There are two parts that I am confused with, so any kind help will be greatly appreciated!
1. Can 4.56(with bar on top of 6) be expressed as a fraction? (If it can be, then it's a rational number right?)
If you haven't yet been shown the trick for these, here it is:
Given a decimal expansion such as [imath]0.333\overline{3}[/imath] (specifically, a decimal expansion with a one-digit repetition, and no other digits significant digits coming before the repeated part), you can assign the expansion a name (generally, [imath]x[/imath]), multiply this by [imath]10[/imath], subtract, and then divide:
[imath]\qquad x = 0.33333...[/imath]
[imath]\qquad 10x = 3.33333...[/imath]
[imath]\qquad \begin{array}{r} 10x = 3.33333\ldots \\ \underline{\phantom{10}x = 0.33333\ldots} \\ 9x = 3\phantom{.33333\ldots}\end{array}[/imath]
[imath]\qquad \dfrac{9x}{9} = \dfrac{3}{9}[/imath]
[imath]\qquad x = \dfrac{1}{3}[/imath]
In your case, you have [imath]4.5666\overline{6}[/imath]. There is just one digit in the repeated part, so multiply by [imath]10[/imath]. This will give you:
[imath]\qquad \begin{array}{r} 10x = 45.6666\ldots \\ \underline{\phantom{10}x = \phantom{5}4.5666\ldots} \\ 9x = 41.1\phantom{666\ldots}\end{array}[/imath]
Where does this lead? In particular, does it lead to a fraction?
2. Does the "56" in ".566666..." makes the number non-recurring? (Because an irrational number is non-terminating and non-recurring)
All that matters is that, eventually (in this case, right after the "5"), you get the same digit repeating forever. And that "dot, dot, dot" at the end of the "6666" means "forever after, in this manner". So the sixes never end. It's non-terminating, but it is certain not non-recurring.
