Solution 3.2:2b

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The inequality <math>0\leq \mathop{\rm Re} z \leq \mathop{\rm Im} z \leq 1</math> is actually several inequalities:
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The inequality <math>0\leq \mathrm{Re}z \leq \mathrm{Im}z \leq 1</math> is actually several inequalities:
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<math>\begin{align}
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:*<math>0 \leq \mathop{\rm Re} z \leq 1\,</math>,
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0 &\leq \mathrm{Re}z \leq 1,\\
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:*<math>0 \leq \mathop{\rm Im}z \leq 1\,</math>,
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0 &\leq \mathrm{Im}z \leq 1,\end{align}</math>
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:*<math>\mathop{\rm Re}z \leq \mathop{\rm Im}z\,\textrm{.}</math>
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<math>\mathrm{Re}z \leq \mathrm{Im}z
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</math>.
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The first two inequalities in this list define the unit square in the complex number plane.
The first two inequalities in this list define the unit square in the complex number plane.
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[[Image:3_2_2_b1.gif|center]]
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The last inequality says that the real part of <math>z</math> should be less than or equal to the imaginary part of <math>z</math>, I.e. <math>z</math> should lie to the left of the line <math>y=x</math> if <math>x=\mathrm{Re} z</math> and <math>y = \mathrm{Im} z</math>.
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The last inequality says that the real part of <math>z</math> should be less than or equal to the imaginary part of <math>z</math>, i.e. <math>z</math> should lie to the left of the line <math>y=x</math> if <math>x=\mathop{\rm Re} z</math> and <math>y = \mathop{\rm Im} z</math>.
[[Image:3_2_2_b2.gif|center]]
[[Image:3_2_2_b2.gif|center]]
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Current revision

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.

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