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Topic: Trapezoid definition
Replies: 26   Last Post: Oct 7, 2004 11:51 PM

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John Conway

Posts: 2,238
Registered: 12/3/04
Re: Is a rectangle a square?
Posted: Oct 29, 2003 9:26 PM
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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.


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