All you must know about the History of Mathematics

History of Mathematics:

The history of Mathematics is long and magnificent, which can be traced back to the start of humanity itself. Be it the early man’s notched bones, the settled agriculture in Egypt and Mesopotamia or the revolutionary developments of the Hellenistic empire of ancient Greece, every event speaks of mathematics’s highly innovative and charming history. Great innovations were made in different parts of the world, with The East contributing an array of significant developments, mainly in India, China and the medieval Islamic empire. Europe saw some marvelous advances and innovations during the late middle age. From the 16th century, a series of discoveries have been taking place till date, forming the great history of Mathematics.

What is Mathematics?

• Counting, measuring and identifying the shapes of objects evolved the science of order, relation and structure, which began to be known as Mathematics. It has been a fundamental base for engineering, science and philosophy developments.
• The application of imagination, logic and abstraction has led to more complex developments in the discipline over the years. Moreover, it has broadened the scope of Mathematics which now covers logical reasoning, quantitative calculation, number theory, geometry, algebra, calculus, probability, statistics and many more fields of study or specialized areas.

Role of Mathematicians:

There have been great mathematicians over the centuries who were great thinkers. They have gifted the world with inspiring moments by innovating and devoting their lives to Mathematics. Today, we will unfold the history of mathematics and give an insight into the innovations as to how, when, and by whom they were discovered. It is vital to know the math importance and its rich history that has shaped the world into what it is today.

Prehistoric Mathematics or The Origin:

Basic Number System:

• Mathematics owes its foundation to counting, which humanity began almost 40,000 years ago, and its creation was not as simple as it may seem. Though mankind had an idea of the difference between “one” and “two”, with an understanding regarding amounts, it took several ages to invent a word or symbol for the abstract knowledge of “two”.
• The patterns in the environment led to mathematical thoughts of shapes. The hunter-gatherer societies knew no word for numbers greater than two. In the absence of agriculture and trade, there was no need for a standard number system.

Prime Numbers and Shapes:

• The Ishango bone demonstrations represent the earliest sequence of prime numbers. It is argued that prime numbers evolved after the concept of division which dates back to 10,000 BC.
• Geometrical shapes and designs are claimed to have been developed during the 5th millennium BC by the pictorial representation done by the Predynastic Egyptians. Various shapes, like circles, Pythagorean triples and ellipses, were found on the megalithic monuments located in England and Scotland during the 3rd millennium BC. Though these represented mathematical designs, they were mere art and decorations.

Civilisation and Math Importance:

• With the onset of agricultural trade amongst civilisations (Sumerian and Babylonian of Mesopotamia {Iraq today} being the pioneers), the need for measurement of land, taxation policies, the price of trade etc., increased and paved the way for the proper development of Mathematics. Several ancient mathematical sources bear evidence of basic geometric and arithmetic systems.

Mathematics in Sumer and Babylon (Ancient Mesopotamia) [1st and 2nd Millennium BC]:

Sumer, modern Iraq, is known as the Cradle of Civilization for its pioneering activities like writing, irrigation, agriculture, the plough and many more. They used wedge-shaped symbols inscribed on the clay tablets, which were baked under the sun, known as a cuneiform script which marks the innovation of a pictographic writing system. Increased agricultural and trade activities led to the development of a proper arithmetic system. They described large specific numbers through symbols and developed their lunar calendar. Later these symbols were replaced by cuneiform equivalents.

Arithmetic System:

• They followed a base 60 or sexagesimal number system, which could be calculated with 12 knuckles on the one hand and five fingers on the other. In addition, they used an actual place-value system where larger values were represented by digits written in the left column. The number representation was similar to Roman numerals.
• The modern-age time system of 60 seconds in one minute, 60 minutes in one hour and 360 degrees in a full circle are all inspired by the Babylonian system.
• The fact that 60 has multiple divisors like 1, 2, 3, 4, 5, 6, 10 etc., made it the base of their arithmetic system. Similarly, the number 12 with multiple factors like 1, 2, 3, 4 and 6 have been used extensively, such as 12 inches, 12 months, 2*12 hours etc.

