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A Glimpse of the History of Mathematics (1). Mathematics Education. Mathematics and Civilizations. The history of mathematics is interwoven with the history of human civilization.

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Mathematics Education

Mathematics and Civilizations

• The history of mathematics is interwoven with the history of human civilization.

• World’s best known civilizations have contributed, in some way or other, a lot in the development of mathematics education.

• Now there is one predominant international mathematics, and this mathematics has quite a long history. It has roots in ancient Babylonia, China, Egypt, Greece and Indian subcontinent.

• Attempts to improve the teaching and learning of mathematics were going on from time immemorial

Mathematics Education

Mathematics and Civilizations

From Where and When?

A Glimpse of the History of Mathematics (3)

Counting was carried out long before the dawn of any civilization. But mere counting (i.e., early counting of the prehistoric period) is no mathematics.

Once we accept this, the origin of the first mathematics can be traced back to some 5000 years or so.

"Mathematics began with the creation or invention of number symbols or numerals.”

A Story of Complete Fabrication(1)

A Glimpse of the History of Mathematics (4)

• Mathematics has long been considered an invention of European scholars, as a result of which the contributions of non-European countries have been severely neglected in the histories of mathematics. Worse still, many key mathematical developments have been wrongly attributed to scholars of European origin. This has led to so-called Eurocentrism.

‘Mathematics is a European invention’

A Story of Complete Fabrication(2)

A Glimpse of the History of Mathematics (5)

‘Mathematics is a European invention’

Even simple arithmetical operations were not known or could not be performed in the European number system before the introduction of the Hindu-Arabic decimal system some 1000 years ago.

• How ridiculous is their claim of making very complicated computations during those days when they had only the primitive type of number symbols or numerals only ? (Refer to the Greek, Roman and other numerals of the 1st century A.D.)

A Glimpse of the History of Mathematics (6)

‘Mathematics is a European invention’

Eurocentric chronology of mathematics history.

Modified Eurocentric chronology of mathematics history.

A Glimpse of the History of Mathematics (7)

The Ancient World(1)

Babylonian Mathematics

The Babylonian civilization replaced that of the Sumerians from around 2000 BC. So, Babylonian Mathematics, inherited from the Sumerians, cannot be older than that of Sumerian mathematics. Counting in Sumerian civilization was based on a sexagesimal system, that is to say base 60. It was a positional system one of the greatest achievement in the development of the number system Babylonians used only two symbols to produce their base 60 positional system.

Number names, number symbols, arithmetical computations, traditional decimal notation go back to the origin of Chinese writing.

The number system which was used to express this numerical information was based on the decimal system and was both additive and multiplicative in nature.

Mathematics Education

A Glimpse of the History of Mathematics (8)

The Ancient World(2)

Chinese Mathematics

Mathematics Education traditional decimal notation go back to the origin of Chinese writing.

A Glimpse of the History of Mathematics (9)

The Ancient World (3)

Egyptian Mathematics

About 3000 BC the Egyptians developed their hieroglyphic writing (picture writing) to write numerals This was a base 10 system without a zero symbol. It was not a place value system. The numerals are formed by putting together the basic symbols .

The Egyptian number systems were not well suited for arithmetical calculations.

Mathematics Education traditional decimal notation go back to the origin of Chinese writing.

A Glimpse of the History of Mathematics (10)

The Ancient World(4)

• Greek Mathematics

• In the first millennium BC, the Greeks had no single national standard numerals.

• The first Greek number system is an acrophonic system. The word 'Acrophonic' means that the symbols for the numerals come from the first letter of the number name. The system was based on the additive principle.

• A second ancient Greek number system is the alphabetical system. It is sometimes called the 'learned' system. As the name 'alphabetical' suggests, the numerals are based on giving values to the letters of the alphabet .

The evidence of the first use of mathematics in the Indian subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

Mathematics Education

A Glimpse of the History of Mathematics (11)

The Ancient World (5)

Hindu Mathematics

(?)

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A New Wave of Fabrication

A Glimpse of the History of Mathematics (12)

• Not only the fundamental concepts of Ganit (Mathematics) such as those of counting numbers, zero and infinity but also various arithmetical and algebraic operations are being claimed to have been present in the Hindu Granth Vedah some 6000 years ago a time when there was nothing like Hindusthan, Hindu, Hindi and the Devanagari script verson of Vedah. This is a total lie. If there is anything that is worth mentioning, they are the ones found in the excavation of Mohenjadaro and Harrapa which did not belong to what is known today as India. (Refer to the Brahmi numerals of the first century A.D.)

‘Mathematics is a Hindu creation’

Nepalese Mathematics (1) subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

Mathematics Education

A Glimpse of the History of Mathematics (13)

The Ancient World (6)

Record written in Bramhi and Nepal Bhasa (Bhujimol) scripts and in the brick found while reconstructing the Dhando Stupa at Chabahil (Kathmandu) 2003 testifies that counting numbers were used in Nepal as early as 3rd century B.C.

