Here is an interesting conversation I had with a student I tutor. I used Pi to explain Rational and Irrational Numbers to her.
So, this is how I introduced Irrational Numbers to a student of mine. I asked her “Do you know Pi?” she said, “Yes, my teacher said Pi is equal to 3.14”. I said “Is that it, just 3.14? “ she said, “Yes, that’s what she said.”
I shared my screen and typed pi’s value in Google. I found a wiki article and asked her to read it. And she was quite surprised by the number of digits that follow 14. And I also told her about Emma Haruka Iwao, a Google employee from Japan, who calculated Pi to a new world record length of 31 trillion digits with the help of the company’s cloud computing service. She was even amused to know that is still an approximation of Pi.
After this interesting conversation, I told her to observe two Numbers
1.851851851 And 3.141592653589793 (pi)
I asked her to figure out a pattern among the fractional parts of both Numbers. She was quick to find a Pattern with the first one and couldn’t come up with one with the second one.
I told her that finite and recurring decimals are indeed rational Numbers as they can be written as a ratio of two integers. And we discussed writing Pi as a ratio of 2 integers (I told her 22/7 is an approximation) and found it is not possible to come up with one.
She quickly understood the difference between Irrational and Rational Numbers.
I also came up with some open-ended questions at the end of the lesson like,
if Pi is irrational, why are defining it has a ratio of Circumference to the Diameter of any Circle?
I also asked her “Do you think it is possible for Pi to be a rational Number as we have not yet computed it completely?”
Well, I am pretty sure, I didn’t turn her into a “Pi is Rational” Zombie.