The Famous Clay Tablets:

• Many Babylonian clay tablets have been discovered that cover various topics such as multiplication, division, square tables and roots, cube roots, algebra, linear, quadratic and cubic equations and much more.
• Buildings and dice designs were based on geometrical shapes. Calculating the areas of various shapes and volumes of cylinders is some of the other geometrical activities that took place in this period.
• The popular Plimpton 322 clay tablet bears evidence of the knowledge of the principle of the right-angled triangle much before the Pythagoras theorem came to light. However, this fact has been disputed as many suggest that the tablets were used for academic representations and do not interpret anything else.

Mathematics in Egypt [2000 – 6000 BCE]:

Settled along the Nile valley, early Egyptians started to note the lunar phases, agricultural and religious reasons. Measurements were based on body parts, and a decimal numeric system was in place depending on our ten fingers. They had no concept of place value but used a stroke for units, a heel-bone symbol for 10s, a coil of rope for 100s and a lotus plant for 1000s.

Famous Texts Available:

• The most prominent Egyptian mathematical text available from 2000-1800 BCE is the Rhind Papyrus. It contained several formulae for geometry, division, multiplication, knowledge about prime and composite numbers and solutions to linear equations, etc.
• Moscow Papyrus is another text from the same period that contained word problems, making mathematics more entertaining.
• Evidence of solving a second-order algebraic equation is available in the Berlin Papyrus.

Egyptian Pyramids and Rules of Triangles:

• Egyptian pyramids depict the marvellous wit of Egyptian mathematicians. The pyramids suggest that the mathematicians knew the perfect formula for the volume of a pyramid as these structures observed the golden ratio of 1:1.618.
• They knew the rules of a triangle, and Egyptian builders yielded perfect right-angled triangles.

Greek Mathematics [4th – 7th Century BC]:

The Greek empire spread enormously and conquered many societies. They were wise to adopt useful and mathematical elements from those societies, including the Babylonians and the Egyptians. Slowly they made innovations on their own and brought about a revolution in the world of Mathematics. A fully developed Greek numeral system (Attic or Herodianic numerals similar to the Roman system) was used.

The Great Greek Mathematicians:

• Thales, a Greek mathematician and one of the Seven Sages of Ancient Greece, laid down the foundation guidelines for the abstract invention of geometry. He gave the world Thales’s Theorem and the Intercept Theorem.
• Another legendary mathematician who is the dawn of Greek mathematics is Pythagoras, believed to have coined the terms “philosophy” and “mathematics”. He has given the Pythagoras Theorem, which is undoubtedly one of the best mathematical theorems ever known.
• Democritus was the first to note that a cone has 1/3rd the volume of a cylinder with the same height and base.
• Other renowned mathematicians include Plato, Aristotle, Archimedes, Euclid and Pappus of Alexandria, who provided theorems with a logical approach.

Influence on Geometry:

• Greek geometry has seen three major geometrical problems (the squaring of the circle, the doubling of the cube, and the trisection of an angle), which have influenced the future of geometry and resulted in many fruitful developments.

• The solutions to these problems were compiled in the 19th Century. Greeks were pioneers in introducing the concept of infinity with Zeno’s Dichotomy Paradoxes.

Mathematics in India [5th – 12th Century]:

The history of Mathematics in India can be traced to very early stages. The time from the 5th to 12th Century is said to be the Golden Era of Indian Mathematics. Several mathematical developments took place in India simultaneously with the West, which led to a few claims of plagiarism by a few prominent European mathematicians.

Arithmetic System:

• The Vedic period (before 1000 BCE) has mantras including powers of 10 from 100 and bears evidence of the application of arithmetic solutions like addition, subtraction, division, multiplication, cubes, squares, fractions and roots.
• Indians perfected the decimal place value number system like the Chinese, which is one of the most valued intellectual innovations to date.

Famous Texts Available:

• A CE Sanskrit text as old as the 4th Century shows Buddha mentioning different number systems and estimating the number of atoms in the universe.
• Sulba Sutras, also known as Sulva Sutras, is an 8th Century BCE text that has laid down several simplified statements of the Pythagoras theorem. It is believed that Pythagoras got his inspiration from this text. It also contains linear and quadratic geometric solutions and an accurate result for the square of 2.
• Ancient Buddhist literature and Jain mathematicians of the 3rd or 2nd Century recognised the different types of infinities.

The Greatest innovation of all times – The invention of Zero:

• The invention of Zero has been instrumental in the history of Mathematics and is solely attributed to the Indian mathematician, Brahmagupta of the 7th Century.
• He laid down the principles for using zero and negative numbers. In addition, he has done pioneering work in quadratic equations and the concepts of algebra.
• It was Bhaskara II of the 12th Century who corrected the wrong result of division by zero. He explained that dividing 1 by zero will result in infinite pieces.

