If $\Bbb R$ means all real number, then what does $\Bbb R^2$ mean? [closed]
I am sort of baffled by this thing, already real number has every thing in it why is this concept of $\Bbb R^2$ ? What does it mean? What is its advantage?
$\endgroup$4 Answers
$\begingroup$$\Bbb R$ is the set of real numbers. That is, $\Bbb R=\{x:\text{$x$ is a real number}\}$.
$\Bbb R^2$ is the set of pairs of real numbers. That is, $\Bbb R^2=\Bbb R\times\Bbb R=\{(x,y):\text{$x$ and $y$ are real numbers}\}$.
$\endgroup$ $\begingroup$the Cartesian product of the reals with itself is a common meaning.
$\endgroup$ $\begingroup$We usually use $\mathbb{R}$, the set of real numbers, to refer to what we picture as the number line.
Thus, $\mathbb{R}^2$, the set of pairs of real numbers, is what we picture as the $xy$-plane, coordinatized by two number lines.
If you want to talk about space with $n$ dimensions, then you want a "copy" of $\mathbb{R}$ for each dimension, to give you independent coordinates.
$\endgroup$ 0 $\begingroup$$\mathbb R $ doesn't have everything. For example, it doesn't have an $x $ such that $x^2 = -1$. The complex numbers contain such objects.
$\mathbb R^2$ can be used for a lot of things. The most obvious is to describe points in a plane.
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