Is a rectangle a square?
Pamela Paramour
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. :)
Walter Whiteley
texts get this tangled up.
The way mathematicians work, think, and use these words,
we are 'inclusive'. So something with four Equal sides
DOES have two pairs of opposite sides equal.
That means is also a rectangle.
Look at it in reverse.
If you give some definition of a rectangle:
eg. four right angles, then you can start to look
at figures which have this property.
When you happen to find an example with four
equal angles, you will agree it is a rectangle, even
before you check whether the sides are ALSO equal.
If the sides are also equal, it is still a rectangle and is
also a square. The 'image' of this is a collection of
all rectangles as a big circle, and the collection of
all squares as a smaller circle INCLUDED inside the bigger
one.
Take numbers. We can have all even numbers. We can
have all numbers divisible by 10. The second collection
is 'included' in the set of even numbers. They did not stop
being even, they just picked up an extra additional property.
Does that help?
Walter Whiteley
John Conway
> This is a 'standard' difficulty, and some elementary math
> texts get this tangled up.
>
> The way mathematicians work, think, and use these words,
> we are 'inclusive'. So something with four Equal sides
> DOES have two pairs of opposite sides equal.
> That means is also a rectangle.
Quite right! But let me point out that what this
shows is that it's sensible to define "rectangle" so
that it includes "square".
However, the question asked "is a rectangle a square?"
The correct answer to this is "not always". For example,
a 2 x 1 rectangle certainly isn't a square, while a 1 x 1
one is.
John Conway
Mary Krimmel
John Conway answered the question asked as the "subject" of the header.
Walter Whiteley gave what seems to me an excellent explanation in layman's
terms.
Adding my two cents may be more than you want or need, but here goes.
Yes, a square is a rectangle. I do not know just when math textbooks
undertook to make this clear, nor do I know at what level of math it is
explained. It would not surprise me to see kindergarten or preschool
teachers pointing to a shape saying "This is a square," and to another
shape saying "This is a rectangle." They are not wrong, but certainly their
methods lead to misconceptions that need to be set aside as students begin
to learn the way math "works".
The Webster's edition I use gives as definition for "square": "A rectangle
with four equal sides." American Heritage Dictionary's definition is
essentially the same: "A rectangle having four equal sides." Both
dictionaries define a rectangle as "a parallelogram with a right angle".
You did not tell whether you looked up "rectangle" in Webster, but I
suspect that if you do, you will see that by the definition you find there,
a square will be included as a rectangle. In any case, a parallelogram with
four equal sides and four right angles is a rectangle; "rectangle" is not
ruled out.
Check out your daughter's textbook, if she's using one. As Walter Whiteley
said, some elementary texts get this tangled up. Then if the teacher
doesn't make the situation clear, the student loses.
Best wishes for you and your daughter.
Mary Krimmel
------- End of Forwarded Message
John Conway
> 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
Rick Nungester
news:mnp3nvchorqm@legacy...
Is a square a rectangle? Yes.
Is a square is a parallelogram. Yes.
Is a square is a quadrilateral. Yes.
Is any rectangle a square? No.
Are a square and a rectangle the same thing? No.
Think general versus specific categories. A Ford is a car. Not
all cars are Fords. A squares is a rectangle. A rectangle
is a parallelogram. A parallelogram is a quadrilateral.
Each category is more general, including the previously named
category, plus more.
Rick
-----= Posted via Newsfeeds.Com, Uncensored Usenet News =-----
http://www.newsfeeds.com - The #1 Newsgroup Service in the World!
-----== Over 100,000 Newsgroups - 19 Different Servers! =-----
Kit
>Is a square a rectangle? ...
> If you refer to Webster, ...
Here is a nice story realy happend in german tv, sorry for my bad
english.
In the german quiz-show "Wer wird Millionär" (Who becomes a
millionaire) from January, 31 2003 the 8000-Euro question was:
Every rectangle is:
(a) a rhombus
(b) a square
(c) a trapezoid
(d) a parallelogram.
In this show _allways_ exactly one answer is (has to be) correct.
The candidate was so confused, she didn't know if c or d is thw right
answer, so she skipped the question and went home (with "just" 4000
Euro). In the following days the broadcast station got tons of mails,
letters and phone calls. Nearly all "mathematicians" regarded c _and_
d as correct. The broadcast station told, that they looked up in three
different encyclopaedias, all three saying that trapezoids have only
one pair of parallel sides. Taking this definition only d is correct.
