Orienting Yourself: The Use of
1 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
Bhārat. The first systematic use of grids occurred thousands of years
ago — on a massive urban scale— in the Sindhu-Sarasvatī 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. Baudhāyana (c. 800
C.E.), as we have seen, later used East–West and North–South lines for his
deep geometric constructions, developing the Baudhāyana–Pythagoras
Theorem and thus laying the foundation of coordinate geometry.
Putting coordinates on the Earth’s surface later became important
for navigation. Ujjayinī was described in the ancient world — at least
as early as the 4th century BCE in the early Siddhāntas — 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’
(Ujjayinī). Āryabhaṭa (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.
, 2 Ganita Manjari | Grade 9 | Part I
Brahmagupta’s work was translated into Arabic (as the Sindhind),
and the Ujjayinī 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-Bīrūnī (c. 1000 CE) travelled to India, studied the Siddhāntas,
and used Indian trigonometric methods to calculate the coordinates of
various cities across Asia. Al-Bīrūnī also later perfected the ‘astrolabe’,
a handheld device that allowed sailors to find their coordinates by
looking at the stars. Ömar Khayyām (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
1 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
Bhārat. The first systematic use of grids occurred thousands of years
ago — on a massive urban scale— in the Sindhu-Sarasvatī 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. Baudhāyana (c. 800
C.E.), as we have seen, later used East–West and North–South lines for his
deep geometric constructions, developing the Baudhāyana–Pythagoras
Theorem and thus laying the foundation of coordinate geometry.
Putting coordinates on the Earth’s surface later became important
for navigation. Ujjayinī was described in the ancient world — at least
as early as the 4th century BCE in the early Siddhāntas — 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’
(Ujjayinī). Āryabhaṭa (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.
, 2 Ganita Manjari | Grade 9 | Part I
Brahmagupta’s work was translated into Arabic (as the Sindhind),
and the Ujjayinī 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-Bīrūnī (c. 1000 CE) travelled to India, studied the Siddhāntas,
and used Indian trigonometric methods to calculate the coordinates of
various cities across Asia. Al-Bīrūnī also later perfected the ‘astrolabe’,
a handheld device that allowed sailors to find their coordinates by
looking at the stars. Ömar Khayyām (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