The Lichhavian numerals used in the beginning of the last millennium is both additive and multiplicative. It was decimal in nature. There exists a complete analogy between the Lichhavian number system and the 14th Century B.C. Chinese system both in form and technique of writing numbers using numerals.

Nepalese Mathematics (2) subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

Mathematics Education

A Glimpse of the History of Mathematics (14)

The Ancient World (6)

The extremely artistic ways of writing numerals together with the various shaped objects placed symmetrically or asymmetrically in the ancient temples of Nepal do show that Nepalese people had not only the knowledge of various geometric shapes (supposed to have been discovered by Westerners) but also their practical use long long ago. They further show their skill in measurement and power of reproduction of congruent copies.

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A Glimpse of the History of Mathematics (15)

Medieval Mathematics (1)

Britain and European Mathematics

The collapse of Rome and the general chaos that followed has no great advancements in the mathematical community in it.. The Dark ages and then the Middle Ages were upon the land and civilization let alone the science of mathematics was having trouble surviving the times.

Schools were reduced tolittle or no arithmetic, it is doubtful whether few knew more than basic counting and finger reckoning.

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A Glimpse of the History of Mathematics (16)

Medieval Mathematics (2)

• In the period from 300AD to 1600AD there existed two major sub-divisions, the early Middle Ages, or Dark Ages (from 300AD to 1100AD) and the late Middle Ages, just before the Renaissance. In the early Middle Ages mathematics made no progress, but in the late Middle Ages there were a few advances and much of what had been forgotten from the ancient world was rediscovered and re-evaluated

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A Glimpse of the History of Mathematics (17)

Medieval Mathematics (3)

• The so called'classical period', or'Golden era' ofIndian mathematics.

• From the earliest numerate civilisation of the Indus valley, through the scholars of the 5th to 12th centuries who were conversant in arithmetic, algebra, trigonometry, geometry combinatorics and latterly differential calculus, Indian scholars led the world in the field of mathematics. The peak coming between the 14th and 16th centuries in the far South, where scholars were the first to derive infinite series expansions of trigonometric functions

• The work of Bhaskara was considered the highest point Indian mathematics attained, and it was long considered that Indian mathematics ceased after that point.

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

Arabic Mathematics : Forgotten brilliance?

A Glimpse of the History of Mathematics (18)

Medieval Mathematics (4)

• Many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the sixteenth, seventeenth and eighteenth centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier.

• The period after the Late Middle Ages saw the rise of maths again as a science worth a second glance and many educated men throughout Europe took up the challenge of relearning the "Wisdom of the Ancients".

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A Glimpse of the History of Mathematics (19)

• Renaissance Mathematics(1)

In England, Robert Recorde wrote what is thought to be the first series of textbooks in English. These were not intended for the highly educated mathematician but for the common man seeking to improve his understanding of such subjects as the Hindu-Arabic numeral system, conversions between weights and coins, computation with counters which would aid their work in trade and commerce

The quality and the quantity of teaching still varied hugely, and a significant number of students entering Oxford and Cambridge in the 1630's still had no prior knowledge of the Hindu-Arabic numeral system

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A Glimpse of the History of Mathematics (20)

Renaissance Mathematics(2)

Once the European community based their study, research and application on the Hindu-Arabic Number System, their contributions to the theory and application of mathematics grew tremendously during the latter part of the seventeenth century.

During the same period, worldwide usage of the Hindu-Arabic number system proved to be a boon for both mathematics and the whole of human society.

Progress towards the calculus continued with Fermat, who, together with Pascal, began the mathematical study of probability. However the calculus was to be the topic of most significance to evolve in the 17th Century

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A Glimpse of the History of Mathematics (21)

18th – 19th Centuries (1)

• Newton developed the calculus into a tool to push forward the study of nature. His work contained a wealth of new discoveries showing the interaction between mathematics, physics and astronomy. Newton's theory of gravitation and his theory of light take us into the 18th Century.

• Leibniz, much more rigorous approach to the calculus, was to set the scene for the mathematical work of the 18th Century rather than that of Newton.

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A Glimpse of the History of Mathematics (22)

18th – 19th Centuries (2)

• The most important mathematician of the 18th Century was Euler who, in addition to work in a wide range of mathematical areas, was to invent two new branches, namely the calculus of variations and differential geometry. Euler was also important in pushing forward with research in number theory begun so effectively by Fermat.

• Toward the end of the 18th Century, Lagrange was to begin a rigorous theory of functions and of mechanics. The period around the turn of the century saw Laplace's great work on celestial mechanics as well as major progress in synthetic geometry by Monge and Carnot.

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A Glimpse of the History of Mathematics (23)

18th – 19th Centuries (3)

• The 1800s—societal emphasis Mathematics teaching mainly meant arithmetic and basic geometry--skills needed for daily life. Specialized content might be learned on the job or in special academies. There was little formal teacher education until late in the century.

