TITLE: How to explain irrational numbers to laymen?
QUESTION [9 upvotes]: I am trying to describe how irrational numbers, which are all modeled as a series of fractions, can themselves not be fractions, and are instead part of a unique group of "decimal numbers" outside of fractions, called the irrational numbers. I am confused atm.

From Wikipedia, some example irrational numbers include:

$\sqrt 2$
the golden ratio
The sqrt of all natural numbers which are not perfect squares
Logarithms

Then they say:

Almost all irrational numbers are transcendental and all real transcendental numbers are irrational. Examples include $e^\pi$.

Rational numbers are fractions, which are included in the set of irrational numbers. Irrational numbers, however, are decimals and include things that "can't be represented as fractions" it seems.
But where I'm confused is, sqrt 2 can be represented by a series of fractions:
$${\displaystyle {\sqrt {2}}=\prod _{k=0}^{\infty }{\frac {(4k+2)^{2}}{(4k+1)(4k+3)}}=\left({\frac {2\cdot 2}{1\cdot 3}}\right)\left({\frac {6\cdot 6}{5\cdot 7}}\right)\left({\frac {10\cdot 10}{9\cdot 11}}\right)\left({\frac {14\cdot 14}{13\cdot 15}}\right)\cdots }$$
Similarly, $\pi$ can be represented by a series of fractions:
$${\displaystyle 1\,-\,{\frac {1}{3}}\,+\,{\frac {1}{5}}\,-\,{\frac {1}{7}}\,+\,{\frac {1}{9}}\,-\,\cdots \,=\,{\frac {\pi }{4}}.}$$
Finally, the natural logarithm can be written as a series of fractions:
$${\displaystyle \ln(1+x)=\sum _{k=1}^{\infty }{\frac {(-1)^{k-1}}{k}}x^{k}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\cdots }$$
It has been a while since I have added/divided/subtracted/multiplied fractions, but from what I remember doing any of those operations results in a new fraction. So I'm wondering what I'm missing when it comes to understanding irrational numbers. If irrational numbers can represent non-fraction numbers, yet they are themselves represented by a series of fractions, it seems the result of the series would itself be a fraction, and so the irrational numbers are all rational numbers. Looking for an understanding of how to explain the difference between rational and irrational numbers. I tried saying "irrational are decimal numbers you can't represent with a fraction", but then when getting into the definition of a rational numbers (fraction numbers), I was unable to explain how if all irrational numbers are themselves definable as a series of fractions, how they themselves aren't representable as fractions. Thank you for your help.

REPLY [2 votes]: Just because you have a sequence $a_n$ that gets closer and closer to $a$, that doesn't mean $a_n$ and $a$ have to be the same type. For instance, you can have a sequence of polygons that gets closer and closer to a circle, but the circle isn't a polygon. If you have a sequence of staircases where the size of each step gets smaller and smaller, they will get closer and close to a diagonal line, even though none of the staircases is a diagonal line. If you take a number and take half of it, then half of that, and so on, then every number you get will be positive, but you will get closer and closer to zero, which is not positive.