We define two numerical operations labelled T and R.
The effect of T is to add 1 to a number. For example, if we apply the operation T to the number 2 three times in a row, we obtain 3, then 4, then 5.
The effect of R is to find the negative reciprocal of a number. For example, if we apply the operation R to -3/2 we obtain 2/3. Note that R can never be applied to the number 0.
The operations T and R can be combined. For example, we can turn 0 in to 2/5 by successively applying the operations T, T, T, R, T, T, R, T.
a) Starting with 2, list the numbers produced by successively applying the operations T, R, R, T, R, T, R, T, R.
b) Find a sequence of operations which turns 3/4 into 2/3.
c) Find a sequence of operations which turns 3 into 0.
d) Find a sequence of 20 operations that turns 7 into 0.
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To solve these problems, we need to understand how the operations T and R work and how they can be combined.
The operation T adds 1 to a number, while the operation R finds the negative reciprocal of a number (except for 0, which R cannot be applied to).
a) To find the numbers produced by successively applying the operations T, R, R, T, R, T, R, T, R starting with 2, we can follow these steps:
1. Apply the operation T to 2: 2 + 1 = 3
2. Apply the operation R to 3: 1/3
3. Apply the operation R to 1/3: -3
4. Apply the operation T to -3: -3 + 1 = -2
5. Apply the operation R to -2: -1/2
6. Apply the operation T to -1/2: -1/2 + 1 = 1/2
7. Apply the operation R to 1/2: -2
8. Apply the operation T to -2: -2 + 1 = -1
9. Apply the operation R to -1: -1/1 = -1
The numbers produced by these operations starting with 2 are: 2, 3, 1/3, -3, -1/2, 1/2, -2, -1.
b) To find a sequence of operations that turns 3/4 into 2/3, we can follow these steps:
1. Apply the operation R to 3/4: 4/3
2. Apply the operation T to 4/3: 4/3 + 1 = 7/3
3. Apply the operation R to 7/3: -3/7
The sequence of operations that turns 3/4 into 2/3 is: R, T, R.
c) To find a sequence of operations that turns 3 into 0, we need to find an operation that will result in a negative reciprocal of 3. Since the operation R finds the negative reciprocal, we can apply it to 3:
1. Apply the operation R to 3: -1/3
The sequence of operations that turns 3 into 0 is: R.
d) To find a sequence of 20 operations that turns 7 into 0, we need to apply R repeatedly until we reach 0. We can follow these steps:
1. Apply the operation R to 7: -1/7
2. Apply the operation R to -1/7: 7
3. Apply the operation R to 7: -1/7
4. Apply the operation R to -1/7: 7
5. Repeat steps 3 and 4 ten times.
The sequence of 20 operations that turns 7 into 0 is: R, R, R, R, R, R, R, R, R, R, R, R, R, R, R, R, R, R, R, R.