Why is it Euler's 'Totient' Function?












10












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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' ?






number-theory terminology math-history







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  • 2




    $begingroup$
    Here's an answer by Robert Israel.
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    – Mason
    Dec 13 '18 at 15:38






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    The History of Science and Mathematics Stack Exchange seems like a better place for this question.
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    – Blue
    Dec 13 '18 at 16:49






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    $begingroup$
    This has an answer at english.stackexchange.com/questions/23694/…
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    – Tesseract
    Dec 28 '18 at 22:18
















10












$begingroup$


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' ?






number-theory terminology math-history







share|cite|improve this question











$endgroup$








  • 2




    $begingroup$
    Here's an answer by Robert Israel.
    $endgroup$
    – Mason
    Dec 13 '18 at 15:38






  • 2




    $begingroup$
    The History of Science and Mathematics Stack Exchange seems like a better place for this question.
    $endgroup$
    – Blue
    Dec 13 '18 at 16:49






  • 2




    $begingroup$
    This has an answer at english.stackexchange.com/questions/23694/…
    $endgroup$
    – Tesseract
    Dec 28 '18 at 22:18














10












10








10


0



$begingroup$


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' ?






number-theory terminology math-history







share|cite|improve this question











$endgroup$




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' ?






number-theory terminology math-history




number-theory terminology math-history






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edited Jan 3 at 22:34









Pierre-Guy Plamondon

8,90511739




8,90511739










asked Dec 13 '18 at 15:31









HaranHaran

1,139424




1,139424








  • 2




    $begingroup$
    Here's an answer by Robert Israel.
    $endgroup$
    – Mason
    Dec 13 '18 at 15:38






  • 2




    $begingroup$
    The History of Science and Mathematics Stack Exchange seems like a better place for this question.
    $endgroup$
    – Blue
    Dec 13 '18 at 16:49






  • 2




    $begingroup$
    This has an answer at english.stackexchange.com/questions/23694/…
    $endgroup$
    – Tesseract
    Dec 28 '18 at 22:18














  • 2




    $begingroup$
    Here's an answer by Robert Israel.
    $endgroup$
    – Mason
    Dec 13 '18 at 15:38






  • 2




    $begingroup$
    The History of Science and Mathematics Stack Exchange seems like a better place for this question.
    $endgroup$
    – Blue
    Dec 13 '18 at 16:49






  • 2




    $begingroup$
    This has an answer at english.stackexchange.com/questions/23694/…
    $endgroup$
    – Tesseract
    Dec 28 '18 at 22:18








2




2




$begingroup$
Here's an answer by Robert Israel.
$endgroup$
– Mason
Dec 13 '18 at 15:38




$begingroup$
Here's an answer by Robert Israel.
$endgroup$
– Mason
Dec 13 '18 at 15:38




2




2




$begingroup$
The History of Science and Mathematics Stack Exchange seems like a better place for this question.
$endgroup$
– Blue
Dec 13 '18 at 16:49




$begingroup$
The History of Science and Mathematics Stack Exchange seems like a better place for this question.
$endgroup$
– Blue
Dec 13 '18 at 16:49




2




2




$begingroup$
This has an answer at english.stackexchange.com/questions/23694/…
$endgroup$
– Tesseract
Dec 28 '18 at 22:18




$begingroup$
This has an answer at english.stackexchange.com/questions/23694/…
$endgroup$
– Tesseract
Dec 28 '18 at 22:18










5 Answers
5






active

oldest

votes


















2





+50







$begingroup$

The Latin tot is correct enough 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 demonstrate how common these q-t correlatives are.)






share|cite|improve this answer











$endgroup$





















    3












    $begingroup$

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






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      I'm guessing that totient = tot + ient where tot refers to what you said and ient, to the suffix such as quotient. Where is the source for this answer?
      $endgroup$
      – Haran
      Dec 13 '18 at 15:51



















