Solution
(a).
[tex]\begin{gathered} 0.125 \\ Conver\text{t to fraction} \\ 0.125=\frac{125}{1000}=\frac{1}{8} \end{gathered}[/tex]
Therefore, it is a Rational Number (since it can be written as a fraction)
Decimal Form = 0.125
(b). 0.125456...
Notice that the decimal fraction does not have any pattern and it has no end as well.
Therefore, it cannot be expressed as a fraction and thus an IRRATIONAL number
Decimal Form = 0.125456...
(c).
[tex]0.\bar{\text{ 125}}\text{ }=0.125125125125125[/tex]
The bar on 125 indicates that it keeps repeating itself continuously
Indeed, this has a pattern and can be expressed as a fraction
[tex]0.\bar{\text{ 125}}\text{ }=\frac{125}{999}[/tex]
Therefore, it is a Rational Number (since it can be written as a fraction)
Decimal Form = 0.125125125125125...
(d).
[tex]0.1\bar{\text{25}}\text{ }=0.125252525252525252525252525.....[/tex]
The bar on 25 indicates that it keeps repeating itself continuously
Indeed, this has a pattern and can be expressed as a fraction
[tex]\begin{gathered} 0.1\bar{\text{25}}\text{ }=0.1+0.025252525252525252525.... \\ 0.1\bar{\text{25}}\text{ }=\frac{1}{10}+\frac{5}{198} \\ 0.1\bar{\text{25}}\text{ }=\frac{62}{495} \end{gathered}[/tex]
Therefore, it is a Rational Number (since it can be written as a fraction)
Decimal Form = 0.1252525252525...
