Friday, April 17, 2015

Infinity: Why I'm Pursuing The Master's Degree That I Am

Yesterday one of my students, one who almost never engages me in conversation, asked me a question right before class was over:  he wanted to know how "infinity" could have different sizes.

My degree was in applied math, not theory.  I did calculus, differential equations, partial differential equations, separable differential equation, numerical solutions to differential equations, some math modeling (probably with differential equations), etc.  Math history, number theory, set theory, graph theory--those weren't the classes I took in college.

Interestingly, they're covered in varying degrees in the courses I'm taking for my master's degree.  Yes, I could have gone to National University and in 10 months picked up a Master's in Education with an Emphasis on Curriculum and Instruction and gotten a mambo-sized pay raise, but instead I chose to pursue a degree that would make me a better math teacher rather than just a better-paid one.  I can't fault people who did go the National (or similar) route, as they just played by the system's rules, I just want more.

And it's working.  I'm a much better statistics teacher than I was because now I have both a broader and a deeper understanding of what I teach, I can answer the "why" questions and tempt students with a taste of what university math could have in store for them.

So when this student asked me about infinity, I was able to answer his question somewhat.  I told him I'd like to review my notes and to check with me tomorrow, which was today.  Last night I consulted my notes and wrote up 2 pages of commentary and examples to show how the size of the infinity that encompasses the set of natural numbers (1, 2, 3, ...) is the same size as that of the integers or even the rational numbers, but the infinity of the set of real numbers (or even just the numbers between 0 and 1) is larger than the infinity of the natural numbers.  When my instruction was done today, he and I got together and went through the integers and rational numbers but didn't have time to go through the real numbers.  I told him if he couldn't figure out my examples by Monday, we'll meet again then and go through it.

A year ago I wouldn't have been able to answer his question, now I can.  That makes me a better math teacher.  That's why I'm getting the degree I am.


maxutils said...

I REALLY don't mean to be the jerk hear … but wouldn't restricting infinity to the set of natural numbers, by definition, negate the entire idea of infinity? because, if it doesn't than the old playground challenge of I hate you/ I hate you times 1000 / I hate you infinity/ I hate you infinity +1 actually has merit...

Joshua Sasmor said...

And it feels so good to be able to explain that and watch the students eyes light up as they get excited about understanding something brand new, doesn't it? That's the best part about blowing their minds - some of them make the cognitive leaps to get it, and it's amazing to be a part of!

Darren said...

Do the natural numbers go on forever? Then their set is of infinite size.

That some infinities are larger than others, well, that's the fun of it all.

Niels Henrik Abel said...

The natural numbers are infinite.

The whole numbers are also infinite, but they contain exactly one more element than the natural numbers.

Which set is "bigger"?

Also, the set of integers would have the same cardinality as the cardinality of the natural numbers plus the cardinality of the whole numbers.

Now which set is "bigger"? Yet all these sets are infinite sets.

There are more rational numbers than integers, so the set of rationals is bigger than the set of integers, although both are infinite.

All these sets thus far are countable--i.e., can be mapped onto the natural numbers (there exists a 1-1 function between any one of these sets and the set of natural numbers)--and hence "countably infinite;" however, the set of irrationals is "uncountably infinite." It doesn't tax the imagination too much to understand that an uncountably infinite set would have to be in some sense "larger" than a countably infinite set....

maxutils said...

Well, you didn't really answer my question … but, if that's the fun of it all … suppose you had a ox of infinite volume, and you wished to attempt to fill it with the infinite set of natural numbers, in some tangible sense. Could you do it?

Joshua Sasmor said...

maxutils - have you heard of Gabriel's horn? That's the shape formed by rotating the curve y=1/x with x>=1 around the x-axis. The resulting "infinitely long" shape has a surface area of infinity and a finite volume. So I can pour it full of paint, but I can never completely paint the outside...

And the set of transfinite cardinal numbers say that infinity+1 is a perfectly valid concept. :)

Anonymous said...


You have just stumbled into a tiny corner of math that I have spent an unreasonable amount of time thinking about.

One key takeaway about the cardinality of the reals being larger than the cardinality of the integers is that the reals have a sub-set of numbers that you and I and your student will never see -- the "uncomputable numbers" (Wiki has a nice writeup on the Computable numbers).

-Mark Roulo