A note from Mr. “Mean” Thoens to me in my senior yearbook. We never did agree on infinite numbers.

My disdain for infinity and infinite numbers started when I was a senior in high school. My North Bergen High School calculus teacher, Mr. Ray Thoens, (who I called “Mean” Thoens) was teaching us about infinity and the infinite number of points in a line. Okay, I could get that. But then he told us that two lines of unequal lengths would have the same number of infinite points. What!!

I argued with him.

How can a line this long ___________, have the same number of points as a line this long _________________? The lines have a definite beginning and an end. How can they have the same infinite number of points! For my logical mind, one must have more points than the other.

Mr. Thoens and I argued about this all year. Whenever I was upset about something I would just say, “Yes just like those lines and infinite number of points. It just doesn’t make sense.” And I would sometimes add while shaking my head, “that is just wrong.” Other students in my class perhaps agreed with Mr. Thoens, but that did not change my mind.

Senior year, basically the calculus class. I had a lot of hair, but not as much as the boy next to me.

Over the years, the long years, since I graduated high school, I still felt that the information about infinity and lines and infinite numbers of points was a crazy thing and just could not be right. But I kept my point of view to myself all these years. I never took another math class (except statistics), so I did not have to worry about these numbers. And even though my husband studied math and physics for the first two years of his college career, infinite numbers just did not come up.

Until now, when my nephew, my sister’s son, came to stay with us for a few days.

My nephew just earned his master’s degree in mathematics from the University of Kansas. He taught calculus to college freshman for the past few years, and he is staying with me before he leaves for Florida to study for his PhD in math at a university there.

And we got into a math debate.

I am not a hundred percent sure how it started, but we got on to the topic of calculus. I could not help myself, I had to tell him about my disdain for infinite numbers and points in lines.

He said something like, “I will explain it to you. Many people have this problem.”

I said, “You are not going to change my mind. It is not right! I have held this view for 40 years!”

He told me that Mr. Thoens, my high school math teacher was right! Can you imagine that! He told me that my high school teacher was probably trying not to use more advanced math language when he tried to exlain it all those years ago. But he, my nephew had explained this to many students, and he could explain it to me.

The diagrams in my nephew and my debate over infinite points in lines of two different lengths.

He started talking about ‘cardinality’ and how to match numbers. He showed me two sets of numbers, one with three dots and one with five. We could agree that these did not match. Then he added two more dots to make them equal sets. And we could agree that they were now equal.

He made graphs and wrote equation-like things. Who cares? When you look at two lines of unequal length it is intuitive and logical to realize that they do not have the same number of infinite points. ( I spoke to my daughter about this, and she totally agreed! So I must be right.)

I showed him two equal lines, A to B. We agreed that they had the same number of infinite points. Then I added a segment that doubled the size of one line to C. And I said, “This line has more points. It is a longer line.”

And he said, “NO!”

What! How can you say no?

He then told me that “The same way of matching is not going to work.”

Of course it will not. You cannot match the same way because they are different lengths.

And then he went into a silly math concept that showed matching using x/2 (x over 2). In this way the numbers in the longer line matched numbers in the shorter line like this: .3 went with .15 and so on. So! Yes you can make pairs of numbers, but there are always other numbers. He agreed and said something like, “But you never actually get to zero so your cardinality is okay as long as you can keep matching.”

Yes, Mr. Thoens had tried that same trick on me when I was 17. It did not work then and it will not work now.

I appreciate my nephew’s passion for math. I hope he has great success and continues to teach and learn. But I am not changing my mind. Two lines of unequal length and size cannot have the same number of infinite points even if both have an infinite number of points.

And do not tell me that an infinite number of points is an infinite number of points. I know that. But it is something that does not make sense in my mind, and probably will never make sense.

I think I will just go another 40 years believing that learning about infinity and beyond just makes me insane!

Tags: calculus, favorite teachers, high school memories, infinite numbers, math, New Jersey, North Bergen