• Children in U.K. were once again enjoined to go to school, but could leave the educational system once they could read, write and had an elementary knowledge of Arithmetic. It was now generally accepted that some level of understanding of Mathematics was absolutely necessary for modern life, and there were few schools who did not give Mathematics a place in a student's timetable of classes.

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A Glimpse of the History of Mathematics (24)

18th – 19th Centuries (4)

• By 1823, while Augustus De Morgan was at Cambridge, the analytical methods and notation of differential calculus made their way into the course

• The 19th Century saw rapid progress. Fourier's work on heat was of fundamental importance. In geometry Plücker produced fundamental work on analytic geometry and Steiner in synthetic geometry.

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A Glimpse of the History of Mathematics (25)

18th – 19th Centuries (5)

• Progress towards the calculus continued with Fermat, who, together with Pascal, began the mathematical study of probability. However the calculus was to be the topic of most Non-euclidean geometry developed by Lobachevsky and Bolyai led to characterisation of geometry by Riemann. Gauss, thought by some to be the greatest mathematician of all time, studied quadratic reciprocity and integer congruences. His work in differential geometry was to revolutionise the topic. He also contributed in a major way to astronomy and magnetism.

• The 19th Century saw the work of Galois on equations and his insight into the path that mathematics would follow in studying fundamental operations. Galois' introduction of the group concept was to herald in a new direction for mathematical research which has continued through the 20th Century.

• Cauchy, building on the work of Lagrange on functions, began rigorous analysis and began the study of the theory of functions of a complex variable. This work would continue through Weierstrass and Riemann.

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A Glimpse of the History of Mathematics (26)

18th – 19th Centuries (6)

• Algebraic geometry was carried forward by Cayley whose work on matrices and linear algebra complemented that by Hamilton and Grassmann. Cantor invent set theory almost single handedly while his analysis of the concept of number added to the major work of Dedekind and Weierstrass on irrational numbers

• Lie's work on differential equations led to the study of topological groups and differential topology. Maxwell was to revolutionise the application of analysis to mathematical physics. Statistical mechanics was developed by Maxwell, Boltzmann and Gibbs. It led to ergodic theory.

• The study of electrostatics and potential theory. By Fredholm led to Hilbert and the development of functional analysis.

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A Glimpse of the History of Mathematics (27)

Some Numerals of the World (1)

The number system employed throughout the greater part of the world today was probably developed in India, but because it was the Arabs who transmitted this system to the West the numerals it uses have come to be called Arabic ( Hindu-Arabic) .

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A Glimpse of the History of Mathematics (28)

Some Numerals of the World (2)

Roman Numerals:

I = 1, V = 5, X = 10, L = 50, C = 100, D = 500 and M = 1000

Brahmi Numerals:

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A Glimpse of the History of Mathematics (29)

Some Numerals of the World (3)

Until 771, the Egyptian, Greek, and other cultures used their own numerals in a manner similar to that of the Romans.

Thus the number 323 was expressed like this:

Egyptian : 999 nn III ,

Greek : HHH ÆÆ III ,

Roman : CCC XX III

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A Glimpse of the History of Mathematics (30)

Some Numerals of the World (4)

• Modern Hindu- Arabic

• Early Hindu-Arabic

• Arabic Letters

• Early Arabic

• Modern Arabic

• Early Devanagari

• Later Devanagari

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A Glimpse of the History of Mathematics (31)

Some Numerals of the World (5)

! @ # \$ % ^ & * (

1 2 3 4 5 6 7 8 9

Some Numerals used in India

Some Numerals used in Nepal

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A Glimpse of the History of Mathematics (32)

Some Numerals of the World (6)

Ancient Chinese Lichchavian

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A Glimpse of the History of Mathematics (33)

Some Numerals of the World (7)

An Inscribed Statue of the Year 207 From Maligaon, Kathmandu

Rajbanshi Castro/GabiniSamvat Samvat107 207100 200 7 7 4 4

Translation of Castro and Garbini

'Of the great king Jayavarma, on the fourth day of the seventh (?) fortnight of summer, in the year 207'.

According to Rajbanshi the year is 107

Mathematics Education subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding

A Glimpse of the History of Mathematics (34)

Some Numerals of the World (8)

Some Conflicting Interpretations of Inscribed Numerals of Ancient Nepal

Fabrication of Nepal’s History

END OF PART ONE subcontinent was found in the Indus valley and dates back to at least 3000 BC. Excavations at Mohenjodaro and Harrapa, and the surrounding area of the Indus River, have uncovered much evidence of the use of basic mathematics. The maths used by this early Harrapan civilization was very much for practical means, and was primarily concerned with weights, measuring scales and a surprisingly advanced 'brick technology', (which utilized ratios). The ratio for brick dimensions 4:2:1 is even today considered optimal for effective bonding