TEXTBOOK REFERENCE · GANITA MANJARI
End-of-Chapter Q5 · Open official source ↗
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?
AIMS worked solution
AIMS SOLUTION / EXPLANATION
Given:
A single fixed Cartesian frame.
To Find:
Whether non-negative coordinate values suffice for the whole plane.
Property:
Signs distinguish opposite directions from the reference point.
Solution:
With x ≥ 0 and y ≥ 0, only the upper-right region and its bounding rays can be described.
Moving the origin can help with a bounded set of points, but no fixed origin makes every point of the entire plane non-negative.
A different notation could encode directions separately, but that restores the information supplied by signs.
Results:
Negative coordinates are needed to cover the whole plane with the usual fixed Cartesian system.
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