What are the machine numbers immediately to the right and left of 2^m? How far is each from 2^m?
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To find the machine numbers immediately to the right and left of 2^m and determine the distance from 2^m, we need to consider the binary representation of the numbers.
In binary representation, 2^m is represented as a 1 followed by m zeroes. For example, if m = 4, then 2^m = 16 which is written as 10000 in binary.
The number immediately to the left of 2^m can be obtained by taking the binary representation of 2^m and subtracting 1 from it. This is because subtracting 1 from a binary number flips all the bits to the right of the rightmost 1, and if there are any 1s to the left, it leaves them unchanged. So, for example, if m = 4, then the number to the left of 2^m (16) is 1111 in binary, which is 15 in decimal.
Conversely, the number immediately to the right of 2^m can be obtained by taking the binary representation of 2^m and adding 1 to it. So, using the same example, the number to the right of 2^m (16) is 10001 in binary, which is 17 in decimal.
To calculate the distance of each of these numbers from 2^m, we subtract 2^m from each of the numbers. In the example above:
- The distance between 15 and 16 is 16 - 15 = 1.
- The distance between 17 and 16 is 17 - 16 = 1.
So, for any positive integer m, the machine numbers immediately to the left and right of 2^m are 2^m - 1 and 2^m + 1, respectively, and the distance from 2^m to each of these numbers is 1.