It's very interesting to talk about zeros. Notice that the word 'zeros' is a plural, which suggests they are of many kinds.

What exactly are zeros? Zeros in mathematics are different from the 'concept' of zero. Pragmatically, zero refers to nothingness or emptiness. But theoretically, it needs some quality of rigor to formulate it. Zeros don't just spring up into existence, but are a result of careful synthesis of number systems.

First we define a set [math]S[/math] and a binary operation, addition ([math]+)[/math] on it, which is nothing but a function from [math]S^2[/math] to [math]S[/math]. Now we can define the additive identity of [math]S[/math], which we call 'zero of [math]S[/math]', as that element [math]z[/math] of [math]S[/math], for which [math]z+x = x+z = x, \forall x \in S[/math]. We can also denote [math]z[/math] by [math]0_S[/math].

From this, one thing is clear: The type of a zero depends on its containing set and the corresponding addition operator. That is, zeros vary from set to set, provided that set has a structure (an addition operator, for instance). Following are some examples of zeros.

Boolean set:

Set: [math]\mathbb{Z}_2 = \{True, False\}[/math]

Operation: [math]+_{\mathbb{Z}_2}[/math] = The [math]XOR[/math] operator

Zero: [math]0_{\mathbb{Z}_2} = False[/math]

Reason: [math]False ~XOR~ x = x~ \forall x \in \mathbb{Z}_2[/math]

Integers:

Set: [math]\mathbb{Z}[/math]

Operation: [math]+_\mathbb{Z}[/math] = The usual integer addition

Zero: [math]0_\mathbb{Z} = 0[/math] = The usual integer zero

Reason: [math]0 + x = x~ \forall x \in \mathbb{Z}[/math]

Rationals:

Set: [math]\mathbb{Q}[/math]

Operation: [math]+_\mathbb{Q}[/math] = The usual rational number addition

Zero: [math]0_\mathbb{Q} = (0, 1) = 0/1[/math]

Reason: [math](0/1) + (p/q) = (p/q)~ \forall (p/q) \in \mathbb{Q}[/math]

Reals:

Set: [math]\mathbb{R}[/math]

Operation: [math]+_\mathbb{R}[/math] = The usual real number addition

Zero: [math]0_\mathbb{R} = 0.000...[/math]

Reason: ([math]0.000...) + x = x ~\forall x \in \mathbb{R}[/math]

Complex Numbers:

Set: [math]\mathbb{C}[/math]

Operation: [math]+_\mathbb{C}[/math] = The usual complex number addition

Zero: [math]0_\mathbb{C} = (0_\mathbb{R}, 0_\mathbb{R}) = 0_\mathbb{R} + 0_\mathbb{R}i[/math]

Reason: [math](0_\mathbb{R} + 0_\mathbb{R}i) + (a+bi) = (a+bi) ~\forall (a+bi) \in \mathbb{C}[/math]

Power set of a set:

Set: [math]2^S := \{A | A \subseteq S\}[/math]

Operation: [math]+_{2^S}[/math] = The UNION operator, [math]\cup[/math]

Zero: [math]0_{2^S} = \emptyset[/math], the empty set

Reason: [math]\emptyset \cup A = A ~\forall A \in 2^S[/math]

The real plane (2-D vectors):

Set: [math]\mathbb{R}^2[/math]

Operation: [math]+_{\mathbb{R}^2}[/math] = The usual element-wise vector addition

Zero: [math]0_{\mathbb{R}^2} = \left(\begin{matrix} 0_\mathbb{R}\\ 0_\mathbb{R}\end{matrix}\right)[/math]

Reason: [math]\left(\begin{matrix} 0_\mathbb{R}\\ 0_\mathbb{R}\end{matrix}\right) +\left(\begin{matrix} x\\ y\end{matrix}\right) = \left(\begin{matrix} x\\ y\end{matrix}\right) ~\forall \left(\begin{matrix} x\\ y\end{matrix}\right) \in \mathbb{R}^2[/math]

[math]2 \times 2[/math] Matrices:

Set: [math]\mathbb{R}^{2 \times 2}[/math]

Operation: [math]+_{\mathbb{R}^{2\times 2}}[/math] = The usual element-wise matrix addition

Zero: [math]0_{\mathbb{R}^{2 \times 2}} = \left(\begin{matrix}0_\mathbb{R} & 0_\mathbb{R} \\ 0_\mathbb{R} & 0_\mathbb{R}\end{matrix}\right)[/math]

Reason: [math]0_{\mathbb{R}^{2 \times 2}} = \left(\begin{matrix} 0_\mathbb{R} & 0_\mathbb{R}\\ 0_\mathbb{R} & 0_\mathbb{R}\end{matrix}\right) +\left(\begin{matrix} w & x\\ y & z\end{matrix}\right) = \left(\begin{matrix} w & x\\ y & z\end{matrix}\right) ~\forall [/math][math]\left(\begin{matrix} w & x\\ y & z\end{matrix}\right) \in \mathbb{R}^{2 \times 2}[/math]

While you could assume [math]0_\mathbb{Z} = 0_\mathbb{Q} = 0_\mathbb{R} = 0_\mathbb{C} = 0[/math], since [math]\mathbb{Z} \subseteq \mathbb{Q} \subseteq \mathbb{R} \subseteq \mathbb{C}[/math], rest of the zeros are entirely different from each other. Specifically, the zero matrix [math]0_{\mathbb{R}^{2 \times 2}} = \left(\begin{matrix} 0 & 0 \\ 0 & 0\end{matrix}\right)[/math] is entirely different from the number [math]0[/math], although both act as additive identities for their respective containing sets.