Why does $0.888888888889 \times 9 = 8$? So I'm teaching myself maths, watching alot of youtube videos about topics way beyond my head. I'm trying to unlearn the rigid way school taught maths, such as the rules and procedures to solve specific problem, which I hated. Feynman said maths is 'too abstract' and some book somewhere said  'math is about logical reasoning and pattern recognition', which I enjoy. I like to think logically and abstractly but dont have a way to show it or express it (except with coding and basic social skills). Anyway here's my question: Given
$0.888888888889 \times 9 = 8$
$0.88888888889 \times 9 = 8.00000000001$
$0.8888888889 \times 9 = 8.0000000001$
...
$0.8889 \times 9 = 8.0001$
$0.889 \times 9 = 8.001$
$0.89 \times 9 = 8.01$
First question why are eleven $8$'s the magic number for it to be a whole number again? What is the relationship between $8$'s and the $0$'s on the output when there are less than eleven $8$'s? Why are they the same when less than eleven? Then why is more the eleven $8$'s still equal $8$?
Any sources would be great, like what field of mathematics deals with these kinds of questions, its because the calculator blah blah cant calculate that high, the history behind it showing how these things work etc.
Also if this is too dumb of a question, how? (do you see things in your head that just makes sense of these symbols?) What are good math questions?  Why did you get into maths?
Again I just want to learn maths, working as business rule translator (software developer) is kinda boring and I feel like I'm just coasting, if i can understand maths better then hopefully I at least have a chance to learn/work on more interesting topics like theoretical computer science (algorithms specifically), theoretical physics (how the universe works), dynamical systems (how unpredictable phenomena work), bioinformatics etc.
 A: \begin{align}
& \big( 0.\underbrace{88888\ldots8888}\,9 \big) \times 9 \\[8pt]
= {} & {} + 7.2 \\
& {} + 0.72 \\
& {} + 0.072 \\
& {} + 0.0072 \\
& {} + 0.00072 \\
& {} + 0.000072 \\
& \,\,\vdots \\
& {} + 0.00000\ldots0072 \\
& {} + 0.00000\ldots00072 \\
& {} + 0.00000\ldots000081
\end{align}
Add these from right to left. In the righmost place you get a $1.$
Then you get $8+2=10,$ so you put down the $0$ and carry the $1.$
Then you get $1+7+2,$ so you put down the $0$ and carry the $1.$
Then you get $1+7+2,$ so you put down the $0$ and carry the $1.$
And so on.
You get
$$
8.\underbrace{00000\ldots0000}\,1
$$
There could be a million $0$s and then a $1.$ And a calculator doesn't show you a million $0$s. Calculators round when necessary.
A: You have discovered a calculator truth, not a mathematical truth.  Calculators and computers(when using the standard floating point numbers) only store numbers with a fixed number of significant figures.  Any difference smaller than that disappears in rounding.  If I do your calculation in Alpha I get $0.888888888889\times9=8.000000000001$ which is exactly correct.  Your calculator is rounding that off to $8$.  The field of numerical analysis studies these problems.  A large one is loss of precision when you subtract two similar numbers.  If you store decimal digits to six places and compute $3.1416-\pi$ you probably get $0.00001$, which has only one place of accuracy.  It has become much less of a practical problem with the change to $64$ bit computing because floating point numbers have much more precision.  The problems are still possible, but you need a much closer agreement between the numbers you are subtracting to have a problem.
