##### Mathematics is like a universal language, helping us understand patterns in the world and solve problems. It’s an art form woven into human Knowledge, revealing our journey from ancient times of modern discoveries. Math is simple yet deep, revealing life’s mysteries

##### By Mohan Kanda

- Sumerians, in 3400 BCE, created the first numeral system and measures. Egypt followed with the earliest decimal system
- In the 4th century CE, Hypatia, a prominent philosopher, astronomer, and mathematician, defied societal sexism but tragically faced a violent death
- Da Vinci showcased mathematical prowess, crafting depth illusions on flat surfaces. “The Annunciation” stands as a classic testament to this technique
- Nature’s geometric wonder, the hexagon, with six equal sides, repeats in our surroundings. A prime instance is the beehive

**“Mathematics, rightly viewed, possesses not only truth, but supreme beauty, – a beauty cold and austere, like that of sculpture ……… sublimely pure, and capable of stern perfection, such as only the greatest art can show”**

Bertrand Russell, British philosopher and mathematician.

I have always felt that the importance of mathematics is often underrated. After all, applied mathematics is physics, applied physics is chemistry, and applied chemistry is zoology, which deals with living beings. Mathematics, in other words, is life itself!

As the great astrophysicist Neil deGrasse Tyson said, “Maths is the language of the universe”. Every year on December 22, India observes National Mathematics Day in honour of Srinivasa Ramanujan, one of the greatest mathematicians the world has ever seen. This tradition has been upheld since 2012, when the then Prime Minister, Dr Manmohan Singh, made an official announcement to that effect.

Internationally, the Day of Mathematics is celebrated on March 14, also known as ‘Pi Day’ because the mathematical constant π (Pi) can be rounded off to 3.14.

##### FROM ANCIENT ROCKS TO ALGORITHMS

The history of mathematics is a chronicle of hundreds of important milestones. The events are also densely packed in time. Further, what constitutes a momentous discovery or invention, and what can be passed off as an ordinary development, is also a matter for subjective judgement. I have, therefore, attempted a concise sequence of developments which, in my opinion, can serve as a satisfactory presentation of the timeline of events of the subject. The timeline spans the period from 70,000 BCE to the present. According to one estimate, it begins with the adorning of ochre rocks, with geometric patterns, in South Africa. The Ishango bone, belonging to around 20,000 BCE, discovered in the Nile Valley, is probably the earliest reference to prime numbers and Egyptian multiplication. Later on, in 3400 BCE, the Sumerians, in Mesopotamia, invented the first numeral system and a system of weights and measures. That was followed, in Egypt, by the earliest known decimal system. The next significant development came when, in Egypt, in 2400 BCE, an astronomical calendar was prepared.

##### GREEK MATHEMATICIANS CONTRIBUTIONS

Among the important developments that followed thereafter, was the study of geometry as an axiomatic system, in 300 BCE, in the book ‘Elements’ by Greek mathematician Euclid. That was also the period during which Archimedes, the most famous mathematician and inventor of ancient Greece, known as the ‘Father of Mathematics’ lived. Subsequently, Greek philosopher Aristotle in 40 BCE discussed logical reasoning in his book, ‘Organic’. Then came the description of the concept of infinity, in the Hindu Vedas in the 8th century BCE. Around the same time, Yagnavalkya, the Hindu sage, described the motions of the sun and the moon. Pythagoras – the Greek mathematician and philosopher who lived in the 6th BCE was familiar with the Upanishads and learnt his basic geometry from the Shulba Sutra. An early statement of what is commonly known as the Pythagoras theorem is to be found in ‘Baudhāyana sūtras’ a work by 8th century Indian mathematician Baudhayana.

The history of mathematics begins with the adorning of ochre rocks, with geometric patterns, in South Africa. The Ishango bone, belonging to around 20,000 BCE, discovered in the Nile Valley, is probably the earliest reference to prime numbers and Egyptian multiplication

In the 4th century ADE, Hypatia was the greatest philosopher, astronomer and mathematician of her time, who spectacularly overcame the profound sexism of her society, and met a violent death at the hands of ignorant zealots.

