# Zeros of complex functions - simple question

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.

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

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.

• “Complex zero” usually refers to the domain. For example, $f(x) = x^2 +1$ has no real zeros, but two complex zeros ($i$ and $-i$). Commented Dec 19, 2022 at 7:30

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 $$x$$.