That's the problem. Who is right: More than 90 percent of the
mathematicians saying a parallelogram is also an trapezoid or three
encyclopeadias saying the opposite?
The Solomonian solution. In the next week the candidate got a "new"
8000-Euro-question.
David W. Cantrell
This is actually rather complicated.
First, we should know precisely which German word was used for (c). Was it
Trapez or Trapezoid? One of my German-English dictionaries says
Trapez means trapezoid or trapezium
and that
Trapezoid (German) means quadrilateral.
Of course, I can't say for sure that my dictionary is correct. But I can
tell you that the English words "trapezoid" and "trapezium" have different
meanings on different sides of the Atlantic. One of my mathematics
dictionaries (which happens to have editors from both Canada and the UK)
says that
(1) A quadrilateral having two parallel sides of _unequal_ length is
called a trapezium in the UK and a trapezoid in North America.
and that
(2) A quadrilateral with neither pair of sides parallel is called a
trapezoid in the UK and a trapezium in North America.
I must add that, as a North American mathematician myself, I disagree
with (1). For me, a trapezoid is simply a quadrilateral having two parallel
sides. Those sides need not be of unequal length. Furthermore, note that
I only said "two", _not_ "exactly two". Thus, for me, both (c) and (d) are
correct.
[BTW, I also question the correctness of (2), but I don't often use the
word "trapezium".]
But if (1) and (2) are used precisely as given above, then regardless of
which side of the Atlantic an English speaker resides on, (c) must be
wrong.
David Cantrell
Neal Silverman
You can't really say "who's right." It's just a question of how one
defines "trapezoid." In all American textbooks (except the University
of Chicago geometry textbook), a trapezoid is defined as a
quadrilateral with exactly one (or at most one) pair of parallel
sides. A parallelogram is defined as a quadrilateral with 2 pairs of
parallel sides.
Professor Conway, on the other hand, defines a trapezoid as a
quadrilateral with "at least" on pair of parallel sides. Using that
definition, the set of parallelograms is clearly a subset of the set
of trapezoids. Thus, using that definition, every rectangle is a
trapezoid, and also a parallelogram (as is the case with the usual
definition).
A similar problem exists with the definition of the kite. Most
writers say it is a quadrilateral in which AT MOST one diagonal is the
perpendicular bisector of the other. Conway would say that it is a
quadrilateral in which AT LEAST one diagonal is the perpendicular
bisector of the other. So using Conway's definition, every rhombus is
a kite.
There have been many, many message threads here on this issue.
Logically, there is a great deal to be said for Prof. Conway's
position.
Mary Krimmel
. . .
. . .
>Professor Conway, on the other hand, defines a trapezoid as a
>quadrilateral with "at least" on pair of parallel sides. Using that
>definition, the set of parallelograms is clearly a subset of the set
>of trapezoids. Thus, using that definition, every rectangle is a
>trapezoid, and also a parallelogram (as is the case with the usual
>definition).
>
>A similar problem exists with the definition of the kite. Most
>writers say it is a quadrilateral in which AT MOST one diagonal is the
>perpendicular bisector of the other. Conway would say that it is a
>quadrilateral in which AT LEAST one diagonal is the perpendicular
>bisector of the other. So using Conway's definition, every rhombus is
>a kite.
>
>There have been many, many message threads here on this issue.
>Logically, there is a great deal to be said for Prof. Conway's
>position.
Amen. Most of the other conflicting definitions defy logic.
Donna W
found an answer I'm satisfied with.
A rectangle is a geometric figure with four straight sides, opposite
sides parellel, and each corner angle a right or 90 degree angle.
A square is simply a rectangle with all four sides of equal length.
So, by definition, a square is a rectangle, however, a rectangle is
not always square.
I wish they would include parent refresher guidelines with our
children's schoolbooks so we could really help them instead of seeming
so dumb. A lot of what I learned I have never really used or thought
about since I was their age.
Donna W.
I have a similiar situation and my question is "If a rectangle is
actually a square, why were we ever taught about a shape called a
rectangle in the first place. If the shape fits the description of a
square, it's a square. No rectangle I have ever seen has looked like
or fit the mathematical description of a square and vice-versa.