##### RISE OF MATHEMATICS IN ANCIENT INDIA

Then, in the 5th century ADE, came Aryabhatta, the first of the major mathematician-astronomers from the classical age of Indian mathematics and astronomy, who correctly insisted that the earth rotated about its axis daily, contrary to then, prevailing view that the sky did so. His calculations on Pi, the circumference of the earth and the length of the solar year were remarkably close approximations to modern calculations. He became famous for having authored ‘Aryabhatiyam’, a compendium of mathematics and astronomy and Arya-Siddhanta, a lost work on astronomical computations, is known through the writings of Aryabhata’s contemporary, Varahamihira.

Varahamihira, a 6th-century astrologer/astronomer, is also credited with having been the first to claim that some force might be keeping bodies attached to earth, the force which is now called gravity. In addition to developing the properties of zero and negative numbers, he also made important contributions to meteorology, writing about weather patterns, cloud formations, rainfall and how to predict weather using astronomical observations. Varahamihira authored the ‘Pancha-Siddhantika’, a compendium merging Egyptian, Roman, and Indian astronomy.

Among his notable works, the ‘Brihat Samhita’ stands out as a comprehensive exploration of diverse subjects. Covering architecture, temples, planetary motions, eclipses, timekeeping, astrology, seasons, cloud formation, rainfall, agriculture, mathematics, gemology, fragrances, and more, it remains one of Varahamihira’s enduring contributions to knowledge.

Another milestone in the history of the discipline followed, in the 9th century AD, when Al-Khwarizmi, Persian mathematician, known as the Father of Algebra, wrote ‘Kitab al-Jabr wa-l-Muqabala’ or algebra, in which he introduced the Hindu – Arabic decimal number system to the western world. His algebra treatise gives us the word and can be considered as the first book to be written on algebra.The word algorithm is also named after him.

The next leading light in the history of mathematics was 12th century ADE Indian mathematician, Bhaskaracharya, who came from a long-line of mathematicians and was head of the astronomical observatory at Ujjain. He authored several important mathematical texts including ‘Lilavati’ and ‘Bijaganita’ and the ‘Siddhanta Shiromani’, an astronomical text, and also introduced the systematic use of the decimal numbers system. His daughter, Lilavati, was also a distinguished mathematician, who wrote ‘Lilavati Ganitam’ which included topics such as arithmetic, algebra, and geometry. The book was written in verse and was widely popular during that time.

##### FROM NAPIER’S LOGARITHMS TO LOVELACE’S LEGACY

Towards the beginning of the 17th century, Scottish mathematician and scholar John Napier invented logarithms, an invention that had a lasting impact on various activities, such as simplifying and increasing the speed of complex calculations, measuring the magnitude of earthquakes, levels of noise and detection of radioactivity.

Isaac Newton, 17th century mathematician, is known not merely for discovering the laws of gravity, but also for working out many of the principles of visible light and the laws of motion, and contributing to calculus. In a letter to a friend, in 1675, he made the famous statement: “If I have seen further it is by standing on the shoulders of Giants” He also authored ‘Philosophiæ Naturalis Principia Mathematica’, in which he expounded the laws of motion and the law of gravitation.

Leibniz, a German philosopher, mathematician and logician, is probably most well known for having invented the differential and integral calculus independently of his contemporary, Isaac Newton. His notation of calculus is commonly favoured by mathematicians to this day. He also invented the ‘Leibniz Wheel’, the first mathematical calculator.

No narration of the history of mathematics will be complete without a special mention of Ada Lovelace. A giant in the world of science, technology, engineering, and mathematics (STEM), she not only holds the honour of being one of the most famous women in mathematics history but is also recognized as the first-ever computer programmer of any gender.

Florence Nightingale, who also belonged to the same time, is well known as an inspiring and courageous woman and is regarded as the founder of modern nursing. Known as the ‘Lady with the Lamp’, she is remembered for her great contribution to the sick and the wounded during the Crimean War in 1854. She was also a pioneer in the field of statistics and went on to become the first female member of the Royal Statistical Society.

