"Where Am I?": The Need for Coordinates
Why two numbers are enough to find any spot on the floor of your room — an insight 5,000 years old
AI Generation Prompt
Ultra-wide cinematic banner (16:5 ratio). An aerial view at first light of an ancient Sindhu-Sarasvati civilisation city like Dholavira or Mohenjo-daro. Streets meet at perfect right angles in a north-south, east-west grid. Mud-brick houses line each block at uniform intervals. A central plaza or great bath sits at the city's heart. Long shadows stretch from the rising sun, picking out the precision of the layout. The atmosphere conveys: thousands of years ago, before any modern map, humans already knew how to organise space using a grid. Warm dawn light, dusty haze, painterly cinematic illustration in the style of epic Indian art. Dark background. No text, no labels.
A friend calls you from a city you have never visited. 'I'm lost,' they say. 'Help me find the bookshop.' You cannot see what they see. You cannot point. All you have is words and numbers. What is the *minimum* information your friend needs from you to walk straight to the shop?
They already know which city they are in. They just don't know how to move within it.
The Verse on One Truth, Many Names
एकं सद्विप्रा बहुधा वदन्ति।
(ekaṃ sad viprā bahudhā vadanti)
'सच्चाई एक ही है — विद्वान लोग उसे अलग-अलग नामों से बुलाते हैं।'
'Truth is one — the wise call it by many names.'
The idea you are about to study — that any point in space can be located by a few numbers — was discovered independently by builders in Bhārat 5,000 years ago, by sailors in Greece, by mathematicians in Persia, and finally written down formally in France. The same truth. Many names. Many languages. One mathematics.
What does it mean to *describe* where something is?
Imagine you are walking along a long, straight road. There is just one road. No turnings. No branches.
If a friend asks 'where exactly are you?' — you can answer with a single number.
'I'm 200 metres from the school.'
One road. One number. That's enough.
Now imagine you are inside a city — full of cross-streets, lanes, shops, parks. The same friend asks the same question. One number is no longer enough. Why not? Because a single number can tell you how far, but not in which direction. You now need at least two numbers.
'I'm 200 metres east of the railway station, and 350 metres north of it.'
Notice what just happened. We picked one place — the railway station — as our reference point. From there, we measured two distances: one along an east-west line, one along a north-south line. With those two numbers, your friend can walk straight to you. There is no other place in the city that fits both numbers.
This is the entire idea behind a coordinate system. A coordinate system is a structured framework — a fixed reference point and a set of directions — that lets us pin down any location with just a few numbers.
In a city, two numbers are enough. (East-west, north-south.) On a long road, one is enough. In a multi-storey building, you would need three — east-west, north-south, and how high. The number of measurements depends on the kind of space you live in. We will spend this whole chapter on the case of two numbers — the 2-D coordinate plane.
The world's first coordinate system
The earliest people we know of who used this idea on a large scale lived in the Sindhu-Sarasvatī Civilisation — the ancient cities of Mohenjo-daro, Harappa, Dholavira, Lothal, Rakhigarhi, and many others — about 5,000 years ago.
Walk through the ruins of Dholavira (in modern Gujarat) today and you can still see it. The streets do not curve, do not wander, do not branch unpredictably. They run in two directions only — north–south and east–west — and they are spaced about 10 metres apart, with surprising regularity.
Why is this remarkable? Because it means the city was designed using a coordinate system. A merchant in ancient Mohenjo-daro could find any shop or warehouse by counting blocks east-west and blocks north-south from a known landmark — exactly the same logic you used to direct your friend to the bookshop.
No paper. No pen. No graph paper. No equations. Just a clear understanding that two numbers locate a point in a flat space. That insight is at least 5,000 years old. The mathematics we will write down in this chapter is younger. The intuition is ancient — and it began here.
AI Generation Prompt
Top-down schematic of an ancient Sindhu-Sarasvatī grid city, viewed as if from a bird's eye. Streets run strictly north-south and east-west, perfectly perpendicular, evenly spaced — about a dozen blocks across. A bold marker labelled 'CENTRE' sits in the middle. Two thick orange arrows extend from the centre — one pointing east, one pointing north — each labelled with an example distance like '4 blocks east, 3 blocks north'. A single highlighted shop is shown at the tip, demonstrating that two numbers locate any point. Style: clean technical illustration, faintly stylised to evoke ancient brick-block geometry. Dark background, orange accent labels, clean technical illustration style.
Loading simulator…
Threads of Curiosity
Walk down a Dholavira street today and you'll find the stones aligned within a few degrees of true north. There are no compass markings preserved on the bricks. No instructions. No surveyor's manuals. We don't know exactly how the builders fixed those directions so accurately 5,000 years ago.
Three students each give directions to the same hidden treasure inside a park. - **Aanya:** 'It's 50 metres from the gate.' - **Bhavya:** 'Go 30 metres east of the gate, then 40 metres north.' - **Charu:** 'Walk towards the big neem tree until you feel like stopping.' Which student is using a coordinate system?
Bridging Science and Society
Every time the Google Maps app on your phone shows a blue dot for 'you are here', it is using exactly this idea. The phone receives signals from satellites of the GPS (and India's own NavIC system, run by ISRO). From the signals, it computes two numbers — your latitude and longitude — which are coordinates on the curved surface of the Earth.
