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My textbook has the following to say about the Triangle Inequality:enter image description here

Now my question is regarding the general form. It says that the equality sign holds iff the ratio of any 2 non-zero terms is positive. But what does that mean? Is it talking about the terms on the RHS? If so, they are always positive so that can't be it.
But if it is talking about the terms in the LHS what exactly does a positive ratio mean? The ratio of any two complex numbers will be another complex number and as far as I can tell, through information on the net, there is no concept of negative and positive with regards to complex numbers.
I would really appreciate some help in understanding this.

Anili
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  • Need $z_i\overline{z_j}$ to be real and non-negative. Geometrically, the vectors are aligned in a straight line, because the angles cancel (think of complex numbers written as $re^{I\theta}$ and the compute the ratio. – Rishi Sonthalia Dec 29 '21 at 15:54
  • I'm sorry but I dont quite understand what exactly you mean. I haven't yet gotten to the part of my textbook where it talks about the notation you are referring to. I will get back to you when I do read it. Thanks for the help though. – Anili Dec 29 '21 at 16:01
  • @Anili "Positive" here stands for "real and positive" (since non-real complex numbers do not have a sign), so $,\dfrac{z_i}{z_j} \in \mathbb R^+,$ for all non-zero $,z_i, z_j,$. – dxiv Dec 29 '21 at 19:26
  • @dxiv Ah okay. So the ratio must simplify to a purely real complex number (i.e. a real number) and it must be +ve. Many thanks! – Anili Dec 30 '21 at 03:42

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