Question

I know this might involve many things, but can someone give me a brief explanation on why are there more irrational numbers than real numbers if both are infinite?

Answer 1

same reason \[e^{i \pi} = -1\]

Answer 2

Could you go further?

Answer 3

lol that actually didnt make sense..

Answer 4

irrational numbers ARE real numbers...

Answer 5

irrationals are a subset of the real numbers

Answer 6

@TuringTest

Answer 7

But someone told me that or something like that... I don't get it

Answer 8

you'd rather believe that guy you met in real life rather than a random person in the internet with a profile pic of a green owl!!? o.O i think you should trust the random owl more..

Answer 9

I think it has something to do with this: "rational numbers are what is known as a countable set, one that can be matched with the natural numbers. However, the irrational numbers are uncountable, which is a larger amount."

Answer 10

Go with the Owl...

Answer 11

C'mon guys, this is a serious question. I just don't know how to prove it.

Answer 12

@MelindaR there is going to be an argument that given any two "closest" rational numbers you can always find another one (necessarily irrational) between. It will have something to do with Dedekind. And the idea will be that if you just magnify the number line, you can do that as many times as you like and you still have a number line. Populated mainly by irrationals.

Answer 13

But it's been answered. All real numbers includes all irrational numbers...so the number of irrational numbers cannot be larger than the number of real numbers.

Answer 14

I don't believe this. Are you absolutely sure.

Answer 15

Both go to infinity yes...but by definition, irrational numbers are a subset of real numbers.

Answer 16

@mahmit2012

Answer 17

OK I see, I misread the problem. Real = irrational + rational. But maybe the question means rational v. irrational. I get why you all think it's simple.

Answer 18

But there are more irrational than rational, right?

Answer 19

Yes

Answer 20

But the question says, real

Answer 21

Oh, I meant rational.

Answer 22

Maybe you should post it again, correctly?

Answer 23

Why are there more irrational numbers than rational numbers?

Answer 24

I'm not sure of the proof. I'm looking for someone who remembers it. But basically, there is a limit to the number of rational numbers in any interval. So you just blow up the number line until those marks get far apart. And there are lots of irrationals in between. (Dedekind, as I said). But I don't know the precise formulation. In particular, how to show that the rationals are limited within an interval. @jim_thompson5910

Answer 25

you cant, there are an infinite number of rationals in any interval

Answer 26

so the statement "there is a limit to the number of rational numbers in any interval" isn't true

Answer 27

the difference between the rationals and irrationals is that the rationals are countable while the irrationals are not

Answer 28

So there are not more irrational than rational in a given interval?

Answer 29

the rational set is countably infinite while the irrational set is uncountably infinite

Answer 30

there are more irrational (so to speak)

Answer 31

How so?

Answer 32

@MelindaR I'm not satisfied. But I'll have to get back to you.

Answer 33

if the two were the same, then we can say that the set of irrationals were countably infinite...which just isn't true

Answer 34

Well, you guys were making fun of me because of my typo... That's actually a hard question. I've tried reading about it but I didn't understand much.

Answer 35

i didn't mean to make fun

Answer 36

oh are you talking about previous posters?

Answer 37

Yes, it's not about you, Jim. Thanks a lot for your explanation

Answer 38

I don't think anyone was making fun of anyone

Answer 39

alright, just checking...you're welcome

Answer 40

@MelindaR I just meant, if it has an error, it's better to start fresh. Maybe more people would look. I think it's a great question.

Answer 41

Alright, Telliott. Thanks

Answer 42

Quote from
http://answers.yahoo.com/question/index?qid=20070906084238AAuj5ww

It turns out that there are actually more irrational numbers than rational numbers. The rationals are COUNTABLY infinite; the irrationals are UNCOUNTABLY infinite. This means that the set of irrational numbers has a cardinality called the "cardinality of the continuum," which is strictly greater than the cardinality of the set of natural numbers (i.e., the set {1,2,3,4,...}). The set of rational numbers has the same cardinality (number of elements) as the set of natural numbers, so there are more irrationals (numbers like pi and e) than rationals (numbers like 1/2, 3/4, etc).

But don't believe everythng you read on the internet!

Answer 43

@mahmit2012 any comment?

Answer 44

One more quote:
http://en.wikipedia.org/wiki/Irrational_number

Informally, this means that an irrational number cannot be represented as a simple fraction. Irrational numbers are those real numbers that cannot be represented as terminating or repeating decimals. As a consequence of Cantor's proof that the real numbers are uncountable (and the rationals countable) it follows that almost all real numbers are irrational.[1]