Inclusive vs. Exclusive Definitions
Date: 01/24/2002 at 15:44:09 From: Logan Rhyne Subject: Quadrilaterals 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 so. Please help!
Date: 01/24/2002 at 16:37:55 From: Doctor Peterson Subject: Re: Quadrilaterals 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 (assumed) Vulgar Latin exquadra, from exquadrare to square, from Latin ex- + quadrare to square -- more at QUADRATE 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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