What's between an irrational and rational number? [closed]

There is a rational number between two irrational numbers, and an irrational number between two rational numbers. So what's between an irrational and rational number?

I know about rational numbers being found between 2 real numbers but I don't know how it applies to this.

• between distinct REAL numbers there are infinitely many rational and irrational numbers. – sranthrop Jul 18 '17 at 22:16

Rational numbers and irrational numbers are real numbers.

For any $x,y\in{\bf R}$ with $x<y$, there exists rational $a$ and irrational $b$ such that $a,b\in(x,y)$ by density:

What does it mean for rational numbers to be "dense in the reals?"

Proof that the set of irrational numbers is dense in reals

Both rational and irrational numbers are dense in the real line. This means for any two real numbers (rational or irrational), there is at least one rational number and at least one irrational number between them. It follows from this fact that there are infinitely many rational and irrational numbers between each pair of real numbers.

This can be a difficult concept to grasp, but as mentioned in the answers above, if you pick any two real numbers, there will be a rational number between them. Moreover, there will also be an irrational number between them.

This also implies that you can find a rational number as close to an irrational number as you want (without being equal). For examples, the number $$3.14159265358979323846264338327950288$$ is a rational number which is less than $1 \times 10^{-35}$ bigger than $\pi$. Using an approximation of $\pi$ using even more decimal, you would also find even more rational numbers that are between that number and $\pi$. In fact, you could also imagine more irrational numbers between them. Here's a process how to find one. You take the number $\pi$ which as an infinite decimal expansion that never repeats, and you change $1$ digit really far in the expansion, but fixing everything else. In that way, you would obtain an other irrational number between the number above and $\pi$.

As you might have realized, there is nothing special about the numbers that I picked above, any two numbers will have that property. This is due to the fact that both rationals and irrationals are dense in the real number.

However, on result which is even more interesting in this area, and I hope you'll soon see and understand the result is that even though both sets are infinite, there is immensely more irrational numbers than there are rational numbers.

You can actually see a longer discussion on that subject on the following post:

Cardinality of the Irrationals
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