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.
impatient much? reposting after 3 minutes?
The duplicate assignment and one other long, complicated assignment of bucky's have been removed. If bucky really wants help, s/he'll indicate what S/HE has done so far and/or where s/he gets stuck.
a. To list 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:
Step 1: Apply T to 2 -> 2 + 1 = 3.
Step 2: Apply R to 3 -> -1/3.
Step 3: Apply R to -1/3 -> 3 (negative reciprocal of -1/3 is 3).
Step 4: Apply T to 3 -> 3 + 1 = 4.
Step 5: Apply R to 4 -> -1/4.
Step 6: Apply T to -1/4 -> -1/4 + 1 = 3/4.
Step 7: Apply R to 3/4 -> 4/3.
Step 8: Apply T to 4/3 -> 4/3 + 1 = 7/3.
Step 9: Apply R to 7/3 -> -3/7.
The sequence of numbers produced by applying the operations T, R, R, T, R, T, R, T, R starting with 2 is:
2, 3, -1/3, 3, -1/4, 3/4, 4/3, 7/3, -3/7.
b. To find a sequence of operations which turns 3 into 0, we can follow these steps:
Step 1: Apply R to 3 -> -1/3.
Step 2: Apply T to -1/3 -> -1/3 + 1 = 2/3.
Step 3: Apply R to 2/3 -> 3/2.
Step 4: Apply T to 3/2 -> 3/2 + 1 = 5/2.
Step 5: Apply R to 5/2 -> -2/5.
Step 6: Apply T to -2/5 -> -2/5 + 1 = 3/5.
Step 7: Apply R to 3/5 -> 5/3.
Step 8: Apply T to 5/3 -> 5/3 + 1 = 8/3.
Step 9: Apply R to 8/3 -> -3/8.
The sequence of operations that turns 3 into 0 is:
3, -1/3, 2/3, 3/2, 5/2, -2/5, 3/5, 5/3, 8/3, -3/8.
c. To explain how any positive integer n can be turned back into 0 by applying 3n-1 operations, we can use the following steps:
Step 1: Start with the positive integer n.
Step 2: Apply T to n successively for n times.
Step 3: Apply R to the result of Step 2.
Step 4: Repeat Steps 2 and 3 a total of (3n-1) times.
Since each application of T adds 1 to the number, after n applications of T, the number will be n. Then, applying R to n gives the negative reciprocal of n. By repeating Steps 2 and 3 (3n-1) times, we ensure that we are alternating between applying T and R to the number, eventually leading back to 0.
d. To explain how 0 can be turned into any negative integer, we can use the following steps:
Step 1: Start with 0.
Step 2: Apply R to 0.
Step 3: Apply T to the result of Step 2.
Step 4: Repeat Steps 2 and 3 a total of (|n|-1) times, where n is the negative integer we want to turn 0 into.
Since applying R to 0 is not defined, we cannot directly turn 0 into a negative number. However, we can start with 0 and alternate between applying R and T to obtain the negative reciprocal of 0 (which is undefined) and then add 1 repeatedly to get closer to the desired negative integer. By repeating Steps 2 and 3 (|n|-1) times, we ensure that we are alternating between applying R and T and eventually reaching the desired negative integer.