How to explain irrational numbers to laymen? 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.

Cut a string so that its length is that of the diameter of the wheel. Wrap the string along the wheel's edge and make a mark at each endpoint. Then use it to measure the curve's length from the clockwise endpoint, going further in the clockwise direction a distance equal to the string's length, and make the next mark. And the next in the same way. And so on. After measuring $22$ times the length of the string, you've gone around the circle approximately seven times and you're close to your starting point, but not exactly at the same point. Keep going. Will you ever hit the same point twice? That $\pi$ is irrational just means that you won't.
