Consider the following number:
$$x=0.23434343434\dots$$
My question is whether this number is rational or irrational, and how can I make sure that a specific number is rational if it was written in decimal form.
Also, is $0.234$ rational or irrational?
I believe it is more interesting to prove a slighly more general result, which answers the question asked directly :
Any number with a finite repeating pattern is rational.
Proof : Say $X = a.u_1u_2u_3...u_n\overline{v_1v_2v_3...v_d}$ where $a$ is a whole number, $u_1,u_2,u_3,...,u_n, v_1,v_2,v_3,...,v_d$ are digits and $\overline{v_1v_2v_3...v_d}$ means that the pattern $v_1v_2v_3...v_d$ is repeating.
Then, $X$ can be written as $a+\dfrac{u_1u_2u_3...u_n}{10^n}+\dfrac{v_1v_2v_3...v_d}{10^n(10^d-1)}$.
Hence, $X$ is rational.
In your example, in $0.234343434...$, $a=0, u_1 = 2, n=1, v_1v_2 = 34$ and $d = 2$.
So $0.234343434... = 0 +\dfrac{2}{10}+\dfrac{34}{990}$.
Let $y = .3434\ldots$. Then, $100y = 34.3434\ldots$. So $100y-y = 99y = 34$. Hence, $y = \frac{34}{99}$.
Since $0.03434\ldots$ is just the above divided by $10$, we have
$$0.2343434\ldots = 0.2 + \frac{34}{990}$$
which is certainly rational.
We have $100x=23.434343\cdots$ and $10000x=2343.434343\cdots$ so $$9900x=2320$$ hence $x$ is rational and
$$x=\frac{2320}{9900}=\frac{116}{495}$$
$\begin{array}{r}-\\\ \end{array} \begin{align} 1000x & = 234.3434...\\ 10x & = \phantom{00}2.3434...\\ \hline 990x & = 232 \end{align}$
$x = 232/990=116/495$. Thus $x$ can be represented as a quotient of two integers with denominator not equal to zero, so it is a rational number.
Every number which can be written in a decimal notation in such a way that from some point on, there exists a subsequence that repeats itself, is rational.
For example, $1=1.0000000000000000000000...$ is rational as the zero repeats until infinity. $1/3=0.33333333333333333333...$ is also rational, as the three repeats on and on.
Also rational is the number $0.4872823482342934765656565656565656565656$, because the $56$ begins to repeat at some point.
In your specific case, it isn't hard to see that $0.234343434...$ is rational, since, if $t=0.234343434...$, then $100t = 23.434343434...$, so $$99t = 100t - t = 23.2000000000000\dots = \frac{242}{10} = \frac{116}{5}$$ meaning that $$t=\frac{\frac{116}{5}}{99} = \frac{116}{495}$$ which is rational.
$0.2\overline{34} = {1 \over 10} (2+0.\overline{34})$, and $0.\overline{34} = 100(0.\overline{34})-34$, so $99 (0.\overline{34}) = 34$, from which we get $0.2\overline{34} = {1 \over 10} (2+{34 \over 99}) = { 232 \over 990}$. Since $\gcd(232,990) = 2$, we get $0.2\overline{34} = { 116 \over 495}$.
anything that can be written as a fraction of two integers 23/99 or whatever is rational
anything that cannot be written as the fraction of two integers like the sqare root of most numbers most infinite sums eg. pi. the logarithm, and most limits are not rational
$$1.41421...=\sqrt{2} $$
$$1.100100001... = \sum_{n=0}^{\infty} 10^\left (-1 (n^2 )\right )$$
why is that you ask ?
because mathematicians thrive to have the inverse operations to all operations possible,
the inversion for every addition and subtraction, multiplication and division for rational numbers is possible with rational numbers themself
as soon as you look at square numbers, you see that there are big leaps: 1,4,9
so what is the square root of 2,3 or 5,6,7,8 ?
as multiplying and squaring with fractions is multipying numerator with numerator and denumerator with denumerator, which are as defined above also whole number and result in whole numbers, we need to search for a fraction that results in one of the above numbers
but as fractions can be written in shortened form with common primefactors canceled out, it is possible to prove that there can be no fraction that results in 2,3 or any other whole number that is not a square number
