X-Coordinate
The x-coordinate is the first number in an ordered pair, indicating a point's horizontal distance and direction from the origin.
Definition
The x-coordinate (or abscissa) is the first number in an ordered pair $(x, y)$: it tells you how far to move left or right from the origin, positive meaning right and negative meaning left; for a function $f$, it represents the input (independent variable). Geometrically, it is the projection of a point onto the x-axis, formally the result of applying the linear projection $\pi_1 : \mathbb{R}^2 \to \mathbb{R}$ defined by $\pi_1(x, y) = x$, equivalent to the dot product of the point with the unit vector $e_1 = (1, 0)$; in a rotated coordinate system with basis $\{u, v\}$, the "x-coordinate" becomes $p \cdot u / |u|$.
Example
In the point $(6, 2)$, the x-coordinate is $6$, meaning move $6$ spaces right, while in $(-4, 3)$ the x-coordinate $-4$ means move $4$ spaces left. To find the x-coordinate of the midpoint of $(2, 8)$ and $(10, 4)$: $x_{mid} = (2 + 10)/2 = 6$, the average of the two x-coordinates. Under a $45$-degree rotation, the new x-coordinate of point $(3, 1)$ in the rotated frame is $3\cos(45^\circ) + 1\sin(45^\circ) = (3 + 1)/\sqrt{2} = 2\sqrt{2}$.
Key Insight
Always read an ordered pair left to right, x first, y second; think "x comes before y in the alphabet" to remember the order. Geometrically, drop a perpendicular from a point to the x-axis and where it lands is the x-coordinate, an idea that extends to higher-dimensional spaces. Coordinates are not intrinsic properties of points but depend on the choice of basis: changing basis (such as rotating axes) changes all coordinates while the geometric point remains fixed, the core insight distinguishing vectors from coordinates.