Definitions are a part of the mathematical discourse. There are axioms, theorems, undefined terms, but also definitions. The purpose of definitions is to make the discourse manageable and to highlight some mathematical constructs that happen repeatedly and proved useful. The phrase "by definition" refers to a previously given definition.I googled my site for "by definition". The returned list contained 118 entries. Let's consider a couple.
At the beginning of the page there is a definition: ... let there be a circle ω with center O and radius R. Let A be an arbitrary point, except O, whose inverse image in ω is A'. OA·OA' = R². A' is collinear with O and A. Let a be the line through A' perpendicular to OA' (or OA, which is the same.) Then a is called the polar of A, while A is the pole of a, w.r.t. to ω.
Why the difinition was given? It'so happens that the pole/polar correspondence pops up in many geometric constructs and has frequently recurring and otheriwse useful properties. It is more convenient to refer to a polar of point A wrt to ω than to "line through A' perpendicular to OA' where OA'·OA = R², R being the radius of circle ω."
And what is the inverse image? There is a link to another definition: A and A' are inverse images of each other wrt to circle ω if the line AA' passes through the center O of ω and OA·OA' = R². Since it is a frequent construct it was found and agreed upon to give it a name, viz., "inverse images" which is indeed more convenient than writing explicitly "A and A' lie on a line through the center O of circle ω such that OA·OA' = R²."
So later on the page, when proving property #3, I say "Let A' be the inverse image of point A lying outside the circle of inversion. A' then lies inside the circle and, by definition, OA·OA' = R², meaning that there is a definition of the "inverse image" which implies the claimed ("by definition") identity OA·OA' = R².
Printer-friendly copy
Email this topic to a friend
