which is greater, the positive integer or the rational numbers?

Answer 1

Rational numbers. The positive integers is a subset of the rational numbers.

In other words, every positive integer is a rational number. But, every rational number isn't a positive integer.

Answer 2

Actually, the answer is complicated. All of the following are true:

- the set of positive integers and the set of rational numbers are both infinite.

- the set of positive integers is merely a subset of the rational numbers.

- BUT: there is a one-to-one correspondence between the set of positive integers and the set of rationals. So, in a very important sense, the number of rationals and the number of positive integers is EXACTLY the same, i.e. the SAME DEGREE OF INFINITY.

How can this be? First, let me consider only positive rational numbers. Once we get that fixed, including the negative rationals is only a factor of 2 and is really easy to fix.

a) Mapping from integers to rationals:
n => (1/2)^n
0 => 0
For every positive integer, there is one unique rational number. So clearly there cannot be more integers than rational numbers.

b) Mapping from rationals to integers:
If r = p/q, then

r => (2^p)*(3^q)
0 => 0

So clearly, for every positive rational number, there is a unique integer. So clearly, there cannot be more rational numbers than positive integers!

c) You're worried about the negative rationals? No sweat: I'll change the rule:
If r = p/q , where both p and q are positive,
r => (2^p)*(3^q)

If r = - p/q, where both p and q are positive,
r => (2^p)*(5^q)

0 => 0

The mapping is still unique.

Does this mean that ALL INFINITIES are the same? No. For example, it can be shown that the infinity of the real numbers is larger than the infinity of integers. (Real numbers include irrational numbers like sqrt(2) as well as the rational numbers of the form p/q.) I give a reference below.

Answer 3

greater in what sense?

magnitude/value? Quantity?

There are an infinite number of each.

2/3 < 4 < 101/3 < 217 ... its continuous... there are always numbers, both integer (which are rational) and non-integer rational, that will be bigger.

Answer 4

depends which are the numbers.
ex:
2<2.5
2.5=5/2
1>0.5
0.5=1/2

Answer 5

Both are infinite.

Answer 6

It all depends. Please add some more perimeters to your questions next time.