## Tuesday, January 11, 2011

### Maybe this will rile up your geometry class

I was reading this awesome unit on "other dimensions" courtesy of Mathematics Illuminated. In it, they make the following deceptively simple observation:
If we take two points in 1-D space and connect them, we form a line segment. This line segment has a property that no single point has, length. The length of a line segment in 1-D space can be found from the positions of the two endpoints via subtraction.
But if no single point has length (instead, it is defined only by a location), yet a line segment consists only of single points, how is it that a line segment itself has measurable length?

Of course this isn't really a paradox once we introduce infinity and stop trying to interpret infinity as just "a really big number" with all the same properties as numbers. The segment contains infinitely many points, so apparently infinity*0 is not zero (unlike "numbers" we are used to, all of which satisfy the property that: # times zero is zero). A beautiful revelation just waiting to be plucked from an awfully dry geometry standard.

From experience, my kids love talking about infinity in all its forms. Maybe yours will too!