Comprehension Instruction

During Bart Nelson’s calculus class at Snow College we encountered the following function:

f(x) = 1/x

(http://mathworld.wolfram.com/GabrielsHorn.html)

When rotating this function, where x ≥ 1, around the x-axis on a graph you get something which resembles a cornucopia or trumpet.

We were surprised when calculating the volume yielded a finite answer but when solving for the surface area the answer was infinite. He explained that this means you could never finish painting the inside of it but you could pour it full of paint.

(http://blog.plover.com/math/gabriels-horn.html)

This seeming paradox led to a discussion about what infinity means and to the question, “Just how big is a ‘point’?” Supposedly a point, by definition, is something which has no mass and takes up no space. Another way to think of them (very informally and non-mathematically of course) is that points are bits of nothing. Then, if you stack up enough bits of nothing, you somehow get a line which has infinite length. These infinite lengths of nothing are then connected side by side to make the plane represented by our graph. If the premise upon which our geometry is based seems paradoxical we should not be too surprised when our mathematics yields paradoxical results.

Additionally, the results are dependent on allowing the possibility of a function with infinite length. Just how big is infinity? If you take the biggest number in the world and add 1 to it then you have a larger number. So infinity as a number can’t happen but we want it to anyway so we are going to invent the largest possible number and call it infinity.

I did not realize it at the time but infinity is actually not a number.

(https://www.khanacademy.org/math/recreational-math/vi-hart/infinity/v/9-999-reasons-that-999-1)

At the beginning of the course, we talked about numbers and how they do not exist. Up until that point my experience was that math was logical and always made sense. Why would he say that numbers do not exist? He then asked, “Have you ever seen a 'two'?” We said that of course we had seen a 'two' and he asked for examples. Someone wrote the word "two" on the board and someone else wrote the number ‘2’. He asked which one was the 'two' and whether what we had written was a ‘two’ or just a representation of a 'two'? What about the roman numeral (II) or what if we write it in French (deux) or Spanish (dos)? Is what we have written a 'two'? What if we made a statue in the shape of a ‘2’? Would that be a 'two'? Is ‘two’ concrete or abstract? Is ‘two’ something tangible? Does ‘two’ actually exist or is it just a concept? We finally agreed that 'two' does not actually exist but is a concept to describe quantities. ‘Two’ may describe a property of a set of items which do exist but the number ‘two’ does not exist as a tangible thing.

This was the beginning of my understanding of how math was a language. Having my assumptions challenged helped me keep an open mind later when we encountered functions such as “f(x) = 1/x”.

If you think about language, words do not physically exist either as tangible entities separate from the things they describe. “Apple” is the name which we use for a particular fruit but the word “apple” is not itself an apple. Numbers are names for quantities of things. You might say there are two sheep in a field but the word “two” is not a sheep. Numbers are words. Math is a language.