Chapter 1 Notes: Orienting Yourself: The Use of Coordinates

A coordinate system is a structured framework that uses numbers to describe exact locations of points or objects in space — just like grid lines on a map or graph paper. The idea of grid-based thinking has ancient roots in Bhārat. The Sindhu-Sarasvatī Civilisation built city streets in precise North–South and East–West directions about 10 metres apart — an early real-world coordinate system. Baudhāyana (c. 800 CE) used these directions for geometric constructions and developed the Baudhāyana–Pythagoras Theorem. Ujjayinī was used as the central longitude meridian (zero reference) in early astronomy. Āryabhaṭa (c. 499 CE) introduced 'sines' to calculate star and city coordinates. Brahmagupta (c. 628 CE) formalised zero and negative numbers — both essential for the modern four-quadrant coordinate plane. Eventually, René Descartes (1637 CE) formalised that any point in a 2-D plane can be defined by just two numbers.

The 2-D Cartesian coordinate system uses two number lines placed at right angles (perpendicular) to each other to locate any point in a flat (2-D) plane. The horizontal line is called the x-axis and the vertical line is called the y-axis. They meet at a point called the Origin O, whose coordinates are (0, 0). Distances to the RIGHT of O on the x-axis are POSITIVE; distances to the LEFT are NEGATIVE. Distances ABOVE O on the y-axis are POSITIVE; distances BELOW are NEGATIVE. The coordinates of a point are written as an ordered pair (x, y), where x is the x-coordinate (distance from y-axis) and y is the y-coordinate (distance from x-axis). The ORDER matters: (x, y) ≠ (y, x) unless x = y.

[DIAGRAM NEEDED: Cartesian plane showing x-axis and y-axis, origin O(0,0), and example points B(4.5,0) on x-axis, G(0,–4.5) on y-axis, H(0,4) on y-axis, and E(–2.9,0) on x-axis, with positive and negative directions clearly labelled]

The coordinate axes divide the Cartesian plane into four regions called quadrants, numbered I to IV in an anti-clockwise direction starting from the top-right region. The sign of the coordinates tells us which quadrant a point lies in.

[DIAGRAM NEEDED: Cartesian plane divided into four quadrants labelled Quadrant I (top-right), Quadrant II (top-left), Quadrant III (bottom-left), Quadrant IV (bottom-right), with the sign pattern (+,+), (–,+), (–,–), (+,–) shown in each quadrant, and example points Q(–5,3) in QII and S(3,–5) in QIV marked]

To find the distance between any two points in the coordinate plane, we use the Baudhāyana–Pythagoras Theorem. Given two points A(x₁, y₁) and D(x₂, y₂), we imagine a right-angled triangle where: the horizontal leg has length |x₂ – x₁| (the difference in x-coordinates) and the vertical leg has length |y₂ – y₁| (the difference in y-coordinates). The distance AD is the hypotenuse of this right triangle. It does not matter whether the differences are positive or negative since we square them. This formula works for points in any quadrant, including those with negative coordinates.

Example: A(3, 4) and D(7, 1) • Horizontal distance = 7 – 3 = 4 • Vertical distance = 4 – 1 = 3 • AD = √(4² + 3²) = √(16 + 9) = √25 = 5 units

[DIAGRAM NEEDED: Right-angled triangle formed by points A(x₁,y₁), D(x₂,y₂), and a helper point F(x₂,y₁), showing horizontal leg (x₂–x₁), vertical leg (y₂–y₁), and hypotenuse as the distance AD with the distance formula labelled]

This chapter introduced the 2-D Cartesian coordinate system, its historical origins in Bhārat, and its essential concepts including axes, quadrants, coordinates, and the distance formula. The coordinate system allows us to locate any point precisely using two numbers and to calculate distances using algebra and geometry together.