Hard dividing an irrational number, which Diophantus once showed (3)
I believe the answer is:
phi
'which diophantus once showed' is the definition.
(Greek letter which appears in the name Diophantus)
'hard dividing an irrational number' is the wordplay.
'hard' becomes 'h' (abbreviation used in pencil classifications).
'dividing' is an insertion indicator.
'an irrational number' becomes 'pi' (pi is a kind of irrational number).
'h' placed inside 'pi' is 'PHI'.
(Other definitions for phi that I've seen before include "Greek letter, spoken as f" , "Foreign character" , "Greek character" , "Greek letter; sounds like expression of outrage" , "Greek letter; sounds like expression of disgust" .)