I am inexperienced with math and I need help with absolute values and complex numbers.
Given the following: $$ |a||b|=|ab| $$
Assuming $a$ and $b$ are real, is the following statement true? $$ (|a|+|b|)|a+b|=|a||a+b|+|b||a+b|=|a(a+b)|+|b(a+b)|=|a^2+ab|+|b^2+ab| $$
How can this be simplified?
How about the complex case (say $a=re^{i\phi}$ and $b=se^{i\theta}$)?
Edit: I am trying to prove the following: $$ |a+b|\ge\frac{1}{2}(|a|+|b|)\left|\frac{a}{|b|}+\frac{b}{|b|}\right| $$
I have gotten this far but am stuck: $$ 2|a+b|\ge\left|r\left(1+e^{i(\theta+\phi)}\right)\right|+\left|s\left(1+e^{i(\theta+\phi)}\right)\right| $$
I would ask this as a separate question but math.stackexchange is prohibiting me from doing so because I am a new user and prohibited from asking more than 2 questions within 48 hours.