The picture below describes a method of listing all positive rational numbers. W

Question

The picture below describes a method of listing all positive rational numbers. We simply follow the indicated path, listing each rational number in the order we hit it. Let us call the resulting sequence an. Thus, a1 = 1, a2 = 2, a3 = 1/2, a4 = 1/3, etc. What is the first n such that an = 5/3? This is not a one-to-one correspondence. Demonstrate this by finding values of n and m with n m such that an = am = 2/3. We may obtain a one-to-one correspondence by omitting all non-reduced fractions. What is the 14th fraction in this reduced list? Describe a listing of the set of all rational numbers, including negatives and 0.

Explanation / Answer

(a) n = 25

(b) for n = 8 value is 2/3

for n = 39 value = 4/6 = 2/3

So it is not one-to-one correspondence

(c)( 14th fraction in reduced list = 5

Similar to above diagram start from 0 and write all positive numbers on right side and all negative numbers on left side and similar to above procedur proceed in both directions to get all rational numbers

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