Why are there more irrational numbers than rational numbers?

[koldapu]

i already searched the forums and found a similar thread, but the answers REALLY confused me

im only in an algebra 2 class and am doing an extra credit assignment on whether there are more irrational or rational numbers

from countless websites, i know that there are more IRRATIONAL, but can anyone please explain why?

thank you

 

[Tide]

First, two sets have the same "size" (cardinality) if and only if there is a one to one correspondence between the elements of each.

Second, imagine that you could list ALL the real numbers. For our purposes, let's just work with numbers that contain only 0's and 1's and only consider numbers in the range 0 to 1. Your list might look like this:

(1) 0.11010100010001011110 ...
(2) 0.10010100101110001001 ...
(3) 0.11111111011111111110 ...

and so on (remember, each number has an infinite number of digits).

The left column (numbers in parentheses) are the natural numbers which set has the same cardinality (size) as the rational numbers and we have attempted to list ALL the real numbers in the right column.

Now the kicker is that no matter how hard you try there must always be numbers that you did not include in your list! To see this, we use the ingenius work of Georg Cantor (called Cantor's Diagonalization) and it works like this:

We're going to find a number that is NOT contained in the list. Look at the FIRST digit (after the decimal point) in the number corresponding to (1). It's a 1 so for our mystery number, call it M, select a 0 for its first digit. Whatever M turns out to be it is absolutely guaranteed NOT to be the same as the first number! So far we know M looks like 0.0? ... Now look at the SECOND digit of the SECOND number. It's a 0 so choose 1 for the second digit of M which now looks like 0.01?... and M is absolutely guaranteed NOT to the same as the second number in our list. Repeat for the THIRD digit of the THIRD number which is a 1 so select 0 for the third digit of M = 0.010?... and again it's guaranteed NOT to be the same as the third number.

Repeat this process for all the numbers in your supposed complete list of real numbers and what results is a number M that is guaranteed not to be ANY of the numbers in your list!

You could attempt to revise your list to include the new number M but then you could go through exactly the same arguments and find yet ANOTHER number that you left out! In other words, it is IMPOSSIBLE to find a one to one correspondence between the natural numbers and the set of real numbers so they cannot have the same cardinality (size)!

Voila!

Of course, the same argument works with numbers in our base ten system using all the digits 0-9.

 

[koldapu]

okay, a little confusing at first, but i got it

thanks!

 

[JonF]

The reason there are more irrationals then rational is you could find infinite irrationals between any two rational. For example.

Let’s take 1 and 2 for example

Irrationals between 1 and 2: 3/2, 4/3, 5/3, 5/4, 7/4, 6/5, 7/5, 8/5, 9/5…

 

[Tide]

The reason there are more irrationals then rational is you could find infinite irrationals between any two rational. For example.

Let’s take 1 and 2 for example

Irrationals between 1 and 2: 3/2, 4/3, 5/3, 5/4, 7/4, 6/5, 7/5, 8/5, 9/5…

But you could also find an infinity of rationals between any pair of irrationals too! Incidentally, the numbers you listed are rational!

 

[JonF]

yarg got mixed up on my terms... forgive me, it is late. I mean really rational, whole same idea really

 

[JonF]

Here is why the infinitely more irrationals then rationals makes sense to me. (note this might not be the most mathematically correct way to look at things)

For any rational number you can take the square root of that number. This will give an irrational number for any rational that isn’t composed of a perfect squares.

Well you can also take the cube root of any rational, and this will give an irrational for any number that isn’t a perfect cube.

You can take the 1 through nth root of an rational and get very large number of irrationals if n is very large. And if n is infinite you will get an infinite number of irrationals for each rational.

Then you can multiply this infinitely larger collection of irrationals by a each term in that infinitely collection of irrationals and get an even bigger collection of infinitely large irrationals.

Note: with the exceptions of 0 and 1 of course.

 

[Tide]

You're right - it's not quite mathematically correct! :-)

One thing you left out is raising irrational numbers to various powers. I don't believe, for example, that anyone even knows whether a number like $\pi ^{\pi}$ is rational or irrational. Raising irrational numbers to irrational powers may result in rationals as often as your scenario. And raising numbers to powers aren't the only operations you could perform.

 

[matt grime]

Here is why the infinitely more irrationals then rationals makes sense to me. (note this might not be the most mathematically correct way to look at things)

For any rational number you can take the square root of that number. This will give an irrational number for any rational that isn’t composed of a perfect squares.

Well you can also take the cube root of any rational, and this will give an irrational for any number that isn’t a perfect cube.

You can take the 1 through nth root of an rational and get very large number of irrationals if n is very large. And if n is infinite you will get an infinite number of irrationals for each rational.

Then you can multiply this infinitely larger collection of irrationals by a each term in that infinitely collection of irrationals and get an even bigger collection of infinitely large irrationals.

Note: with the exceptions of 0 and 1 of course.

erm, no that's not a good example either, since all you're describing is some subset of the algebraic numbers which are countable too. you also at one point take the infinite'th root of a number.of course as we all prove in lesson one of countability the countable union of countable sets is countable.

 

[nanajuice]

So, just as of recently I've been learning this and I'll try to explain it to you in such a way that it makes sense :) Ok?

So, if we were to randomly chose a real number...absolutely ANY real number...what are the chances of it being rational/irrational? Well , we'd be tempted to say 50-50 just because there are only two options, rational or irrational, but this is incorrect.

Let's try this...if we were to partner up with someone and sit there and ask them to randomly spout out numbers between 0-9 (of course without telling them what it is for) , if we were to just have them guessing numbers for an infinite amount of time, what are the chances of us getting a rational number? Not very likely! For us to get a rational number, we would have to randomly get a repetition in the decimal expression of this random real number! So most likely, you will get an irrational number that has no predictable repetition in it's decimal expression.

So , if we were to randomly chose a real number then we are much more likely to get an irrational number due to their excessive presence in the mathematical world!

So...based on what I've explained above there are obviously more irrationals! :)

If you have any more questions please let me know!
Grace Sulaiman

 

[JonF]

Oh the silly things I thought over six years ago...

 

[disregardthat]

I think most intuitive explanations that does not explicitly show that the reals are uncountable is going to fail, because any such argument most likely applies to the computable reals as well, which are countable. So any appeal to arbitrary constructions of real numbers will be a mathematically false explanation.

 

[Tac-Tics]

It depends what your teacher means by "more". Honestly.

There is an infinity of both. In this sense, there are an "equal" number of each.

A popular measure of the "size" of a set is cardinality, which (roughly speaking) talks about trying to pair up members of each set. The cardinality for finite sets is the same as the size of the set. There are also various cardinalities for infinite sets. Most sets we deal with are either the same cardinality as the integers or the same cardinality as the real numbers.

http://en.wikipedia.org/wiki/Cardinality

The rationals have the cardinality of the integers, while the irrationals have the cardinality of the reals. In this sense, there are "more" irrationals than rationals.

Cardinality is important in set theory, but is rarely considered in other branches of math.
Either way, this is WAAAY beyond algebra 2 :P