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.

Representations that use ... to denote an infinite sequence often trick the mind into thinking the infinite sequence will behave like a finite one. They don't.
The fraction series are a neat way to give you an idea of the value of an irrational. If cut the sequence somewhere and compute the result of that finite sum, you will get an approximation of the irrational number you are looking for. If you add the next term of the sequence, you will obtain a (usually) better approximation. This process can be repeated as many times as you like and you will get arbitrarily close to the value of the irrational. The limit of the sum to infinity is equal in the strictest sense to the irrational number. However, whether an infinite sum is equal to its limit is more a philosophical question than a mathematical one. This is often defined to be true for convenience.
In general the limit of a sequence isn't always defined because the sequence might not converge. Maybe this makes it easier to accept that sequences might converge towards things that have properties absent of the elements of the sequence.
