Aus Online Mathematik Brückenkurs 2

The inequality \displaystyle 0\leq \mathop{\rm Re} z \leq \mathop{\rm Im} z \leq 1 is actually several inequalities:

• \displaystyle 0 \leq \mathop{\rm Re} z \leq 1\,,
• \displaystyle 0 \leq \mathop{\rm Im}z \leq 1\,,
• \displaystyle \mathop{\rm Re}z \leq \mathop{\rm Im}z\,\textrm{.}

The first two inequalities in this list define the unit square in the complex number plane.

The last inequality says that the real part of \displaystyle z should be less than or equal to the imaginary part of \displaystyle z, i.e. \displaystyle z should lie to the left of the line \displaystyle y=x if \displaystyle x=\mathop{\rm Re} z and \displaystyle y = \mathop{\rm Im} z.

All together, the inequalities define the region which the unit square and the half-plane have in common: a triangle with corner points at \displaystyle 0, \displaystyle i and \displaystyle 1+i.