This document provides an overview of the mathematical evolution that occurred in the Indus Valley Civilization between 3500-2500 BC. It discusses evidence found through archaeological excavations that indicates the Indus people had knowledge of geometry, measurement systems, arithmetic operations, and more. Specific examples highlighted include standardized weights and measures like the "Indus inch"; geometrically shaped bricks; artistic depictions showing understanding of circles and symmetry; and hemispherical constructions revealing grasp of three-dimensional concepts. Overall, the document argues the extensive ruins and artifacts from this early civilization demonstrate mathematical concepts existed and helped enable their large urban planning and technological accomplishments.

1. GRD Journals- Global Research and Development Journal for Engineering | Volume 5 | Issue 3 | February 2020
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An Overview of Mathematical Evolution in Indus
Valley Civilization (3500-2500 BC)
Sandip Ghanta Dr. Sankar Prasad Mukherjee
Department of Engineering Maths Department of Engineering Maths
Seacom Skills University, West Bengal Seacom Skills University, West Bengal
Abstract
This paper is based on refinding / assessment of the mathematical evolution during ancient civilization of Harappa and Mohenjo-
daro during 3500-2500B.C. period. Due to lack of written documentation, any such effort has to have dependence on the relics
and remains of any civilization, from the excavated areas. Sources of excavated remains viz civil structures, sanitary system,
metalled coins and utensils, the inscription on walls, tablets and alike relics were abundance proof of the concept of mathematics
existed at that point of time. Through a scientific analytical observation a trial may be made which would reveal that idea of
symmetric geometrical figures as well as numeric operation existed in that civilization in their own way / concept. This paper is
an endeavour to assess how mathematical knowledge in Indus Valley Civilization can be considered as initialization of
Mathematical evolution.
Keywords- Indus, Harappa, Mohenjo-Daro, Lothal, Dholavira, Indus Inch, Harappan Weights, Harappan Bricks
I. INTRODUCTION
The relics of Harappa and Mohenjo-daro as found from the remains of excavations may be presumed to be during 3500 – 2000
BCE, termed as Indus Valley Civilization, spread over more then million square kilometer whose geographical area encompasses
Pakistan and northwestern India. The Indus culture was characterized by extensive urbanization with large planned cities, as seen
from the ruins of Harappa and Mohenjo-daro. There were evidences of mainly craft specialization with varied geometrical
figures which existed and long-distance trade with Mesopotamia and Central Asia.
The findings of research of any civilization can only be ascertained from various available remains and their conjoint
analysis and examinations. Those are the civil structural relics; use of metal in coins and utensils, the town planning inclusion of
sanitary planning and the road ways, etc and most importantly the conception of time measurement as well as other measurement
of materials with yard sticks.
It‟s quite natural as well as rational thinking that if such relics are viewed through quantity and qualitative angle of
measurement, every aspect of physical measures (i.e. linear, two dimensional, three dimensional) and technological known-how
involved therein reflect the existing mathematical concept in that period. Planning of civil structure, use of metallic and
nonmetallic elements and various alike metallurgical remains if viewed systematically in an integrated manner will help us in
assessing the real picture of that civilization, where in the existence of mathematically based engineering is reflected.
This research work mainly oriented towards refinding and revealing a connective process of mathematical evolution
during Indus Valley Civilization. We can‟t ignore the prehistoric contributions, which act as a shadow knowledge in establishing
its contributory preliminary mathematical evolution during any Civilization.
Eventually, such accounts of prehistoric contribution definitely help the later mathematicians in their research work on
history of the subject. So the different excavated sites of Indus Valley Civilization of Mohenjo-daro and Harappa are of immense
value as a starter of history of the subject. This civilization uncovered a very important fact that “practical mathematics” was
existing at that point of time and our ignorance about the same is due to non-availability of any proper written document and
knowledge on those accounts.
A. Ascending Order of Weights and Measures of Indus Valley Civilization
Important innovations of this civilization include standardized weights as weighting measures. Early evidence of enumeration /
counting is found among the material ruins of the Indus Valley civilization. Archaeological remains include an elaborate system
of ascending weights and measures used at that point of time. In the Indus Valley civilization we have found 500 plumb-bobs of
uniform geometrical sizes and weights. Two series were found. One is present integral method 1, 2, 4, 8, 16, 32, 64 leading to
decimal system of moderate time 10, 20, 40, 160, 200, 300, 640, 1600, 6400, 8000 and 12,800. Equivalent weights had been in
use in parts of India until recently, with conversion rates conform to the above ratios and thus help to conceptualise the basis
of an elaborate system of exchange of one commodity from another. Such measurement system, their unit of weight is
approximately 28 grams of present system which approximated to English weight “ounce”. [2]

