Example 4: Algebraic Rules

Click here to open an associated Mathcad worksheet:

d = 2w - 1

Algebraic expressions, like the English expressions "I dress for the weather" and "If it's raining I wear my black shoes; otherwise I wear either my red shoes or my blue shoes," are another way of expressing the relation between two variable quantities. Compared to natural languages, the language of algebra often speaks more explicitly, more succinctly, and, ultimately, more clearly – at least when it comes to numerical relationships.

Imagine that you have numbered your shoes  d  = 1, 2, 3, ... according to some personal, obsessive scheme, such as the order in which they were purchased, or the order in which they are lined up in the closet. If  w  again represents the outside temperature (rounded to the nearest degree), then the rule associates a unique pair of shoes to some (but not all) outside temperatures.

What is the lowest temperature,  w = wmin, for which the rule works?

How does the highest temperature for which the rule works,  w = wmax, depend on the total number of shoes N?

Which shoes will never be worn?

What would have to be true about the shoes for this rule to be an extension of the rule  g  in Example 2?