Question

I need help learning how to do this step by step for each one

Answer 1

(a).

`\begin{gathered} 0.125 \\ Conver\text{t to fraction} \\ 0.125=(125)/(1000)=(1)/(8) \end{gathered}`

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).

`0.\bar{\text{ 125}}\text{ }=0.125125125125125`

The bar on 125 indicates that it keeps repeating itself continuously

Indeed, this has a pattern and can be expressed as a fraction

`0.\bar{\text{ 125}}\text{ }=(125)/(999)`

Therefore, it is a Rational Number (since it can be written as a fraction)

Decimal Form = 0.125125125125125...

(d).

`0.1\bar{\text{25}}\text{ }=0.125252525252525252525252525.....`

The bar on 25 indicates that it keeps repeating itself continuously

Indeed, this has a pattern and can be expressed as a fraction

`\begin{gathered} 0.1\bar{\text{25}}\text{ }=0.1+0.025252525252525252525.... \\ 0.1\bar{\text{25}}\text{ }=(1)/(10)+(5)/(198) \\ 0.1\bar{\text{25}}\text{ }=(62)/(495) \end{gathered}`

Therefore, it is a Rational Number (since it can be written as a fraction)

Decimal Form = 0.1252525252525...