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Fig. 1: Harappan Weights of special geometrical measures in ascending order
B. Indus Inch Measures and Scales
The scales and instruments for measuring length have been excavated at major urban centres of this civilization, mainly in
Mohenjo-Daro, Harappa and Lothal. The Mohenjo-Daro scale [Figure2] is a part of a scale having length of 66.2mm, with nine
carefully sawn, equally spaced parallel lines, on average of 6.7056mm apart as from the excavations of the Great Bath in
Mohenjo-Daro. One of the lines is marked by a hollow circle, and the sixth line from the circle is indicated by a large circular
dot. The distance between the two markers is 1.32 inches (33.53mm), and has been named the „Indus inch‟. There are certain
interesting connections/correlation between the unit of length of that time and Sumerian unit of length, called „sushi‟ as exactly
equal to half of an „Indus inch‟. [2]
Fig. 2: Ruler found at Lothal (Indus Inch scale)
C. Harappan Bricks
A notable feature of the Harappan culture was its extensive use of “specific geometrical shaped” kiln-fired bricks and the
advanced standard of its making technology. In Harappa we found 15 different sizes of rectangular piped shaped Harappan
bricks (figure 3), with standard ratio of the three dimensions (length, breadth and thickness) of each brick was close to 4:2:1.
Hence knowledge of mathematical proportion did exist. Now, in the modern age this is taken into account as the optimal ratio for
efficient bonding required for a powerful brick technological erection. It is presumed that these measurements were used in the
buildings and other urban structures by the Indus people with great exactness. But Pythagorean triple theorem did not use in the
dimensions of the bricks in Indus Valley. Divakaran said that “rather than using the Pythagorean principle to generate right
angles, they might have used the property that the line through the intersection points of two circles is perpendicular to the line
joining their centres”. [4]
Fig. 3: Harappan Bricks
D. Geometrical Concept of Indus Valley Civilization
Aside measurements, the discoveries of the archaeological remains of the Indus Valley civilization in various parts of Pakistan
and northwestern India during the course of latest century revealed a culture having a esthetic sophisticated idea and
understanding of Geometrical figure which helped to date back the “Sulbasutras” by more than a thousand years. The Indus cities
well-laid out with elaborate street plans and their accurate geometrical orientation along the cardinal axes with origin have long
been considered as evidences that the Indus people had at least a working knowledge of practical Geometry stable for its
longevity. On the basis of earlier studies we can suggest that not only did these people have a practical grasp of mensuration, but
at the same time they also had an understanding of the basic principles of geometry. It can reasonably said that the people of the
Indus civilization were the self-inventors and had well versed acquaintance with the geometry of linear as well as circular

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dimension and techniques of approximating their measurement of areas. The Indus civilization paid special attention to the circle
circular curvature and its variants in various geometrical designs that they made on many artifacts models having added
esthetical look (figure-4). So we can safely say that the uses of geometrical structures in shapes of cones, cylinders, barrels,
hexahedra structure, etc were in commanding use in that civilization. Carvings of them had concentric and intersecting circles
and triangles. [4]
Fig. 4: Harappan Artifacts are the proof of symmetrically geometric shapes
E. Idea of Hemispherical Dimension
Seven hemispherical constructions were found at “Dholavira”, of which two could be excavated in details, which were
constructed over large rock cut chambers, having a circular plan. These were big hemispherical elevated mud brick
constructions. One of the excavated structures was designed in the form of a wheel having spokes. The other was a wheel
without spokes. These hemispherical structures bear similarity to early Buddhist stupas. The Archaeological Survey of India,
who conducted the excavation, opines that "the kind of design that is of spoked wheel and unspoked wheel also remind one of
the Sararata-chakra-citi and sapradhi-rata-chakra-citi mentioned in the Satapatha Brahmana and Sulba-sutras". [5]
Fig. 5: Construction at Dholavira revealing knowledge of hemisphere
F. The Numerals System and Arithmetic
The people of the Indus valley civilization achieved many notable advances in Applied Mathematical technology, with great
accuracy in their numerical systems. The oldest evidence of mathematical knowledge of Indians is found in the Indus valley
civilization. The seals and pictographic inscriptions found in the excavations of Mohenjo Daro and Harappa, indicated that the
people of that civilization had knowledge of numbers in their own way. The Harappan civilization was the mother of
mathematics from where both the concept of numbers and numerical system were born. The numerical system which was first
used by the Harappan found it‟s later development into other ancient civilizations. Therefore, it can be presented on the basis of
those relics, to be first use of mathematics in the Indian subcontinent as found in the Indus valley and dates as far back as 3500