Q1.How many numbers do you need to specify any point on a flat 2-dimensional surface (like a sheet of paper or the floor of a room)?
AI Generation Prompt
Ultra-wide cinematic banner (16:5 ratio). An aerial view at first light of an ancient Sindhu-Sarasvati civilisation city like Dholavira or Mohenjo-daro. Streets meet at perfect right angles in a north-south, east-west grid. Mud-brick houses line each block at uniform intervals. A central plaza or great bath sits at the city's heart. Long shadows stretch from the rising sun, picking out the precision of the layout. The atmosphere conveys: thousands of years ago, before any modern map, humans already knew how to organise space using a grid. Warm dawn light, dusty haze, painterly cinematic illustration in the style of epic Indian art. Dark background. No text, no labels.
A friend calls you from a city you have never visited. 'I'm lost,' they say. 'Help me find the bookshop.' You cannot see what they see. You cannot point. All you have is words and numbers. What is the *minimum* information your friend needs from you to walk straight to the shop?
They already know which city they are in. They just don't know how to move within it.
The Verse on One Truth, Many Names
एकं सद्विप्रा बहुधा वदन्ति।
(ekaṃ sad viprā bahudhā vadanti)
'सच्चाई एक ही है — विद्वान लोग उसे अलग-अलग नामों से बुलाते हैं।'
'Truth is one — the wise call it by many names.'
The idea you are about to study — that any point in space can be located by a few numbers — was discovered independently by builders in Bhārat 5,000 years ago, by sailors in Greece, by mathematicians in Persia, and finally written down formally in France. The same truth. Many names. Many languages. One mathematics.
What does it mean to *describe* where something is?
Imagine you are walking along a long, straight road. There is just one road. No turnings. No branches.
If a friend asks 'where exactly are you?' — you can answer with a single number.
'I'm 200 metres from the school.'
One road. One number. That's enough.
Now imagine you are inside a city — full of cross-streets, lanes, shops, parks. The same friend asks the same question. One number is no longer enough. Why not? Because a single number can tell you how far, but not in which direction. You now need at least two numbers.
'I'm 200 metres east of the railway station, and 350 metres north of it.'
Notice what just happened. We picked one place — the railway station — as our reference point. From there, we measured two distances: one along an east-west line, one along a north-south line. With those two numbers, your friend can walk straight to you. There is no other place in the city that fits both numbers.
This is the entire idea behind a coordinate system. A coordinate system is a structured framework — a fixed reference point and a set of directions — that lets us pin down any location with just a few numbers.
In a city, two numbers are enough. (East-west, north-south.) On a long road, one is enough. In a multi-storey building, you would need three — east-west, north-south, and how high. The number of measurements depends on the kind of space you live in. We will spend this whole chapter on the case of two numbers — the 2-D coordinate plane.
The world's first coordinate system
The earliest people we know of who used this idea on a large scale lived in the Sindhu-Sarasvatī Civilisation — the ancient cities of Mohenjo-daro, Harappa, Dholavira, Lothal, Rakhigarhi, and many others — about 5,000 years ago.
Walk through the ruins of Dholavira (in modern Gujarat) today and you can still see it. The streets do not curve, do not wander, do not branch unpredictably. They run in two directions only — north–south and east–west — and they are spaced about 10 metres apart, with surprising regularity.
Why is this remarkable? Because it means the city was designed using a coordinate system. A merchant in ancient Mohenjo-daro could find any shop or warehouse by counting blocks east-west and blocks north-south from a known landmark — exactly the same logic you used to direct your friend to the bookshop.
No paper. No pen. No graph paper. No equations. Just a clear understanding that two numbers locate a point in a flat space. That insight is at least 5,000 years old. The mathematics we will write down in this chapter is younger. The intuition is ancient — and it began here.
AI Generation Prompt
Top-down schematic of an ancient Sindhu-Sarasvatī grid city, viewed as if from a bird's eye. Streets run strictly north-south and east-west, perfectly perpendicular, evenly spaced — about a dozen blocks across. A bold marker labelled 'CENTRE' sits in the middle. Two thick orange arrows extend from the centre — one pointing east, one pointing north — each labelled with an example distance like '4 blocks east, 3 blocks north'. A single highlighted shop is shown at the tip, demonstrating that two numbers locate any point. Style: clean technical illustration, faintly stylised to evoke ancient brick-block geometry. Dark background, orange accent labels, clean technical illustration style.
Loading simulator…
Threads of Curiosity
Walk down a Dholavira street today and you'll find the stones aligned within a few degrees of true north. There are no compass markings preserved on the bricks. No instructions. No surveyor's manuals. We don't know exactly how the builders fixed those directions so accurately 5,000 years ago.
Three students each give directions to the same hidden treasure inside a park. - **Aanya:** 'It's 50 metres from the gate.' - **Bhavya:** 'Go 30 metres east of the gate, then 40 metres north.' - **Charu:** 'Walk towards the big neem tree until you feel like stopping.' Which student is using a coordinate system?
Bridging Science and Society
Every time the Google Maps app on your phone shows a blue dot for 'you are here', it is using exactly this idea. The phone receives signals from satellites of the GPS (and India's own NavIC system, run by ISRO). From the signals, it computes two numbers — your latitude and longitude — which are coordinates on the curved surface of the Earth.
Q1.How many numbers do you need to specify any point on a flat 2-dimensional surface (like a sheet of paper or the floor of a room)?