3.2222222... and 1.589 are rational numbers.
What are rational numbers?
The numbers that can be represented in [tex]\frac{p}{q}[/tex] form, where p and q are integers and q ≠ 0.
3.222222.... can be represented as [tex]\frac{p}{q}[/tex] form as:
let
x = 0.222222...
∴ 10x = 3.222222...
now,
10x - x = 3.22222... - 0.22222...
9x = 3
x = [tex]\frac{3}{9}[/tex] = [tex]\frac{1}{3}[/tex] (this is a [tex]\frac{p}{q}[/tex] form)
It means, 2 + x also can be written as [tex]\frac{p}{q}[/tex] form:
3 + [tex]\frac{1}{3}[/tex] = [tex]\frac{10}{3}[/tex]
Therefore 3.22222... is rational number.
Similarly 1.589 can be written in the form of [tex]\frac{p}{q}[/tex] form i.e., as:
1.589 x [tex]\frac{1000}{1000}[/tex] = [tex]\frac{1589}{1000}[/tex] (this is a [tex]\frac{p}{q}[/tex] form)
But 0.112123123412345... has not a finite decimal value, therefore its [tex]\frac{p}{q}[/tex] form can't be found.
Hence 3.2222222... and 1.589 are rational and 0.112123123412345... is an irrational real number.
To learn more about rational numbers, refer to the link:
https://brainly.com/question/17450097
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