    3












    $begingroup$

    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-






    share|cite|improve this answer









    $endgroup$













    • $begingroup$
      (+1) from me. However, is there any connection with relatively prime numbers and his etymology?
      $endgroup$
      – Haran
      Dec 31 '18 at 10:25










    • $begingroup$
      Not as far as I can tell. The name "totient" doesn't seem to refer specifically to relative prime numbers. It is not something like "coprime totient function".
      $endgroup$
      – Ferred
      Dec 31 '18 at 11:04



















    2












    $begingroup$

    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






    share|cite|improve this answer









    $endgroup$





















      0












      $begingroup$

      I want to give a reference of this. I have searched the internet after seeing your question and I felt that I can't add anything new with this answer. So instead of copying, I can give you this reference; I hope this will help you.






      share|cite|improve this answer









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        5 Answers
        5






        active

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        5 Answers
        5






        active

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        active

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        active

        oldest

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        2





        +50







        $begingroup$

        The Latin tot is correct enough 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 demonstrate how common these q-t correlatives are.)






        share|cite|improve this answer











        $endgroup$


















          2





          +50







          $begingroup$

          The Latin tot is correct enough 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 demonstrate how common these q-t correlatives are.)






          share|cite|improve this answer











          $endgroup$
















            2





            +50







            2





            +50



            2




            +50



            $begingroup$

            The Latin tot is correct enough 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 demonstrate how common these q-t correlatives are.)






            share|cite|improve this answer











            $endgroup$



            The Latin tot is correct enough 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 demonstrate how common these q-t correlatives are.)







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Jan 3 at 20:19

























            answered Jan 3 at 19:18









            Danny MDanny M

            863




            863























                3












                $begingroup$

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






                share|cite|improve this answer









                $endgroup$













                • $begingroup$
                  I'm guessing that totient = tot + ient where tot refers to what you said and ient, to the suffix such as quotient. Where is the source for this answer?
                  $endgroup$
                  – Haran
                  Dec 13 '18 at 15:51
















                3












                $begingroup$

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






                share|cite|improve this answer









                $endgroup$













                • $begingroup$
                  I'm guessing that totient = tot + ient where tot refers to what you said and ient, to the suffix such as quotient. Where is the source for this answer?
                  $endgroup$
                  – Haran
                  Dec 13 '18 at 15:51














                3












                3








                3





                $begingroup$

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






                share|cite|improve this answer









                $endgroup$



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







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 13 '18 at 15:38









                David G. StorkDavid G. Stork

                12k41735




                12k41735












                • $begingroup$
                  I'm guessing that totient = tot + ient where tot refers to what you said and ient, to the suffix such as quotient. Where is the source for this answer?
                  $endgroup$
                  – Haran
                  Dec 13 '18 at 15:51


















                • $begingroup$
                  I'm guessing that totient = tot + ient where tot refers to what you said and ient, to the suffix such as quotient. Where is the source for this answer?
                  $endgroup$
                  – Haran
                  Dec 13 '18 at 15:51
















                $begingroup$
                I'm guessing that totient = tot + ient where tot refers to what you said and ient, to the suffix such as quotient. Where is the source for this answer?
                $endgroup$
                – Haran
                Dec 13 '18 at 15:51




                $begingroup$
                I'm guessing that totient = tot + ient where tot refers to what you said and ient, to the suffix such as quotient. Where is the source for this answer?
                $endgroup$
                – Haran
                Dec 13 '18 at 15:51











                3












                $begingroup$

                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-






                share|cite|improve this answer









                $endgroup$













                • $begingroup$
                  (+1) from me. However, is there any connection with relatively prime numbers and his etymology?
                  $endgroup$
                  – Haran
                  Dec 31 '18 at 10:25










                • $begingroup$
                  Not as far as I can tell. The name "totient" doesn't seem to refer specifically to relative prime numbers. It is not something like "coprime totient function".
                  $endgroup$
                  – Ferred
                  Dec 31 '18 at 11:04
















