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 2 we obtain – ½, and 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 into 2/5 by successively applying the operations T, T, T, R, T, T, R, T:
0 ---T ---> 1 ---T---> 2 ---T ---> 3 ---R---> -1/3 ---T ---> 2/3 ---T ---> 5/3 ---R ---> -3/5 ---T ---> 2/5.
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 into 0.
c. Notice that 0 can be turned into any positive integer n by applying n successive ts. Explain how any positive integer n can be turned back into 0 by applying 3n-1 operations.
d. Explain how 0 can be turned into any negative integer.
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