You are watching: There are more rational numbers than irrational numbers

I think the many intuitive explanation I have heard is come considering writing down a rational number in decimal form. This way that either it is a repeating decimal or a end decimal, for instance $2.3737overline37$ or $0.42$, i m sorry we will write as $0.4200ar0$. Now, think about the probability that randomly composing down a number. So you have ten alternatives every time you walk to place a digit down. How likely is it that you will just "happen" to acquire a repeating decimal or a decimal where you only have actually zeros ~ a particular point?

*Very*unlikely. Fine those unlikely instances are the reasonable numbers and the "likely" ones are the irrational.

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

That would be true if there was constantly *exactly one* rational in between two irrationals, and also *exactly one* irrational in between two rationals, but that is obviously no the case.

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

And yes, this really becomes less and also less intuitive the more you think about it -- yet it appears to be the just reasonable way hunterriverpei.comematics can fit together nevertheless.

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The wording in the question, "it seems choose it should be separation pretty evenly", is fairly appropriate. I think the lesson to be learned here is that periodically when our intuition says something "seems like" it need to be true, a much more careful evaluation shows that intuition to be wrong. This happens quite regularly in hunterriverpei.comematics, for example space-filling curves, cyclic poll paradoxes, and measure-concentration in high dimensions. It additionally happens in other situations, for instance

I"ve viewed a video clip of a play in an (American) football game where a player, carrying the round forward, throws that back, end his shoulder, to a teammate to run behind him. The referee ruled the this was a forward pass. Intuition claims that throw the ball earlier over your shoulder is not "forward". Yet in fact, as the video clip shows, the teammate captured the ball at a location additional forward than where the very first player threw it. If you"re to run forward with rate $v$ and also you litter the ball "backward" (relative come yourself) with speed $w