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BC. Harappans used decimal numeral system without zero with place value system. There are accounts to present numbers from
1 to 13. The numbers from 1 to 13 have been represented by the vertical strokes (as seen in figure- 6 and table- 1). [3]
Fig. 6: Seal of Indus Valley representing numerology of that period
Table 1: Symbols used in Indus Numeral System ( 3500 BC )
Number 1 2 3 4 5 6 7 8 9 10 11 12 13
Indus Symbol I II III IIII
IIII
I
IIII
II
IIII
III
IIII
IIII
IIII
IIII
I
IIII
IIII
II
IIII
IIII
III
IIII
IIII
IIII
I IIII
IIII
IIII
G. Symbols used in 2500 B.C.
During 2500 BC the Indus number system had a base 10 (or decimal) system, containing different symbols for the numbers 1 to
9; for 10, 100 and 1000 for their multiples. Two types of numerical representation are then identified. The first (as seen in figure-
7 and table-2) contains an ordered sequence of vertical strokes representing numbers 1,2,3,4,6,7,8,9; number 5 represented by a
symbol “ꓵ”, number 10 represented by a symbol “ᴧ” and number 100 represented by a symbol “ᴧᴧ”.
Fig. 7: Seal of Indus Valley representing numerology in 2500 BC
Table 2: Symbols used in Indus Numeral System (2500 BC)
Number 1 2 3 4 5 6 7 8 9 10 100
Indus Symbol I II III IIII ꓵ
III
III
IIII
III
II II
II II
III II
II II
ᴧ ᴧᴧ
The second is more complex in which pictorial symbols in geometric forms represent various numbers. For example
number 4 was represented with four lines, whether it is a square or rectangle or diamond or a cross. Higher numbers were
represented by additional strokes attached to the basic signs. Many of these numbers are identified from their apparent
resemblance to numbers as available in contemporary record or later number systems such as Babylonian, Chinese, Attic Greek,
Kharosthi and Asokan Brahmi. The complexity of the system was increased further as many of the numeral signs are „condensed
in an artistic way‟ or „embellished‟ to look like pictorial depictions.

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II. CONCLUSION
We can conclude from this brief account of assimilated thought process that initial Mathematical knowledge / base existed
during Indus Valley Civilization in the period which reflects the existed standard of Mathematical concepts and its richness. It
will not be out of reasoning to presume that it may have acted as an impetus in furtherance of the evolution.
The story begins with the excavations of Indus Valley civilization. Although, this was a dark period since their script
remains undecoded, and no manuscripts survived to that extent. However, archaeology suggests an urban culture with well-
developed and organized cities that had regular pre plans. Moreover, there is also evidence of uniform measurements of length
and weight as used across different compositions. This uniformity existed despite an apparent absence of a central authority. The
picture of this civilization that evolves is that of a peaceful, prosperous society with agriculture, manufacturing and commerce.
One of the mathematically interesting features of this civilization is a sequence of cubic stone weights forming a regular
series of weights, which are in multiples by two for small weights, evolving into complicated binary and decimal multiples for
higher weights.
Another aspect of mathematical interest is the architecture that consisted of well-designed rectangular structures for
plots and roads. They must have known adequate of geometry.
We therefore conclude that if survey analysis of the relics available from historic excavations alike Indus Valley
civilization can be undertaken, it would add valuable material in framing Mathematical evolution since ancient period.
REFERENCES
[1] B Dutta and A Singh (1938), history of Hindu Mathematics, Volume I & II, Calcutta: Asian Publishing House.
[2] Singh, Rekha (2019), Early Description of Numerical and Measuring System in Indus Valley Civilization. Internate. J.Appl. Soc. Sci. 6(6): 1586-1589.
[3] Puja Kumari Srivastava, K. B. Singh (2018): Originity of the number symbole in the Indus Valley Civilization. IJREAMV0410945002.
[4] https://www-history.mcs.st-and.ac.uk/Projects/Pearce/Chapters/Ch3.html
[5] www.ancientscripts.com/indus.html
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