Significant Contribution to Trigonometry:

• The theory of trigonometry introduced by the Greeks had huge progress in India during the Golden Age of Indian Mathematics. The various functions of sine, cosine and tangent were used to measure lands, navigate through the seas, and calculate the distance between the Moon and the Earth and between the Earth and the Sun.
• The first use of trigonometry is contained in the text named “Surya Siddhanta” by an anonymous author that dates back to 400 CE.
• The great Indian mathematician Aryabhata of the 6th Century CE has provided proper definitions and complete tables of all trigonometry functions.  He specified and used the value of π to give an approximate circumference of the Earth.

Mathematics in Medieval Europe and Renaissance:

During the 4th to 12th Century, when great mathematical innovations were taking place in several parts of the world like China, India and Islamic regions, very little or no progress took place in Europe in the field of intellectual sciences. The focus was majorly on spiritual, literary and philosophical subjects. Their mathematical knowledge was based on the findings and theories of Greek masters like Nicomachus and Euclid. Greek and Roman-based abaci and Roman numerals were used for trade and calculations.

Rise of Mathematics in Europe:

• It was after the 12th Century when Mathematics played a more practical part in the lives of European people. This shift from the academic realm to real-life uses was brought about by the rapid progress of trade and commerce with the East and the West.
• The introduction of the printing press in the mid-15th Century was another catalyst in the process of imparting knowledge.
• Business people were educated on computational methods for trade and commercial needs through various books on arithmetic.

The Great European Mathematicians:

• Leonardo of Pisa, popularly known as Fibonacci, was Europe’s 1st great middle-aged mathematician who is best known for introducing a Fibonacci sequence of numbers. The spread of the Hindu-Arabic numeral system across Europe during the 13th Century is attributed to him. This led to the extinction of the Roman numeral system and opened the doors for great innovations in the field of Mathematics in Europe.
• Frenchman Nicole Oresme of the 14th Century is known for introducing time-speed-distance graphs. He also worked on the system of rectangular coordinates, fractional exponents and the infinite series.
• The great German scholar of the 15th Century, Regiomontanus, has established math importance in the area of trigonometry. He separated it from astronomy and made trigonometry an independent part of Mathematics. His first significant book on trigonometry, “De Triangulis”, exhibits basic trigonometry knowledge and is widely used for high school and college studies.

Mathematics During Renaissance:

During the Renaissance, Italian artists and merchants played an influential role in the development of Mathematics. Both algebra and accounting developed together though there is no direct relationship between them. Arithmetic and algebra were essential in complex bartering operations and compound interest calculations.

The Great Mathematicians:

• During the 14th Century, Piero Della Francesca wrote many books on linear perspective and solid geometry.
• Another great mathematician of this time was Luca Pacioli, whose book Summa de Arithmetica introduced the plus and minus signs for the 1st time. It is also the first book printed in Italy with algebra.
• Scipione del Ferro and Niccolò Fontana Tartaglia of the 16th Century provided solutions for cubic equations.
• Ars Magna, a 15th Century book by Gerolam Cardano, had solutions for the quartic equations.
• The real number system is influenced by Simon Stevin’s “De Thiede”, which showed the 1st systematic dealing of decimal notation.

Mathematics and the Scientific Revolution:

Mathematics in the 17th Century:

The 17th Century, also called the Age of Reason, saw many mathematical and scientific ideas explode across Europe.

The Great Mathematicians and their Innovations:

• The most significant innovation during this period was the introduction of the logarithm by John Napier, which helped Kepler and Newton to perform complex calculations for their innovations in the field of Physics. Logarithms helped the advances in science, astronomy and mathematics by making complex calculations relatively easy.
• René Descartes is known for developing analytic geometry and Cartesian coordinates, which enabled the plotting of orbits of the planets on a graph. He also laid the foundation for calculus.
• Pierre de Fermat and Blaise Pascal are two other great French mathematicians known for establishing math importance and the concept of probability and expected values.
• By laying down the laws of Physics, single-handedly, Sir Issac Newton is often regarded as one of the greatest mathematicians in the history of mathematics. In addition, Newton and Gottfried Leibniz developed two operations called differentiation and integration, which are used widely in different fields of study, highlighting the importance of math.

