctions. A clause will introduce quotiens - how often; the next clause will answer totiens - this often.

ex: quotiens doces, totiens disce. 'Learn as often as you teach.' (literally, 'as often as you teach, learn this often.')

Correlative conjunctions like this are common in Latin. Here's another you'll recognize:

quantum - how much, tantum - this much.

Anyway, it seems to me that the word totient is meant to refer to the abstract notion of saying 'here is how many there are.' It doesn't seem to reference the quality of being relatively prime or any other quality.

(But speaking of 'qualities,' there's also qualis - what kind, talis - this kind, which hopefully goes to show how common these q-t correlatives are.)

Answer 2

It comes from the Latin tot--"that many, so many" (as in "total").

Answer 3

Going with a similar word: quotient

quotiens (how many times) = quot (how many) + tiens (times)

If totient has a similar origin, than it would mean "that many times" or "all the times". It probably refers to "all the numbers" coprime with $n$.

In latin "totus" means "all" or "whole" - see under the IE root teuta-

Answer 4

The word totient is made from two latin words,” tot”, meaning so much, as many, or more archaically from the word “totum”, meaning whole or total (ref 1), and a suffix “ient” meaning, more or less, the process by which the desired total is obtained. That process is defined by the ф (n) equation you mentioned, and that is where the relatively prime instructions reside within the name Totient.
Ref 1 The Oxford Dictionary of Word Histories. 2002, pages 516-517 Oxford University Press

Answer 5

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