Cardinality - Activity 4

Ivan: The challenge I had to solve was to generate all the proper fractions. This looked like impossible task at first: It is easy to point the neighbors of every integer but if you take a fraction… Here is what we did with Jenny and George in order to solve this problem.

Jenny: Which are the neighbors of 3?

Ivan: It’s obvious: 2 and 4.

Jenny: Can you point the neighbors of 1/3?

Ivan: The first that occurs to mind is ¼ and 1/5.

Jenny: Order them by size!

Ivan: 1/5, ¼, 1/3

Jenny: Can we find a number between ¼ and 1/3?

Ivan: Let me think…

Jenny: What is the difference between these numbers?

Ivan: It is equal to 4/12 – 3/12, i.e. 1/12.

Jenny: So, can we find a fraction greater than 3/12 and smaller than 4/12?

Ivan: Yes, we could add to 3/12 any fraction smaller than 1/12, e.g. 1/24.

Jenny: Good. So, there are many more fractions between ¼ and 1/3.

Do you think it is possible to generate all the fractions then?

Ivan: I am not sure. Let me see what Ken suggests in the activity sheet.

In Task A he wants us to build a robot (Next Numerator) that generates all proper fractions whose denominator is 100.

First I did this without the scale and after that added it just as it was in the instructions. When I tested my robot with numerator 1 and denominator 50 (Task B) it produced all the proper fractions with denominator 50:

1/50, 2/50,…..49/50 (just I expected)

In general when given a concrete number as a denominator my robot generates all the proper fractions with this denominator.

Then in Task C Ken expects us to train another robot (Next Denominator), which would increase the denominator by one when Next Numerator finishes its work. I tested the two robots as a team by giving them different inputs (1/5, 5/5).

Jenny: If your team of robots runs forever do you think it will generate two identical fractions?

Ivan: Not as numbers of the numerator and the denominator but it would generate fractions of equal value, e.g. ½, 2/4, 3/6, etc.

Jenny: Do you think it will generate all the proper fractions? Try to convince Yana!

Ivan: If we start with a denominator 1 there is no proper fraction with this denominator. The Next Denominator robot makes this number 2. Then the team produces all the proper fractions with a denominator 2 – only ½ in this case. The next denominator is 3 and the proper fractions the team produces are 1/3 and 2/3, etc. There is no danger to miss a fraction!!!

George: Let us combine your robot with Ken’s No Copies robot in order to remove the duplicates.

Ivan: I will try its action on my robot directly.

Ah, the numbers it generates are decimals…It would have been easier for me to check for duplicates if they were fractions.