Mathematics in the 18th Century:

The Great Mathematicians and their Innovations:

• The Bernoulli of Basel and Leonhard Euler are the two prominent mathematicians who dominated the field during the 18th Century. They worked on calculus, probability, number theory, geometry, trigonometry and algebra.
• Christian Goldbach has proposed the Goldbach Conjecture and proved many number theory theorems, such as the Goldbach-Euler Theorem on perfect powers.
• Abraham de Moivre is known for de Moivre’s formula, which links trigonometry and complex numbers. In addition, his work on Newton’s famous binomial theorem, analytic geometry and probability theory is of great value in the history of mathematics.
• Joseph Louis Lagrange is credited with the four-square theorem and is also popular for Lagrange’s Theorem or Lagrange’s Mean Value Theorem.
• Pierre-Simon Laplace, also known as “the French Newton”, is famous for his work on “Celestial Mechanics”.
• Adrien-Marie Legendre’s contributions to abstract algebra, statistics, number theory and mathematical analysis deserve special attention.

Mathematics in the 19th Century:

The Great Mathematicians and their Innovations:

• Joseph Fourier studied infinite sums where the terms are trigonometry functions. He introduced the Fourier Series, which became a powerful tool in applied and pure mathematics.
• Jean-Robert Argand represented complex numbers on geometric diagrams using trigonometry and vectors, which are popularly known as the Argand Diagrams.
• Évariste Galois advocated that polynomial equations with any degree greater than 4 had no general algebraic solution. He worked in areas such as group theory, rings, vector spaces, algebraic geometry and non-commutative algebra.
• Carl Friedrich Gauss, also named the “Prince of Mathematics”, is one of the greatest in the history of mathematics who worked in various fields of study.

Mathematics in the 20th Century:

Mathematics moved towards generalization and abstraction during this period, and it became a profession in teaching and other industries with a specialization in fields of study.

The Great Mathematicians and their Innovations:

• G.H. Hardy and Srinivasa Ramanujan are note-worthy mathematicians of this time who tried solving the Riemann hypothesis.
• The “Principia Mathematica”, a joint work by Bertrand Russell and A.N. Whitehead, greatly influenced mathematics and philosophy.
• The other great mathematicians who highlighted math importance include Johann Gustav Hermes, David Hilbert, Kurt Gödel, Alan Turing, John von Neumann, Claude Shannon, Andrey Kolmogorov, André Weil, Paul Erdös, Paul Cohen, Julia Robinson and Yuri Matiyasevich.

Mathematics in the 21st Century:

Many online versions and print versions of mathematical journals were made available recently. In addition, open-access publishing is gaining momentum. With computers becoming more powerful and significant, the subject is growing, and the application of mathematics to bioinformatics is increasing at an alarming pace.

Math importance in JEE Main 2023:

JEE Main is one of India’s highly competitive exams for engineering aspirants.  Preparing for such exams can be challenging, and proper knowledge of all subjects can greatly help score good marks in JEE Main.

Question Paper Pattern:

The questions are from 3 major subjects Physics, Chemistry and Mathematics. The question paper pattern has different sections of 100 marks each, with each section having 25 questions.

Important Topics:

The Mathematics section can be the scoring, but adequate practice is the only way to success. Mentioned below are the main topics and chapters for Mathematics for JEE Main 2023:

• Binomial Theorem
• Conic Sections
• Complex Numbers
• Circles
• Differentiation
• Differential Equations
• Integration
• Limits and Continuity
• Matrices and Determinants
• Probability
• Permutation and Combination
• Quadratic Equations
• Straight Lines
• Sets, Relations and Functions
• Sequences and Series
• Trigonometry
• Three Dimensional Geometry
• Vector Algebra

Other topics that are less important and can be looked upon after preparing the above-mentioned important topics include the Application of Derivatives, Statistics, Heights and Distances and Inverse Trigonometry.

Quick Tips for Preparation:

• The students should make a proper plan based on subjects to crack the JEE main exam. Having a thorough knowledge of the syllabus is essential.
• A lot of practice is indispensable. Therefore, solving some practice papers, including the previous year’s question papers and focusing on the NCERT mathematics textbook, can significantly help.
• Revising regularly will help you stay on track with multiple topics and improve memory. A minimum of one month should be set aside for revision while preparing the study plan.
• Developing speed during calculations will do wonders in time management during the exam. It can be achieved through regular practice in core topics.