One of your middle school students was interested in infinite repeating decimals. He noticed that when he tried to convert N = .999 ... to a rational of form § he got the following:
10 = 9.999
10N - N = 9
9N = 9
N = 1
He asked you how .999 ... could be 1.
a. How would you answer this student?
I would explain to the student that the concept of repeating decimals can sometimes lead to counterintuitive results. In this case, when we convert the decimal .999... to a rational number, it appears to be equal to 1. However, it is important to understand that .999... is not exactly equal to 1, but it is very close to 1.
To help the student understand, I would use a visual approach. I would draw a number line and place 1 on it. Then, I would ask the student to plot the point representing .999... on the number line. They would notice that it falls exactly on the same spot as 1.
This occurs because .999... is an infinite decimal that continues with a series of 9s. It means that the number gets infinitely close to 1, but it can never quite reach it. However, in our mathematical system, we treat .999... as equal to 1 for convenience and consistency.
We can also use algebra to explain it further. The equation the student used, 10N - N = 9, is correct. However, when we solve for N, we get N = 1, which seems to contradict the initial assumption that N was equal to .999... In other words, the decimal .999... and the rational number 1 represent the same quantity, even though they may have different notations.
This is a fascinating topic in mathematics that may seem counterintuitive at first, but by understanding the properties of repeating decimals, we can accept that .999... is equal to 1.