Intuitive explanation for how could there be "more" irrational numbers than rational? [duplicate]

I've been told that the rational numbers from zero to one form a countable infinity, while the irrational ones form an uncountable infinity, which is in some sense "larger". But how could that be? There is always a rational between two irrationals, and always an irrational between two rationals, so it seems like it should be split pretty evenly.

Solution 1:

I think the most intuitive explanation I have heard is to considering writing down a rational number in decimal form. This means that either it is a repeating decimal or a terminating decimal, for example $2.3737\overline{37}$ or $0.42$, which we will write as $0.4200\bar{0}$. Now, consider the probability of randomly writing down a number. So you have ten options every time you go to place a digit down. How likely is it that you will just "happen" to get a repeating decimal or a decimal where you only have zeros after a certain point? Very unlikely. Well those unlikely cases are the rational numbers and the "likely" ones are the irrational.

Solution 2:

There is always a rational between two irrationals, and always an irrational between two rationals, so it seems like it should be split pretty evenly.

That would be true if there was always exactly one rational between two irrationals, and exactly one irrational between two rationals, but that is obviously not the case.

In fact there are more irrationals between any two (different) rationals than there are rationals between any two irrationals -- even though neither set can be empty one is still always larger than the other.

And yes, this really becomes less and less intuitive the more you think about it -- but it seems to be the only reasonable way mathematics can fit together nevertheless.

Solution 3:

The wording in the question, "it seems like it should be split pretty evenly", is quite appropriate. I think the lesson to be learned here is that sometimes when our intuition says something "seems like" it should be true, a more careful analysis shows that intuition was wrong. This happens quite often in mathematics, for example space-filling curves, cyclic voting paradoxes, and measure-concentration in high dimensions. It also happens in other situations, for example [skip the next paragraph if you want only mathematics]:

I've seen a video of a play in an (American) football game where a player, carrying the ball forward, throws it back, over his shoulder, to a teammate running behind him. The referee ruled that this was a forward pass. Intuition says that throwing the ball back over your shoulder is not "forward". But in fact, as the video shows, the teammate caught the ball at a location further forward than where the first player threw it. If you're running forward with speed $v$ and you throw the ball "backward" (relative to yourself) with speed $w<v$, then the ball is still moving forward (relative to the ground) with speed $w-v$. So the referee was quite right.

In my opinion, the possibility of contradicting intuition is one of the great benefits of mathematical reasoning. Mathematics doesn't merely confirm what we naturally know but sometimes corrects what we think we know.

Intuition is a wonderful thing. It's fast and usually gives good results, so it's very valuable when we don't have the time or energy or ability for a more thoughtful analysis. But we should bear in mind that intuition is not infallible and that more careful thought can provide new insights and corrections.

Solution 4:

I think the key is what Henning mentioned, that is,

There are more irrationals between any two different rationals than there are rationals between any two irrationals.

However, to gain some more intuition imagine binary expansion of a rational and irrational numbers, for example

\begin{align} r_1 &= 0.00110100010\overline{0010101010} \\ i_1 &= 0.001101000100010101010 ??? \\ r_2 &= 0.00110100010\overline{0011111010} \\ i_2 &= 0.001101000100110101010 ??? \end{align}

The expansion of rationals are periodic, so every rational can be specified with finite amount of information (bits). You can make it as long as you want, but it still will be finite. However, even if some irrationals can be specified using finite amount of information (e.g. $0.101001000100001000001\ldots$), there are infinitely many more of those that carry infinite amount of information (there are infinitely many more of those even between any two rationals). We cannot specify them$^\dagger$ (how would we?), but we know they exists and exactly those unspecifiable numbers make that there are more of irrationals than rationals.

I hope this helps $\ddot\smile$

$^\dagger$ Here I'm assuming that we are using a language with finite or even countable number of symbols. On the other hand, for example, if we were to treat physical measures like "length of this stick" or "velocity of that bird" as specifications, and given that our world is continuous instead of discrete, then, almost surely, any such definition would be one of irrational number.

Solution 5:

See Cantor's diagonal argument. There are infinitely more reals between each "lattice point" of a rational. This is the basis for having multiple levels of infinity.