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 Discussion: All Topics in Geometry Topic: Is a rhombus a kite?

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 Subject: RE: Is a rhombus a kite? Author: gsw Date: Jun 6 2005
On May 20 2005, Mathman wrote:

> No joking:  There is no good reason to *exclude*
> the rhombus any more than there is reason to exclude the integers
> from the set of rational numbers, which contains the fractions, and
> they are in fact not excluded.

The rhombus/kite problem is - of course - a matter of definition, but it relates
to an important and non-arbitrary mathematical habit which is undermined by
the exclusive definition. Kids always complain when you tell them that a square
is a rectangle,and usually want to add an exclusive clause to the definition of
rectangle when you point out to them that the normal definition includes
squares. But this is a great opportunity, an important discussion to have - and
a useful early example of discovering/exploring the (unintended) consequences of
mathematical definitions. Realizing that squares are (a subset of) rectangles is
a great piece of congitive dissonance, and a significant step in ones
mathematical education, the rhombus/kite example is a good opportunity to
reinforce the lesson.

Strictly speaking however, the integers are not a subset of the rationals -
though they do map one-to-one onto such a subset. A rational number is the
set of all equivalent pairs of integers (with equivalence defined as (a,b) equ.
(c, d) iff. ad = bc)