Cannot get the absolute value of a complex number

Questions
1

I am trying to evaluate an expression using “Abs” but for some reason it does not simplify beyond a point. Am I doing something wrong in here? My command is:

Refine[Abs[k1*I + 1], Element[k1, PositiveReals]]

The output I am getting:

Abs[1 + I k1]

I expected:

Sqrt[1+k1^2]
2

Try:

 Simplify[Abs[k*I + 1] // ComplexExpand, Assumptions -> Element[k, PositiveReals]]
 (* Sqrt[1 + k^2] *)
3

You can also just use ComplexExpand because it assumes all variables are real unless specified as complex:

ComplexExpand[Abs[k1*I + 1]]

Mariusz’s post is better if there are further simplifications from the fact that k is Positive (which in your particular example does not matter).

Regards.