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### Inclusive vs. Exclusive Definitions

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Date: 01/24/2002 at 15:44:09
From: Logan Rhyne

Dr. Math,

My geometry teacher says that a square is not also a rhombus, a
rectangle, and a parallelogram. I cannot convince him that this is not
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Date: 01/24/2002 at 16:37:55
From: Doctor Peterson

Hi, Logan.

Both "inclusive" and "exclusive" definitions are used for such things;
you and I agree that the inclusive definition (rectangles include
squares) is more useful than the exclusive definition (rectangles must
have unequal length and width) that is often taught to children. We
can either convince your teacher of this judgment, using arguments
like those here:

Inclusive and Exclusive Definitions
http://mathforum.org/dr.math/problems/hawes.04.05.01.html

or we can just show that both definitions are valid, so that you are
at least not wrong. Try a dictionary definition like this, from
Merriam-Webster (m-w.com):

Main Entry: rect.an.gle
Function: noun
Etymology: Medieval Latin rectangulus having a right angle, from
Latin rectus right + angulus angle -- more at RIGHT, ANGLE
Date: 1571
a parallelogram all of whose angles are right angles; especially
one with adjacent sides of unequal length

This says that the word can be taken in general of any right-angled
parallelogram, or more specifically of one that is not a square.
Similarly,

Main Entry: rhom.bus
Function: noun
Inflected Form(s): plural rhom.bus.es or rhom.bi  /-"bI, -"bE/
Etymology: Latin, from Greek rhombos piece of wood whirled on a
string, lozenge, from rhembein to whirl
Date: circa 1567
a parallelogram with four equal sides and sometimes one with no
right angles

More clearly this time, the word only _sometimes_ excludes right
angles.

But notice that both of these are defined as parallelograms! There's
no doubt that that is defined inclusively.

Finally, check out "square":

Main Entry: square
Function: noun
Etymology: Middle English, from Middle French esquarre, from
Date: 13th century
...
2 : a rectangle with all four sides equal
...

So a square _is_ a rectangle.

Now, dictionaries don't always get math terms right, but they do
carefully research how words are actually used. The fact that they
give first the definition we prefer, which is probably more popular
among mathematicians than in the general public, seems to support our
contention. Now let's look at a scientific dictionary and check their
view:

Harcourt Academic Press Dictionary of Science and Technology
http://www.harcourt.com/dictionary/def/9/7/2/5/9725100.html

square   Mathematics.  1. a quadrilateral having all four sides
and all four angles equal; equivalently, a rectangle with equal
sides or a rhombus with a right angle.

- Doctor Peterson, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Definitions
High School Geometry
High School Triangles and Other Polygons
Middle School Definitions
Middle School Geometry
Middle School Triangles and Other Polygons

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