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Learning Math Home
Number and Operations: Solutions
Session 2 Part A Part B Part C Homework
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Session 2 Materials:

A B C 


Solutions for Session 2 Homework

See solutions for Problems: H1 | H2 | H3 | H4 | H5

Problem H1

1/1 = 1
1/2 = 0.5
1/3 = 0.333...
1/4 = 0.25
1/5 = 0.2
1/7 = 0.142857142857...
1/8 = 0.125
1/9 = 0.111...
1/10 = 0.1

Some rational numbers when expressed in decimal form will terminate, such as 0.5 or 0.125. Others will have repeating, non-terminating decimals where the repeating part can be a single digit, such as in 1/3 = 0.333..., or six digits, such as in 1/7 = 0.142857...

To further explore why this happens and why some rational numbers terminate and others don't, go to Number and Operations, Session 7.


Problem H2

If we think of division as repeated subtraction, in essence, we are subtracting groups of 0 from the number we are dividing. It is easy to see that you could subtract groups of 0 infinitely many times and never exhaust the number you started with. Therefore, dividing by 0 is not defined.

<< back to Problem H2


Problem H3

The easiest solution would be to move everyone in the hotel one room over. This way, the first room would be freed up for the traveler. Notice that here you've added one element to an infinitely countable set. The result is still an infinitely countable set:

<< back to Problem H3


Problem H4

To put everyone into a room, you would alternate the guests (G) already there and the marching band guests (MB). In other words, the guests who were already in the hotel would be moved into rooms with odd numbers:

By combining the two infinite sets in this way, you still get a countably infinite set that can be put into one-to-one correspondence with the counting numbers.

<< back to Problem H4


Problem H5

By writing down all the rooms in all the hotels in an infinite two-dimensional matrix (see below), they can all then be put into one-to-one correspondence with the counting numbers, each of which corresponds to a particular room:

Notice that here you have multiples of infinitely countable sets combined into one set. The new set is also countably infinite, which you've shown by putting it into one-to-one correspondence with the counting numbers.

<< back to Problem H5


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