# modulus of a complex number

#### arangu1508

In one of the complex number problems which I was working with

It was given as follows:

|z^3 + z^-3| <= |z|^3 + 1/[|z|^3]

How is it?

I am not able to figure it out.

guidance is required.

with regards

aranga

#### Plato

MHF Helper
It was given as follows:
$$\displaystyle |z^3 + z^{-3}| <= |z|^3 + 1/[|z|^3]$$
How is it?
If you are studying complex numbers, then you know that for all numbers $$\displaystyle z~\&~w$$:
the triangle inequality holds $$\displaystyle |z+w|\le|z|+|w|$$,
$$\displaystyle |z^3|=|z|^3$$, and $$\displaystyle \left|\dfrac{1}{w}\right|=\dfrac{1}{|w|}(w\ne 0)$$.
By using those simple properties we can get your result.

• arangu1508 and topsquark

#### arangu1508

Thank you very much. It is very useful.

If you are studying complex numbers, then you know that for all numbers $$\displaystyle z~\&~w$$:
the triangle inequality holds $$\displaystyle |z+w|\le|z|+|w|$$,
$$\displaystyle |z^3|=|z|^3$$, and $$\displaystyle \left|\dfrac{1}{w}\right|=\dfrac{1}{|w|}(w\ne 0)$$.
By using those simple properties we can get your result.