Infinite Decimals Worksheet 1: Rules for infinite Decimals
Answer these questions based on "common sense."
1.
Can you add .121212… to .454545…? What result do you get?
2.
Can you add .123123123… to .454545… ? What result do you get?
3.
Can you add .787878 … to .44444….? What result do you get?
4.
Can you subtract .123123123… from .58585858…?
5.
Can you subtract .181818181818…. from .2323232323…?

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6.
What should be the result if you multiply .33333…. by 2?
7.
What if you multiply .4444444…. by 3?
8.
What if you multiply .123123123… by 10? By 100?
9.
Can you multiply .125125125… by .6363636….?
10.
What is one third of .66666666….?

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11.
Summarize any rules for arithmetic using infinite decimals that you believe are reasonable. Specify situations where the rules apply and also tell when they do not apply.

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Explanation.
1.
Based on the worksheet, you should have some general idea of operations that make sense for infinite decimals. But "common sense" is not reliable in this setting. To see
this, consider the following three T/F statements: a.
.333… = 1/3 b.
We can add infinite decimals in the obvious way, that is, digit by digit, at least if the two decimals have the same size repeating block and if no carrying is required c.
.999… = 1
Most people will agree that a and b are true but c is false. The problem is if a and b are true that c must also be true. Just add the decimals for 1/3, 1/3, and
1/3 and you find that the total, 1, must be .999... . To avoid self contradictory beliefs we need a deeper understanding of these infinite decimals.
Since we already have some idea of what the rules of arithmetic should be, and of what finite decimals are, our goal is to develop a clear understanding of what infinite decimal notation is going to mean . And then we need to be confident that our desired rules of arithmetic are consistent with that meaning.
One obvious interpretation is that an infinite decimal is really an infinite addition problem. For example, we can interpret .333… as the result of adding up
.3 + .03 + .003 + .0003 + … where the calculation has infinitely many terms. We understand the meaning of each individual addend or term, but how do you perform an infinite calculation? As we will see later, we can use the idea of self-similarity again.
Another approach is to start not with an infinite decimal, but with a fraction, say 1/3, and consider how to express this using a variant of decimal notation , as in the next work sheet. In following this path, we are using another strategy for effective thinking: understand simple ideas deeply. Here, we will begin with the simple idea of evenly dividing something (let’s say a square brownie) among several people.

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Dividing a brownie among 6 people using quartering.
It is fairly easy to cut something in half. If we have a square brownie, by cutting it in half top to bottom and then again side to side we can divide it in quarters. But that doesn’t help much to share the brownie among 6 people. But let’s divide each quarter into four pieces. Now we have 16 pieces, and we can give each person 2 of them. That leaves 4 of the pieces unused. Next, quarter each of those four leftovers.
Then we again have 16 pieces and can give each person 2 of them and have four left over again. In this way we can successively split up more and more of the brownie evenly among the 6 people.
To keep track of this process, it is helpful to use diagrams and an abbreviated notation. Start with the original brownie, shown below as a square, and divide it into
4 quarters. Use Q to stand for “quarter”. Then we divide each Q into four squares.
1 Brownie 4 Q’s 16 QQ’s
Those are quarters of quarters, or QQs. We have 16 of them in all.
Now our division process can be described as follows. Divide the brownie into
16QQ’s. Give each person 2QQ’s (using up 12) and leaving 4QQ’s remaining.
Next quarter each of the 4QQ’s. We end up with 16QQQ’s. Each person gets two of those, and 4 are left over. So far, everyone has received 2QQ’s and 2QQQ’s and we have 4QQQ’s left over. We can continue this process as many times as we wish, and we see that there is an obvious pattern. Each person gets 2QQ’s and then 2QQQ’s and then 2QQQQ’s and so on. In order to use up the entire brownie, we would have to repeat this process an infinite number of times, because after any finite number of repetitions there is always a little bit left over. In this way, we see that the idea of an infinite addition problem arises: each person’s full share should be
2QQ + 2 QQQ + 2 QQQQ + 2QQQQQ + etc and that should add up to exactly 1/6 of the brownie. On the next worksheet you are asked to carry out this same kind of process for several other division problems. To make the notation even easier, let’s agree to write Q3 for QQQ, Q4 for QQQQ, etc.

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Infinite Decimals Worksheet 2: Dividing a Brownie by Quartering
1. Use the concept of quartering to divide a brownie among 7 people. Record in the grid below the number of Q’s, QQ’s, QQQ’s, etc that each person should receive. The grid is partially completed for you – with a diagram below showing how the divisions were done. Continue the division process until you discover a pattern that shows how the process will continue. At the bottom of the page, use this analysis to express 1/7 as the addition of infinitely many parts.
Q’s QQ’s QQQ’s Q4’s Q5’s Q6’s Q7’s Q8’s Q9’s Q10’s Q11’s Q12’s
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2. Use the concept of quartering to divide a brownie among 11 people. Record in the grid below the number of Q’s, QQ’s, QQQ’s, etc that each person should receive and use diagrams to show how you do the divisions. Continue the process until you discover a pattern that shows how it will continue. At the bottom of the page, use this analysis to express 1/11 as the addition of infinitely many parts.
Q’s QQ’s QQQ’s Q4’s Q5’s Q6’s Q7’s Q8’s Q9’s Q10’s Q11’s Q12’s
0 2 1