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I know this is easy one but I can't get to the bottom of it. A zero of a real function is when its value is equal to zero. But what is a complex zero? When the real part of its value is equal to zero OR when its imaginary part is equal to zero OR both must be zero? Thanks in advance.

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    $\begingroup$ A complex zero is a number $z$ such that $f(z) = 0$, where we mean $0 = 0 + 0i.$ $\endgroup$ Dec 18, 2022 at 21:53
  • $\begingroup$ You've answered my question! Thanks! So it's both. $\endgroup$
    – Uddie
    Dec 18, 2022 at 21:57
  • $\begingroup$ You can post your answer in the main section so I can approve it. $\endgroup$
    – Uddie
    Dec 18, 2022 at 22:10
  • $\begingroup$ Simple, but good question. $\endgroup$
    – K.defaoite
    Dec 18, 2022 at 23:06
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    $\begingroup$ “Complex zero” usually refers to the domain. For example, $f(x) = x^2 +1$ has no real zeros, but two complex zeros ($ i$ and $-i$). $\endgroup$
    – Martin R
    Dec 19, 2022 at 7:32

2 Answers 2

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I was reluctant to post an answer, but I guess I can give more details than I did in my comment.

We have to remember that the complex number system has two parts - the real and imaginary parts. Whenever we write the symbol $x$ to mean a real number in this system, we regard it as the number $x + 0i,$ that is, the number whose real part is $x$ and imaginary part is $0.$

So, when we refer to a complex zero of a function, we mean a number $z$ such that $f(z) = 0$. While this doesn't seem to fit, we need to remember that the symbol $0$ is really the complex number that serves as the additive identity in $\mathbb{C}.$ So, we mean $0 = 0 + 0i$ as the complex number whose real and imaginary parts are both zero.

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    $\begingroup$ “Complex zero” usually refers to the domain. For example, $f(x) = x^2 +1$ has no real zeros, but two complex zeros ($ i$ and $-i$). $\endgroup$
    – Martin R
    Dec 19, 2022 at 7:30
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I don't quite like the other answer. We do not merely "regard" a real $x$ as $x+0i$. Neither should we think of "$0$" as merely a symbol representing "the complex number that serves as the additive identity in $ℂ$".

In actual fact, when we construct $ℂ$ we want $ℝ ⊆ ℂ$ literally. The claim is that there exists a field $ℂ$ that directly contains $ℝ$ and has some $i∈ℂ$ such that $i^2+1 = 0$. Here, both the $0$ and $1$ are exactly the same zero and one in $ℝ$! That is the whole point of complex numbers! It is non-trivial, but such a field exists and when we write "$ℂ$" we literally mean such a field!

So $0 = 0+0i$, and there is no "interpretation" or "regarding" at all. We define a complex zero of a function $f$ on $ℂ$ to be some $z∈ℂ$ such that $f(z) = 0$. I want to emphasize again that this $0$ is exactly the same zero as the natural number $0$, or the real number $0$.

The only reason we call it "complex zero" is to distinguish it from "real zero", which refers to some $x∈ℝ$ such that $f(x) = 0$. Still the same $0$! The only difference is that we are looking for real $x$.

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