Modulus z - 1 Less than Modulus z + 1 iff Real z Greater than Zero

Theorem

Let $z \in \C$ be a complex number.

Then:

$\cmod {z - 1} < \cmod {z + 1} \iff \map \Re z > 0$

$\ds \cmod {z - 1}$

$<$

$\ds \cmod {z + 1}$

$\ds \leadstoandfrom \ \ $

$\ds \cmod {z + 1}$

$>$

$\ds \cmod {z - 1}$

$\ds \leadstoandfrom \ \ $

$\ds \paren {x + 1}^2 + y^2$

$>$

$\ds \paren {x - 1}^2 + y^2$

$\ds \leadstoandfrom \ \ $

$\ds \paren {x + 1}^2$

$>$

$\ds \paren {x - 1}^2$

$\ds \leadstoandfrom \ \ $

$\ds x^2 + 2 x + 1$

$>$

$\ds x^2 - 2 x + 1$

$\ds \leadstoandfrom \ \ $

$\ds 4 x$

$>$

$\ds 0$

$\ds \leadstoandfrom \ \ $

$\ds x$

$>$

$\ds 0$

$\ds \leadstoandfrom \ \ $

$\ds \Re z$

$>$

$\ds 0$

Definition of Real Part

$\blacksquare$

1981: Murray R. Spiegel: Theory and Problems of Complex Variables (SI ed.) ... (previous) ... (next): $1$: Complex Numbers: Supplementary Problems: Miscellaneous Problems: $144$