##### PIONEERS OF MODERN MATHEMATICS AND STATISTICS

The 20th century also saw many interesting developments in the field of mathematics. Prominent among them was the publication, in 1975, by French mathematician Benoit Mandelbrot, of his controversial book on fractals, ‘Les objets fractals : forme, hasard et dimension’. Then there was Andrew Wiles, the English mathematician, who, in 1994, finally, and to the great delight of the entire mathematical world, found the proof for Fermat’s last theorem, a challenge which had defied the skill and imagination of the greatest of mathematicians, for centuries. Simply put, the theorem states that the equation xn + yn = zn has no solution for values of n exceeding 3. Then, in 2000 ADE, the Clay Mathematical Institute, a global organisation dedicated to furthering the beauty, power and universality of mathematical thought, proposed the seven Millennium Prize problems of unsolved important classic mathematical questions. Subsequently, in 2023, Belgian physicist Ingrid Daubechies was awarded the Wolf Prize in Mathematics, becoming the first woman to receive that award.

Coming to the Indian mathematical scene in those days, there was Dr Calyampudi Radhakrishna Rao, a highly regarded global wizard in the field of Mathematics and Statistical applications, an Indian – American mathematician and statistician, described by the American Statistical Association as “a living legend…”. He had over forty-five PhDs from global universities, including Cambridge. His work had far-reaching implications for fields as varied as economics, genetics, anthropology and geology, among many others. In a fitting tribute to his illustrious career, he was honoured in 2023 with the International Statistics Prize, regarded as the statistics equivalent of the Nobel Prize. That was just before his death when he was 102 years old.

##### ROLE OF MATHEMATICS IN THE FINE ARTS

Mathematics is an essential ingredient of the system that makes our lives orderly and prevents chaos. It helps develop the faculties of reasoning, creativity, and abstract thinking. It also improves the ability to communicate and, as its use needs focus and concentration, helps us remain mentally active and healthy. Whether we notice it or not, there is some mathematics behind every task we perform in our daily lives, such as solving problems, cooking meals, fixing a leaky faucet at home, planning our careers or taking a stroll around a museum. The subject plays an important role, not only in fields such as engineering and architecture, but also in the fine arts, such as painting and music, where parallelism, symmetry and accuracy play an important role. Music, in particular, is highly mathematical as can be seen from the compositions of the legendary composer Bach, creations which were replete with pattern, structures and other precisely crafted features. In painting, mathematics is used for drawing flat objects, which give the illusion of being three-dimensional objects when viewed from certain points of view. The legendary da Vinci demonstrated by creating the illusion of depth on a flat surface. His famous painting “ The Annunciation” is a classic example of this method.

Mathematics also forms the building blocks of the natural world and can be seen, in stunning ways, in many beautiful patterns in nature, such as in symmetry and spirals, which are aesthetically appealing and proportional. Symmetry can be radial, where the lines of symmetry intersect a central point such as a daisy or a starfish. Fractals are another intriguing mathematical shape that we see in nature. They make up many aspects of our world, including the leaves of ferns, tree branches, the branching of neurons in our brain, and coastlines. The hexagon is one of nature’s geometric wonders. A regular hexagon has 6 sides of equal length, and this shape is seen again and again in the world around us. The most common example of nature using hexagons is in a beehive. Another common shape in nature is a set of concentric circles, or circles sharing the same centre, but with different radii, and one inside the other. Common examples are the ripples of a pond, when something hits the surface of the water, concentric circles in the layers of an onion or the rings of trees that form as they grow and age.

