Notes on Cartesian products, coordinate systems, and Euclidean space

Real numbers are points on the real line: each number x ∈ ℝ is a point on the number line.
In the scheme of mathematics, points can be seen as coordinates that encode an order or association of numbers.
The Cartesian product of two sets A and B is the set of all ordered pairs where the first element comes from A and the second from B.
Formal definition:
$A \times B = { (a,b) \mid a \in A,\; b \in B }.$
In particular, the Cartesian product of the real numbers with themselves yields the real plane:
$\mathbb{R} \times \mathbb{R} = { (x,y) \mid x \in \mathbb{R},\; y \in \mathbb{R} }.$
Geometric interpretation: each point in the plane corresponds to an ordered pair (x,y) of real numbers; coordinates provide a two-dimensional representation of the pair.

Points in higher-dimensional spaces are assigned coordinates as tuples: in ℝ^n, a point is represented by an n-tuple (x1, x2, …, x_n).
The standard coordinate system uses sequential axes; correct and consistent orientation is essential.
An intuitive way to think about orientation: if you move along the +z axis and look from above, you should see the x-axis and y-axis arranged in a consistent way (e.g., a right-handed orientation). If you perceive a different arrangement of the x and y axes, the coordinate system may be inconsistent or incorrectly oriented.
A common rule to check orientation in 3D is the right-hand rule: if you point your index finger along +x and your middle finger along +y, your thumb points along +z. This is a way to encode the handedness of the coordinate system.

On the real line (1D): the distance between two points x and y is
$d(x,y) = |x - y| = \sqrt{(x - y)^2}.$
This shows the use of the square root to obtain a nonnegative distance from the squared difference.
In higher dimensions (ℝ^n): the distance between two points p = (x1, …, xn) and q = (y1, …, yn) is the Euclidean distance:
$d(p,q) = \sqrt{ \sum{i=1}^{n} (xi - y_i)^2 }.$
In 2D, this specializes to
$d((x1,y1),(x2,y2)) = \sqrt{ (x1 - x2)^2 + (y1 - y2)^2 }.$
These expressions generalize the Pythagorean idea to higher dimensions.

We can increase dimensions to form ℝ^4, ℝ^5, and in general ℝ^n for any natural number n.
Definition:
$\mathbb{R}^n = { (x1, x2, \dots, xn) \mid xi \in \mathbb{R} \text{ for } i = 1,2,…,n }.$
ℝ^n is called Euclidean n-space, and its points are n-tuples of real numbers.
The phrase from the transcript that each n corresponds to an n-tuple of numbers is captured by the formal definition above.

The orientation of the coordinate system (which axis is which, and their positive directions) affects how we interpret coordinates and distances.
The transcript’s example about looking along the +z axis illustrates that if the axes are not arranged consistently, the picture of the coordinate system can be misleading; maintaining a consistent right-handed orientation is standard practice in mathematics and physics.
The discussion about “putting fingers along the positive direction of the x axis” anticipates the right-hand rule used for cross products to determine the direction of the positive z axis in a right-handed system.

Real numbers can be viewed as points on a line; Cartesian products turn points into coordinate pairs, enabling higher-dimensional spaces.
The two-dimensional plane is naturally identified with the set $\mathbb{R} \times \mathbb{R} = { (x,y) \mid x,y \in \mathbb{R} }$ with coordinates (x,y).
Points in ℝ^n are represented by n-tuples $(x1, x2, …, xn)$ with each xi ∈ ℝ.
Consistent coordinate orientation is essential (often taken as right-handed) and can be checked with the right-hand rule.
Distances generalize from the line to higher dimensions via the Euclidean distance: $d(p,q) = \sqrt{ \sum{i=1}^{n} (xi - y_i)^2 }$ , which reduces to $d(x,y) = \sqrt{(x - y)^2} = |x - y|$ on the real line.
Higher-dimensional spaces are denoted as $\mathbb{R}^n$ , where n is any natural number, and represent n-dimensional Euclidean space.