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I just want to know if it's necessary, I'm not asking for the proof. Can we skip the proof of it if we know that's true in $\mathbb{R}$?

GniruT
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2 Answers2

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No, we can't skip it. We have to know something about the elements in $\mathbb{C}$ that aren't in $\mathbb{R}$, and something about the absolute value sign $|\cdot|$ to conclude. There are certainly functions on $\mathbb{C}$ that are equal on $\mathbb{R}$ but not equal on $\mathbb{C}$, so (without additional information) we can't conclude two functions are equal after restricting them to $\mathbb{R}$.

hunter
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  • (I should maybe add that this result doesn't use very much about $|\cdot|$, so the proof is easy, but your question was whether we could logically "skip" it just because we know it for $\mathbb{R}$.) – hunter Nov 11 '15 at 13:00
  • Exactly, I just wanted to know if we can skip the proof. The fact is that in my book on the chapter about Complex Numbers, sometimes is said that we can skip the proof since we know it in $\mathbb{R}$. This is not tha case. – GniruT Nov 11 '15 at 13:03
  • Are you sure they said "since we know it in $\mathbb{R}$" and not "since we know it in $\mathbb{R}^2$"? – hunter Nov 11 '15 at 13:11
  • I was not thinking about this property. For example it's said that the property $||z|-|w|| \leq |z-w| $ "can be proofed like in $\mathbb{R} $" – GniruT Nov 11 '15 at 13:15
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    They don't mean that you can skip it because it is true in $\mathbb{R}$, or that it is automatically true because it is true in $\mathbb{R}$. They mean that the proof is similar enough to the proof in $\mathbb{R}$ that they are confident that the reader can fill in the details if they need to. – Dylan Nov 12 '15 at 13:39
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Yes, it is necessary. You are asking if $|x|=|-x|$ holds for complex numbers if it holds real numbers, but if you don't know something special about the absolute value function, say $|ab|=|a|\cdot |b|$, then the question is roughly the same as asking "Since $x^2$ is a positive real number whenever the input is a real number, is $x^2$ a positive real number for all complex inputs?"

Knowing what a function does at some collection of points tells you nothing about what it does at other points unless you have additional information about the function.

Aaron
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