                3












                $begingroup$

                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-






                share|cite|improve this answer









                $endgroup$













                • $begingroup$
                  (+1) from me. However, is there any connection with relatively prime numbers and his etymology?
                  $endgroup$
                  – Haran
                  Dec 31 '18 at 10:25










                • $begingroup$
                  Not as far as I can tell. The name "totient" doesn't seem to refer specifically to relative prime numbers. It is not something like "coprime totient function".
                  $endgroup$
                  – Ferred
                  Dec 31 '18 at 11:04














                3












                3








                3





                $begingroup$

                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-






                share|cite|improve this answer









                $endgroup$



                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-







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 31 '18 at 10:22









                FerredFerred

                763




                763












                • $begingroup$
                  (+1) from me. However, is there any connection with relatively prime numbers and his etymology?
                  $endgroup$
                  – Haran
                  Dec 31 '18 at 10:25










                • $begingroup$
                  Not as far as I can tell. The name "totient" doesn't seem to refer specifically to relative prime numbers. It is not something like "coprime totient function".
                  $endgroup$
                  – Ferred
                  Dec 31 '18 at 11:04


















                • $begingroup$
                  (+1) from me. However, is there any connection with relatively prime numbers and his etymology?
                  $endgroup$
                  – Haran
                  Dec 31 '18 at 10:25










                • $begingroup$
                  Not as far as I can tell. The name "totient" doesn't seem to refer specifically to relative prime numbers. It is not something like "coprime totient function".
                  $endgroup$
                  – Ferred
                  Dec 31 '18 at 11:04
















                $begingroup$
                (+1) from me. However, is there any connection with relatively prime numbers and his etymology?
                $endgroup$
                – Haran
                Dec 31 '18 at 10:25




                $begingroup$
                (+1) from me. However, is there any connection with relatively prime numbers and his etymology?
                $endgroup$
                – Haran
                Dec 31 '18 at 10:25












                $begingroup$
                Not as far as I can tell. The name "totient" doesn't seem to refer specifically to relative prime numbers. It is not something like "coprime totient function".
                $endgroup$
                – Ferred
                Dec 31 '18 at 11:04




                $begingroup$
                Not as far as I can tell. The name "totient" doesn't seem to refer specifically to relative prime numbers. It is not something like "coprime totient function".
                $endgroup$
                – Ferred
                Dec 31 '18 at 11:04











                2












                $begingroup$

                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






                share|cite|improve this answer









                $endgroup$


















                  2












                  $begingroup$

                  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






                  share|cite|improve this answer









                  $endgroup$
















                    2












                    2








                    2





                    $begingroup$

                    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






                    share|cite|improve this answer









                    $endgroup$



                    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







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered Jan 2 at 1:04









                    williamowilliamo

                    1025




                    1025























                        0












                        $begingroup$

                        I want to give a reference of this. I have searched the internet after seeing your question and I felt that I can't add anything new with this answer. So instead of copying, I can give you this reference; I hope this will help you.






                        share|cite|improve this answer









                        $endgroup$


















                          0












                          $begingroup$

                          I want to give a reference of this. I have searched the internet after seeing your question and I felt that I can't add anything new with this answer. So instead of copying, I can give you this reference; I hope this will help you.






                          share|cite|improve this answer









                          $endgroup$
















                            0












                            0








                            0





                            $begingroup$

                            I want to give a reference of this. I have searched the internet after seeing your question and I felt that I can't add anything new with this answer. So instead of copying, I can give you this reference; I hope this will help you.






                            share|cite|improve this answer









                            $endgroup$



                            I want to give a reference of this. I have searched the internet after seeing your question and I felt that I can't add anything new with this answer. So instead of copying, I can give you this reference; I hope this will help you.







                            share|cite|improve this answer












                            share|cite|improve this answer



                            share|cite|improve this answer










                            answered Jan 4 at 3:19









                            GimgimGimgim

                            31713




                            31713






























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