Why are there more irrational numbers than rational numbers?

Question

This is a H.W. question from my calc class.

I was trying to search up the answer and i found something that said that rational numbers and natural numbers are countable; while irrational numbers and real numbers are uncountable. I don't quite really get the definition specially if there can be an infinite amount of both

Answer 1

It's indeed true that the rational numbers are countable, meaning that you can literally list them, one by one, in a similar way to how you count the natural numbers: 1, 2, 3, etc. Of course, the list is infinite, but it is tractable.

The real numbers are uncountable, meaning that it's impossible to come up with a list (even an infinitely long one) that contains all the real numbers. This fact is usually proven by Cantor's diagonal argument (see the wikipedia link below).

Since the rational numbers are countable, but the real numbers are not, you can conclude that the irrational numbers are not countable. This is the sense in which there are "more" irrational numbers than rational numbers.

In general, there is an infinite hierarchy of infinities. See the link below on cardinal numbers.

Answer 2

For every rational number, you can create an infinite number of irrational numbers by adding an irrational number to it. using 1/17, for instance, 1/17+PI, 1/17+Euler's Constant, 1/17+sqrt(5), 1/17+sqrt(2), 1/17+sqrt(PI), etc. are all irrational. For every rational number you can think of, there is an infinite number of irrational numbers you can compose that would be unique from the numbers composed from a different starting rational number.

Answer 3

Oh dear, the mathematical concept of "infinity" has brought out the usual rash of completely wrong but reasonable-looking answers from well-meaning but uneducated Yahoo! Answerers. Kasner and Newman observed beautifully that "common sense is amid alien corn, in the land of the infinite".

You need to look at Cantor's "diagonal proof" that you can COUNT the infinite number of rational numbers, but you CANNOT count the infinite number of irrational numbers, therefore they are of DIFFERENT infinite sizes. None of the professional mathematicians of his time wanted to believe this at first, and they thought it was just a piece of trickery with words and diagrams. A decade or two passed before it was generally accepted.

The Mathworld Wolfram page is accurate, but difficult. You may find something easier by googling for "cantor diagonal proof".

Answer 4

I think because for every two consecutive numbers that can be rational, an infinite number of irrational numbers can be placed between them. But even still... one could say there are an infinite amount of BOTH types. THen you have to see which infinity is greater than the other. That stuff makes my head hurt.

Answer 5

Hmm... more? I'm not sure if there are more since both are infinite. I think you cannot establish a biyective function between both sets. You can establish biyective functions between sets that intuitively are of different size, but not between irrationals and rationals (i think).

Example: Function between natural numbers and multiples of 10:

F[n]=10 n, n is a natural number.

Inverse:

G[n]=n/10. n is a natural and multiple of 10.

Yet one set is ten times larger than the other (although both are infinite). Sometimes you cannot do that.

Answer 6

There are infinite amounts of numbers both rational and irrational.

Infinity is not a quantity, but a state of being which makes quantification impossible.

In other words, your question is moot, because they are not inequal in occurance.

Answer 7

how is infinite more than infinite?

In mathematics there's no such thing as "more x numbers than y numbers" because you can always keep counting them. Numbers are infinite, and as such, any particular type of numbers is always an infinite size group.

Answer 8

why are there more spanish irregular verbs than regular?
why did they run out of numbers so they now have imaginary numbers?
we may never know...