
Re: Is a rectangle a square?
Posted:
Oct 29, 2003 9:26 PM


On Wed, 29 Oct 2003, Mary Krimmel wrote:
> At 03:43 PM 10/28/03 0500, you wrote: > > > > Is a square a rectangle? When did the geniuses of the Math world come > > > up with that one? If you refer to Webster, a square is a > > > parallelogram with 4 EQUAL sides and 4 right angles. Wouldn't that > > > rule out a rectangle? Sorry, but I'm no math wiz, just wondering why > > > my daughter got that answer wrong on a math quiz. Would love this > > > explained in Laymons Terms. :) > > John Conway answered the question asked as the "subject" of the header. >
But now he's answering the "square a rectangle?" question.
Clearly one could define "rectangle" either so as to exclude "square" or so as to include it. This is one of many cases where one has a choice between such "exclusive" and "inclusive" definitions. The early geometers usually gave the exclusive definitions, and unfortunately their example is still followedin many elementary textbooks.
Why do I say "unfortunately"? Because books that follow this practice usually get things wrong, as a direct result. For example the book I recall from my own schooldays defined "rectangle" exclusively, but then went on almost immediately to give the "theorem"
a quadrilateral whose diagonals bisect each other is a rectangle.
With the definition the book used, this was FALSE  a correct form woul read a bit more comnplicatedly:
a quadrilateral whose diagonals bisect each other is either a rectangle or a square.
What caused this error? The answer is that the quadrilaterals referred to have the essential property of a rectangle  namely the four right angles  while their sides may or may not be all equal. If they are, it's a square; if not, a rectangle in the exclusive sense. Now experience teaches me that those who give exclusive definitions invariably make lots of mistakes of this kind. This is not only true of school textbooks  it's been true of almost all authors of geometrical works for two thousand years. I've read a lot of them, and I know! What happens, is that although they GIVE the exclusive definitions, they actually USE (like the rest of us) the inclusive ones.
Over the last century, the situation has changed, because the professional mathematicians have switched to giving the inclusive definitions that in fact have always beenthe ones actually used inside geometry. As a result, the theorems they state have more often been correct.
This is not, and should not be, a question about how the words are used in ordinary life. Of course, one wouldn't and shouldn't normally describe a square table as rectangular, since that is not as helpful as describing it as square. But geometrically, it's more helpful to count it as both square and rectangular, since essentially every theorem that holds about rectangles still holds for squares.
If you want your daughter to be able to think clearly and easily, then it's better that she should regard squares as particular cases of rectangles than to see them as different things.
John Conway
PS. Etymologically, of course, the word "rectangle" merely refers to the right angles, and says nothing about the lengths of the edges being not all equal. There is another word "oblong", that in origin does imply this. So what was originally an "oblong rectangle", now usually abbreviated just to "oblong", does properly refer to the kind of rectangle that is not a square.
JHC

