# Thread: Complex numbers - two inequality questions

1. ## Complex numbers - two inequality questions

1)

Prove that

$||z_1|-|z_2||\leq|z_1-z_2|$

2)

Show that if $|z|<1$

then

$|z-1|+|z+1|\leq2$

First one looks like the triangle inequality but i cant go further.

2. Hello, Nrt!

I believe there is a typo in #2.

2) Show that if $|z|\:<\: 1$

then: . $|z-1|+|z+1| \;\;{\color{red}\geq}\;\;2$

Since $|z| < 1,\:z$ is in the unit circle centered at the origin.

. . $\begin{array}{c}|z-1|\text{ is its distance from }A(1,0) \\ \\[-4mm] |z+1|\text{ is its distance from }B(\text{-}1,0) \end{array}\bigg\}\quad\text{ endpoints of a diameter, length 2}$

Code:
                |
o o o
o     | z   o
o       | ∆     o
o       *|   *    o
*   |     *
o  *      |       * o
- B∆ - - - - + - - - - ∆A -
o         |         o
|
o        |        o
o       |       o
o     |     o
o o o
|

From the triangle inequality: . $\overline{zA} + \overline{zB} \:\geq \:\overline{AB}$

. . Therefore: . $|z-1| + |z+1| \:\geq \:2$

3. Thank you all for help.

4. Can someone explain me the first question using complex numbers? I got the idea from Plato's link but couldn't get it done.

5. Originally Posted by Nrt
Can someone explain me the first question using complex numbers? I got the idea from Plato's link but couldn't get it done.
It is the exact same proof for complex numbers as for real real.
You must realize that $|z|$ is just a real number.

The triangle inequality holds for complex numbers: $\left| {z + w} \right| \leqslant \left| z \right| + \left| w \right|\;\& \,\left| {z - w} \right| = \left| {w - z} \right|$.

So you can get $- \left| {z - w} \right| \leqslant \left( {\left| z \right| - \left| w \right|} \right) \leqslant \left| {z - w} \right|$.

At this point you have nothing but real numbers and you use this fact.

If $a\ge 0$ and $-a\le b \le a$ then $|b|\le |a|$ or $\left| {\left| z \right| - \left| w \right|} \right| \leqslant \left| {z - w} \right|$.

See you let $a = \left| {z - w} \right|$ and $b = \left| z \right| - \left| w \right|$ noting that $\left| {\left| {z - w} \right|} \right| = \left| {z - w} \right|$.

6. Thanks mate. Now i can see it clearly.