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?

Step-by-step solution

Idea: Negative coordinates are what describe the points to the left of the y-axis and below the x-axis. Without them, three of the four quadrants disappear.

1. Without negative numbers, both coordinates must be 0 or positive. So the only points we could describe are on the positive x-axis, the positive y-axis, or in Quadrant I.½ mark
2. Points in Quadrants II, III and IV, and on the negative halves of the axes, such as (−3, 0) or (2, −5), could not be written at all.½ mark
3. We could move the origin to the bottom-left corner of a region (as on a computer screen or a page of graph paper), but that covers only that limited region. The plane goes on forever in every direction, so some points would always be left out.½ mark
4. So no, such a system could not locate all the points of the plane. This is why zero and negative numbers are essential for the four-quadrant plane.½ mark

Check: Of the four sign patterns (+, +), (−, +), (−, −), (+, −), only (+, +) survives without negative numbers: one quadrant out of four ✓.

Answer to write in the exam

Common mistakes that cost marks

• Saying “yes, just move the origin”. Moving the origin helps only for a limited region; the plane has no edge.
• Thinking only Quadrant III is lost. Any point with even one negative coordinate is lost: Quadrants II, III and IV.
• Forgetting that 0 is still allowed, so points on the positive halves of the axes can still be described.

How this can come in the exam

Try one yourself

The origin is moved to the point that used to be (−10, −10), and only non-negative coordinates are allowed. What are the new coordinates of the old points (−4, 3) and (0, 0)? Can the old point (−12, 5) be described?

More questions like this

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