What would a system of coordinates be like if we did not have negative numbers? Would this system allow us to locate all the points on a 2-D plane?

Question

Homi · Accepted Answer

Understand the Question

In a standard Cartesian coordinate system, the positive and negative directions of the $x$ -axis and $y$ -axis divide the plane into four quadrants, allowing any point $(x, y) \in \mathbb{R}^2$ to be located.
Without negative numbers, values along the axes can only be non-negative ( $x \ge 0$ and $y \ge 0$ ).
This restricts the coordinate system to only a single quadrant (the first quadrant), meaning points located to the left of or below the origin cannot be identified.

Step 1 · Structure of the System without Negative Numbers

Without negative numbers, the coordinate axes can only extend in the positive directions starting from the origin $(0, 0)$ :

The $x$ -axis only covers values where $x \ge 0$ (to the right of the origin).
The $y$ -axis only covers values where $y \ge 0$ (above the origin).Therefore, the coordinate system would be limited solely to the first quadrant.

Step 2 · Coverage of Points on a 2-D Plane

A full two-dimensional plane extends infinitely in all four directions around the origin:

Quadrant I: $x > 0, y > 0$
Quadrant II: $x < 0, y > 0$
Quadrant III: $x < 0, y < 0$
Quadrant IV: $x > 0, y < 0$

Without negative numbers, points lying in Quadrants II, III, and IV (where either $x < 0$ , $y < 0$ , or both) cannot be represented.

Therefore, such a coordinate system cannot locate all points on a 2-D plane.

Answer

The system would be restricted to only the first quadrant ( $x \ge 0, y \ge 0$ ), and it would not allow us to locate all points on a 2-D plane.

Common Mistakes

Assuming Origin Shift Solves the Problem: While moving the origin shifts which region is covered, any fixed origin on an infinite 2-D plane with only non-negative coordinates will always miss the rest of the plane.
Confusing Cartesian with Polar Coordinates: While polar coordinates $(r, \theta)$ can locate all points with $r \ge 0$ using angles, a standard Cartesian system $(x, y)$ strictly requires negative numbers to cover all four quadrants.

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