TITLE: Why are zeros of functions so important?
QUESTION [6 upvotes]: I realise this is a general question. I am self-teaching mathematics and I have observed that the zeros of real and complex functions are of much interest.
Question: Why are the zeros of real or complex so important?
I was going to say that the zeros completely determine a function because the function can be factorised into factors, with each factor corresponding to one of the zeros. For example, $f(x) = (x-1)(x-3)$ is completely determined by its zeros at $x=1$ and $x=3$.
However this doesn't work for functions like $g(x) = (x-1)(x-3) + 2$.
I appreciate your patience with a question that might be naive.

REPLY [2 votes]: A general remark: Solving any equation
$$A=B$$
is equivalent to solving
$$A-B =0$$
as soon as $A$ and $B$ are anything which at least vaguely resembles mathematical objects that can be subtracted from each other. In particular, if $A-B$ is something which you can express as a function, your task is now to
$$\text{ find the zeros of } A-B.$$
Hence, more often than one thinks, "solving an equation" immediately reduces to "finding zeros".
Now one could ask why solving equations is important, but I trust you don't ask that if you have ever worked in any STEM field. Or you and I trust people who claim that equations like
$$\mathbf F + \mathbf{u} \frac{\mathrm{d} m}{\mathrm{d}t} = m {\mathrm{d} \mathbf v \over \mathrm{d}t}$$
or
$$\oint_C \mathbf{B} \cdot \mathrm{d}\boldsymbol{l} =  \iint_S \left( \mu_0 \mathbf{J}+ \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\right) \cdot \mathrm{d} \mathbf{S}$$
or
$$G_{\mu \nu} = R_{\mu \nu} - \frac{1}{2} R g_{\mu \nu}$$
have importance, and it's useful to solve them occasionally, in order to hit a target or get a motor running or make a GPS useful or estimate how old the universe is.