NCERT Class 9 Ganita Manjari Chapter 01 Orienting Yourself The Use of Coordinates PDF Download

Chapter 1: Orienting Yourself: The Use of Coordinates

1.1 Introduction

A system of coordinates is a structured framework (like the grid lines on a map or graph paper) that enables us to use numbers to describe the exact physical locations of points or objects.

The idea of 'grid-based thinking' and the geometry required to define the locations of points in space-indeed has deep roots in Bharat. The first systematic use of grids occurred thousands of years ago - on a massive urban scale-in the Sindhu-Sarasvati Civilisation, where city streets were constructed with striking precision in North-South and East-West directions at uniform distances of about 10 metres apart. This was a coordinate system in practice: a merchant could find a shop or a warehouse by counting North-South and East-West units of distance from the city centre. Baudhayana (c. 800 C.E.), as we have seen, later used East-West and North-South lines for his deep geometric constructions, developing the Baudhayana-Pythagoras Theorem and thus laying the foundation of coordinate geometry.

Putting coordinates on the Earth's surface later became important for navigation. Ujjayini was described in the ancient world-at least as early as the 4th century BCE in the early Siddhantas-as the point marking the central longitude meridian from which all other locations were measured. The Greek mathematician Ptolemy (c. 150 BCE), building on earlier works including that of Hipparchus, later described the latitudes and longitudes of thousands of locations, including 'Ozine' (Ujjayini). Aryabhata (c. 499 CE) replaced the Greek 'chords' with 'sines', making it much easier to calculate the coordinates of a star or a city. He mapped the sky using Celestial Coordinates, measuring coordinate distances from the ecliptic (the path of the sun).

Brahmagupta (c. 628 CE) formalised the notion and use of zero and the negative numbers as algebraic entities; in modern coordinate systems, the 'origin' is zero and the 'negative axes' represent values less than zero. Without Brahmagupta's work, the four-quadrant Cartesian plane, as we will study in this chapter, would be impossible.

Teacher's Note

Notice how the history of coordinates is deeply rooted in Indian mathematics. Brahmagupta's formalisation of zero and negative numbers was absolutely revolutionary-without it, we couldn't have negative coordinates at all. This is why the Cartesian plane works the way it does: it relies on thinking of numbers both left and right of a central point, which was Brahmagupta's insight.

Brahmagupta's work was translated into Arabic (as the Sindhind), and the Ujjayini meridian entered Arabic geography under the name 'Arin,' serving as the zero-longitude reference for early Arabic maps which also then made use of negative numbers. The influential Arab scholar Al-Biruni (c. 1000 CE) travelled to India, studied the Siddhantas, and used Indian trigonometric methods to calculate the coordinates of various cities across Asia. Al-Biruni also later perfected the 'astrolabe', a handheld device that allowed sailors to find their coordinates by looking at the stars. Ömar Khayyam (c. 1100 CE), who had become an expert in the Indian decimal system and algebraic formalism, was the first mathematician to solve algebraic problems using geometry by interpreting them in terms of coordinates in the plane.

These concepts eventually reached Europe in the 12th century. The final leap occurred when following the related work of Fermat (1636 CE), René Descartes (1637 CE) formalised the fact that any point in a two-dimensional plane could be defined by simply two numbers-representing the point's distances from two perpendicular axes. Points and more complex geometric shapes could then be described precisely using algebra and equations, thus bringing the areas of geometry and algebra even closer together.

In Grades 9 and 10, you will have a chance to study this amazing coordinate system which has such a rich history in human thought and endeavour. You will be able to locate objects with pinpoint accuracy. You will also see how using coordinates enables us to visualise algebraic equations as geometric shapes, and vice versa.

We begin our study of coordinates with a story that will help you understand these new terms better.

1.2 Settling In

It is the beginning of the academic year and Reiaan is both excited and nervous. The family has just moved to a new city. He and his sister, Shalini, will be attending a new school. Today, Shalini will help him settle into the new environment. When someone is not able to see, this can be a very big challenge, but with their mother's transferable job, the siblings have done it often, and it has become easier with each move. Shalini has just completed Grade 9 and this time, she decided to put to use what she has learnt in Coordinate Geometry in Mathematics to guide Reiaan.

Shalini wanted Reiaan to feel the directions, so she used a rectangular grid on which she had fixed pins and threads. This showed the floor of the room. Points in the sketch were marked using pins. Shalini was using a scale of 1 cm : 1 foot. She used pins to mark out various key points of the room. Points representing the corners of objects were connected with thick wool so that Reiaan could feel their positions with his fingers.

[Figure 1.1: Sketch of Reiaan's room. See in your textbook]

Let us examine Fig. 1.1 to understand the layout of the room. Notice that this only shows the map of the floor. Do you see why the position of the windows cannot be marked on this map?

Teacher's Note

The story of Reiaan and Shalini is a great reminder of why coordinates matter in the real world. Maps and grids aren't just abstract maths-they help us locate things precisely and communicate locations to others, even when we can't see them. This is the practical heart of coordinate geometry.

1.3 The 2-D Cartesian Coordinate System

In the chapters on integers, rational numbers, and decimals in earlier grades, you studied the number line which is one-dimensional. The two-dimensional coordinate system uses two lines at right angles to each other to mark points in two-dimensional space (short form: 2-D space). For convenience, we consider one of the lines to be horizontal; it is called the x-axis. The other line is vertical; it is called the y-axis. The point of intersection of the x-axis and y-axis is called the origin O; its coordinates are \( (0, 0) \). Coordinate axes (this is the plural of 'axis') help us to locate any point in 2-D space using the point's 'coordinates'. Distances from O are marked off in equal units, on both the axes. Distances to the right of O or upwards from O are considered positive, and distances to the left of O or downwards from O are considered negative (Fig. 1.2).

Key Points

• A coordinate system is a framework using grid lines and numbers to locate the exact position of points or objects in space.
• The two-dimensional Cartesian coordinate system has a horizontal x-axis and a vertical y-axis that meet at the origin \( (0, 0) \).
• In coordinates, distances to the right or upwards from the origin are positive, while distances to the left or downwards are negative.
• The concept of negative numbers and zero, formalised by Brahmagupta, made the four-quadrant Cartesian plane possible.
• Coordinate systems allow us to describe geometric shapes using algebra and equations, connecting geometry and algebra together.

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