surds

     $\frac{1}{1+\frac{1}{\sqrt2}}$

                     Rewrite the  1  in the denominator as  $\frac{\sqrt2}{\sqrt2}$ .

=  $\frac{1}{\frac{\sqrt2}{\sqrt2}+\frac{1}{\sqrt2}}$

                     Add the fractions in the denominator together.

=  $\frac{1}{\frac{\sqrt2+1}{\sqrt2}}$

                                That is the same as...

=  $1\div\frac{\sqrt2+1}{\sqrt2}$

                                Invert the second fraction and multiply.

=  $1\,\cdot\,\frac{\sqrt2}{\sqrt2+1}$

=  $\frac{\sqrt2}{\sqrt2+1}$

                                Multiply the numerator and denominator by  $\sqrt2-1$ .

=  $\frac{\sqrt2}{\sqrt2+1}\,\cdot\,\frac{\sqrt2-1}{\sqrt2-1}$

=  $\frac{2-\sqrt2}{2-1}$

=  $\frac{2-\sqrt2}{1}$

=  $2-\sqrt2$

Great job, hectictar !. Thank you.