1)
An irrational number is a number that a) can't be written as a fraction of two whole numbers AND b) is an infinite decimal without any sort of pattern.
For the first answer choice, clearly [tex] \frac{1}{3} [/tex] does not pass the first criterion so we look at the second choice.
Let's come back to [tex] \sqrt{2} [/tex] and [tex] \pi [/tex].
[tex] \frac{2}{9} [/tex] doesn't meet our first criterion, and let's skip [tex] \sqrt{3} [/tex] for now.
It is often easier to disprove an irrational number than to prove one. There are a few famous irrationals to know (although there is an infinite number of irrationals). The most common are [tex] \sqrt{2}, \pi, e, \sqrt{3} [/tex]. For now, it's just helpful to know these and recognize them.
So we can check off [tex] \sqrt{2}, \pi[/tex] and [tex] \sqrt{3} [/tex].
2)
For this next question, we know that [tex] \sqrt{64} = 8[/tex]. Clearly this isn't irrational. Likewise, [tex] \frac{1}{2} [/tex] isn't irrational. [tex] \frac{16}{4} = \frac{4}{4} = 1[/tex], which is rational, leaving only [tex] \frac{ \sqrt{20}}{5} = \frac{2 \sqrt{5} }{5}[/tex]. By process of elimination, this is the correct answer. Indeed, [tex] \sqrt{5} [/tex] is an irrational number.
3) This notation means that we have 0.3636363636... and so on, to an infinite number of digits. It is called a repeating decimal.
But it can be written as a fraction because its pattern repeats, unlike for an irrational number.
Let's say [tex]x=0.36363636...[/tex]. Would you agree that [tex]100x=36.36363636...[/tex]? (We choose to multiply by 100 because there are two decimals that repeat. For 1, choose 10, for 3 choose 1,000, and so on.)
Now, let's subtract x from 100x and solve.
[tex]100x=36.36363636\\-x \ \ \ \ \ \ \ -0.36363636\\99x=36\\\\x= \dfrac{36}{99}= \dfrac{4}{11}[/tex]
Voila!