In the 8th century BCE, the concept of infinity emerged in the Hindu Vedas. Yagnavalkya, a Hindu sage, also described the movements of the sun and the moon around that time. Pythagoras, who was a mathematician and philosopher from Greece in the 6th century BCE, knew the Upanishads and used the ‘Shulba Sutra’ as a source of Geometry

##### THE FIBONACCI SEQUENCE : A MATHEMATICAL WONDER

There is a fascinating series of numbers called the ‘Fibonacci Sequence’, which is a set of integers (the Fibonacci numbers) that starts with a zero, followed by a one, then by another one, and then by a series of steadily increasing numbers. The sequence follows the rule that each number is equal to the sum of the preceding two numbers.

Fibonacci numbers can be used to define a spiral and are of interest to biologists and physicists because they are frequently observed in various natural objects and phenomena. The branching patterns in trees and leaves, for example, and the distribution of seeds in a raspberry reflect the Fibonacci sequence. The sequence is often associated with the golden ratio, a proportion (roughly 1:1.6) that occurs frequently throughout the natural world and is applied across many areas of human endeavour.

Both the Fibonacci sequence and the golden ratio are used to guide design for architecture, websites and user interfaces, among other things. The numbers also form a unique shape known as a Fibonacci spiral which, again, we see in nature in the form of shells and the shape of hurricanes.

##### CONNECTION BETWEEN ASTROLOGY AND MATHEMATICS

Yet another area, although often regarded as a pseudoscientific field, in which mathematics has a role, is astrology, which involves the study of planetary movements, positions, and aspects which are measured and calculated, using mathematical principles. For example, astrologers use mathematical techniques such as division, multiplication, addition, and subtraction to calculate the planetary positions and aspects that are used in birth chart analysis. Astrology and mathematics have had a historical connection, particularly in the development of early astronomy. Ancient civilizations used mathematical principles to create calendars, track celestial movements, and make predictions about astronomical events.

The beauty of mathematics is often enhanced by the discovery of neat and simple solutions to problems which, to the average mind, appear to be soluble only by complex and cumbersome procedures. For example, there can often be more than one solution to a given problem in a branch of mathematics. In my experience, I found this to be particularly so in the area of differential equations. In such situations, mathematicians often try to look for what is known as an ‘elegant’ solution, thus called for simplicity and beauty. Great mathematicians have often looked for, and succeeded, in finding such solutions. The great Einstein, for instance, reduced the entire realm of reality of the universe to a set of just ten mathematical equations!

In the 5th century ADE, Aryabhatta was the first among the major mathematician-astronomers of the classical Indian era. He rightly claimed that the earth turned on its axis once a day, instead of the sky, as most people thought. His computations of Pi, the earth’s perimeter, and the length of a year were remarkably precise by modern standards

##### MATHEMATICAL ODYSSEY

Paradoxes and fallacies in mathematics, which often arise from building on defective premises, or choosing erroneous options, also lead to entertaining, and interesting, exercises. A mathematical paradox is a mathematical conclusion so unexpected that it is difficult to accept even though every step in the reasoning is valid. A mathematical fallacy, on the other hand, is an instance of improper reasoning leading to an unexpected result that is patently false or absurd.

Likewise, an interesting, and paradoxical, situation is provided by the story of a barber in a village, whose job it is to shave everyone who cannot shave himself. The question that is posed is, whether the barber shaves himself or not. If he does, he is shaving someone who is capable of shaving himself. If he does not, there is someone who cannot shave himself, whom the barber is failing to shave. Either conclusion leads to a result that is contrary to the original proposition.

A similar situation arises, when one tries to answer the question whether God, as He is infallible, can create objects so big that even He cannot push. As in the earlier case, if it can be done, then God has failed in the task of making a stone which no one can push. If it cannot be done, God is failing to push the stone. An absurd conclusion either way.

^{FIBONACCI’S RABBIT PARADOX}

*I*N the annals of mathematical puzzles, Fibonacci, in the year 1202, posed a captivating conundrum that has intrigued minds for centuries. The enigma revolves around a population of rabbits, their mating habits, and the resulting exponential growth.

A man puts a male-female pair of newly born rabbits in a field. Rabbits take a month to mature before mating. One month after mating, females give birth to a male-female pair and then mate again. Assuming no rabbits die during the period, how many rabbit pairs will there be after one year?

