So far have defined 4 simple sets: $\mathbf{N}, \mathbf{Z}, \mathbf{Q}$, and $\mathbf{R}$, and we can define operations and order based on 4 sets. But the moment we go, say $\mathbf{R} \times \mathbf{Z}$, none of these operations can be covered over. They must be redefined.
Even if $\mathbf{Z}$ is a ring , $\mathbf{Z} \times \mathbf{Z}$ need not be a ring . It will depend on how we define operations and order. For instance, we found that defining these terms as we do for $\mathbf{Q}$, which is defined base on $\mathbf{Z} \times \mathbf{Z}$, actually got "better", if acquired more operations and became a filed with a multiplicative inverse!
A Euclidean space is a real vector space which is equipped with a fixed symmetric bilinear form $\zeta: \mathbf{E} \times \mathbf{E} \rightarrow \mathbf{R}$, which is also positive definite ($\zeta (x,x) > 0$ for every $x \not= 0$).
The standard example of a Euclidean space is $\mathbf{R}^n$, it is defined under the inner product that
$(x_1, \dots, x_n) * (y_1, \dots, y_n)=x_1y_1 +x_2y_2 + \dots + x_ny_n.$
The real number $\zeta (x, y)$ is also called the inner product (or scalar product) of x and y.
$\zeta (x_1 + x_2, y) = \zeta (x_1, y) + \zeta (x_2, y)$
$\zeta (x, y_1 + y_2) = \zeta (x, y_1) + \zeta (x, y_2)$
$\zeta (\lambda x, y) = \lambda \zeta (x, y)$
$\zeta (x, \lambda y) = \lambda \zeta (x, y)$
$\zeta (x, y) = \lambda \zeta (y, x)$
$\mathbf{R} \times \mathbf{R}$ is Euclidean Space and it has addition operation:
'+' $\rightarrow (a_1, a_2) + (b_1, b_2) = (a_1+b_1, a_2+b_2)$
The same applies to Euclidean Space $\mathbf{R}^n$:
'+' $\rightarrow (a_1, a_2, \dots, a_n) + (b_1, b_2, \dots, b_n) = (a_1+b_1, a_2+b_2, \dots, a_n+b_n)$
Addition operation works well for Euclidean Space, how about multiplication operation? Let us define that $(a_1, a_2, \dots, a_n) \times (b_1, b_2, \dots, b_n) = (a_1b_1, a_2b_2, \dots, a_nb_n)$. If we have '0' element, we will get the result of '0' no matter what the multiple is. For example, $(0, a_2) \times (b_1, 0) = (0, 0) $. We cannot get the multiplication inversion. Thus, vector space is not a filed (in a field, only '0' has no multiplication inversion).
Euclidean Space has scalar operation:
$k \times (a_1, a_2, \dots, a_n) = (ka_1, ka_2, \dots, ka_n)$
We did so for $\mathbf{Z} \times \mathbf{Z}$, and we will use the saem mechanism here.
We could define a new field like $\mathbf{Q}$ as, say $\mathbf{B}: {(a,b), \dots}$. If we define $\mathbf{Q}$ as individual elements ($ (a,b) \times (c,d) = (ac+bc, bd)), we could run into issues, that is, defining '1' and '0' and mutliplication inversion.
Question: How to define it over individual elements of $\mathbf{R} ^ 2$?