One number fixes a position on a line. A position in a plane needs two numbers, measured along two perpendicular lines. Negative numbers let those measurements run in both directions.
On a number line, each point has exactly one number, and each number has exactly one point. “I am at −3” tells you exactly where someone is.
A line is one-dimensional: there is only one direction along it (and its reverse). So one number is enough to fix a position.
A sheet of paper is two-dimensional. A point can move left–right and also up–down, and the two movements are separate. “I am at 3” no longer fixes a point, because it does not say 3 in which direction, or how far in the other direction.
Two dimensions need two numbers. On a line, one number gives one point. On a plane, every point at the same distance along one direction fits “3”, so a second number is needed.
You are in a town where all the roads run either north–south or east–west, evenly spaced. How would you tell someone exactly which crossing you are at, using only numbers?
Pick one crossing as the starting point. Count how many roads east of it you are, and how many roads north. Two numbers fix one crossing.
This is a coordinate system. Each number answers a separate question: how far along one direction, and how far along the other.
A position in a plane is fixed by measuring along two reference lines. The lines are drawn perpendicular, meaning they meet at 90°.
The reason is independence. When the lines are perpendicular, moving parallel to one line does not change the distance measured along the other. Each of the two numbers then changes only when you move in its own direction. Together, they identify exactly one point.
The point where the two lines cross is called the origin. Both distances are measured from it.
The Baudhāyana Śulba Sūtra, written in India more than 2,500 years ago, used east–west and north–south lines to lay out altars. It states the relation between the sides of a right-angled triangle that is now called the Baudhāyana–Pythagoras theorem. You will use it in session 4 of this unit to find the distance between two points.
Indian astronomers measured the positions of places on Earth east or west of a reference line through Ujjayinī. In the 11th century, al-Bīrūnī calculated the latitude and longitude of many cities.
The coordinate system used in this unit was set out by René Descartes in 1637 and, independently, by Pierre de Fermat around the same time.
Counting roads from the starting crossing works while you move east or north. Move west or south, and the count needs a way to show the opposite direction.
You could add words such as “west” and “south”. But then every calculation would need a separate rule for each direction. The simpler method is to let the sign of the number show the direction: positive one way, negative the other.
Rules for calculating with zero and negative numbers were written down by Brahmagupta in 628 CE. In a coordinate system, zero marks the origin. Negative values reach the points to the left of it and below it. With signs, a single rule works in every direction.
Suppose negative numbers did not exist. What part of the plane could a coordinate system describe?
Only the points to the right of the origin and above it, with both distances zero or positive. That is one quarter of the plane.
Points to the left or below could only be described with extra words such as “4 west, 3 south”. Each calculation would then need a different rule depending on the direction. With signed numbers, one formula covers all four quarters, as you will see when you calculate distances.
Once every point has a pair of numbers, a shape can be described by an equation. For example, all the points whose two coordinates are equal lie on one straight line through the origin. In the other direction, an equation can be drawn as a shape. Geometry problems can then be solved with algebra, and algebra can be pictured with geometry.
1Why does one number fix a point on a line but not in a plane?
2Why are the two reference lines drawn perpendicular?
3Why do coordinate systems need negative numbers?
4Which of these fix a position using measured numbers along reference directions?
More than one answer is correct.
5The point where the two reference lines cross, from which both distances are measured, is called the ____.
6A friend says “meet me at 4 metres” in a large open ground. Why can you not find the place?
7What becomes possible once every point in a plane has a pair of numbers?
Next: the names of the parts of the plane, and how to read and write coordinates.
Noescio