At the start of January, one pair of juvenile rabbits is introduced into the population. At the start of February, this pair of rabbits have matured and mate. At the start of March, this original pair of rabbits gave birth to a new pair of juvenile rabbits. And so on, as in the following table.

Theoretically, if the rabbits never die, at the end of one year there will be 288 rabbits.

**Month No. of Pairs**

##### MY JOURNEY WITH MATHEMATICS

My own relationship with mathematics has been long and interesting. In 1967, I was undergoing training, as a Probationary Officer, at a branch of the State Bank of India, at Machilipatnam in Andhra Pradesh state, when the call came for the interview for selection to the Indian Administrative Service (IAS). During the interview, the Chairwoman of the Board interviewing me asked me whether my MSc degree in mathematics was useful in my functions. Tongue-in-cheek, I replied, “I have a postgraduate degree in mathematics, not arithmetic!”. Strangely enough, I had reached the stage of the interview, not an account of my prowess in mathematics, but in spite of it! Unlike other subjects such as political science, history, or economics, mathematics has many areas that one can be completely unaware of. This was the case for me when I took two mathematics papers for the civil services examinations, which had many topics that I had never studied before. Luckily, I did well in the general knowledge and English essay papers, which helped me pass the exams.

My interest in chemistry began when I was doing the Pre-University Course at Nizam College in Hyderabad, where I had a brilliant teacher named Seshavataram. He taught chemistry in such a way that I became fascinated by the subject and convinced my father to let me set up a laboratory at home. There, I experimented with various chemicals and equipment, sometimes with unexpected results. Once, I tried to make hydrogen dichloride, but the test tube exploded in my face!

When my father moved to Delhi to practice in the Supreme Court after retiring as a Judge of the Andhra Pradesh High Court, I had to find a college in Delhi to continue my studies. My uncle accompanied me to the Delhi University campus and we visited St. Stephens College first, where I got admission to the B.Sc. (Honours) chemistry course, but I had to wait for two days to confirm it. Then we went to Hindu College, where I was admitted to the same course immediately, but on the condition that I take mathematics as the main subject. My uncle was impatient and urged me to join that course, which dashed my hopes of pursuing chemistry.

I had two thrilling experiences in the final year of MSc Mathematics at Osmania University. One was when Jayant Vishnu Narlikar, a famous astrophysicist and cosmologist, came to the University to give a lecture. He asked for a volunteer from the audience to write down the equations on the chalkboard as he spoke. I raised my hand quickly and had the honour of assisting him as he gave an unforgettable lecture. I was studying the Theory of Relativity as a special subject at that time and I was very interested in astrophysics and cosmology. I was especially intrigued by the Steady-state theory, or the Hoyle-Narlikar theory, of the universe, which Narlikar developed with another astrophysicist Fred Hoyle. Although this theory was later rejected in favour of the more popular Standard Model, I always wanted to meet my hero from college days. I finally got the chance to do so recently, when I visited Pune and met Narlikar for a few minutes.

Another exciting experience was when Shakuntala Devi, the legendary mental calculator known as the ‘Human Computer’, who was in the ‘Guinness Book of World Records’, came to the University to give a lecture and I performed a similar task as I did with Narlikar, feeling proud and happy.

After finishing my graduation at Hindu College, I went back to Hyderabad, where my father had taken up a job as the Chairman of the Central Wage Board for the Cotton textile and Sugar industries. When I applied for the M.Sc. course in chemistry at Osmania University, I was told that I was only eligible for MSc in mathematics, which was my main subject in the Honours course at Delhi University. That was the end of my passion for chemistry.

A professor of astrophysics had just finished a lecture in the town hall of an American City. The house was thrown open to questions from the audience, and a frail and timid looking gentleman from one of the last rows in the lecture hall got up and, in a shaky voice, asked.

“Did you say that the Earth would come to an end in 1 billion years?”

The Professor replied, “No, I said 10 billion years.”

The man heaved a sigh of relief, and sat down, saying, “Thank God”! I thought you said 1 billion years!