Recall the one-to-one and onto mapping of the set of natural numbers to the set of integers.

Question

We used this mapping to show that the integers were countably infinite. Find a function, f(n) that gives the nth integer of the given mapping below. You may find it useful to know that the floor(x) of a number is the largest integer that is less than or equal to x. You may use the floor in your final answer.

Examples: floor(4.3) = 4

floor(10.789) = 10

floor(8) = 8

floor(-2.6) = -3

Mapping:

N Z

1 0

2 1

3 -1

4 2

5 -2

6 3

7 -3

... ...

Answer 1

Notice that the absolute value of the integers is approximately twice as much as the corresponding natural numbers. Thus if we divide the natural numbers by 2 we get approximately the integer set.

N/2 Z

0.5 0

1 1

1.5 -1

2 2

2.5 -2

3 3

3.5 -3

If we take the floor function of the N/2 column, we get the absolute value of the integer set.

floor(N/2) Z

0 0

1 1

1 -1

2 2

2 -2

3 3

3 -3

The integers alternate from positive to negative, thus f(n) = floor(n/2) * (-1)ⁿ