Five ways ancient India changed the world – with maths

It should come as no surprise that the first recorded use of the number zero, recently discovered to be made as early as the 3rd or 4th century, happened in India. Mathematics on the Indian subcontinent has a rich history going back over 3,000 years and thrived for centuries before similar advances were made in Europe, with its influence meanwhile spreading to China and the Middle East.

As well as giving us the concept of zero, Indian mathematicians made seminal contributions to the study of trigonometry, algebra, arithmetic and negative numbers among other areas. Perhaps most significantly, the decimal system that we still employ worldwide today was first seen in India.

The number system

As far back as 1200 BC, mathematical knowledge was being written down as part of a large body of knowledge known as the Vedas. In these texts, numbers were commonly expressed as combinations of powers of ten.

 For example, 365 might be expressed as three hundreds (3x10²), six tens (6x10¹) and five units (5x10⁰), though each power of ten was represented with a name rather than a set of symbols. It is reasonable to believe that this representation using powers of ten played a crucial role in the development of the decimal-place value system in India.

From the third century BC, we also have written evidence of the Brahmi numerals, the precursors to the modern, Indian or Hindu-Arabic numeral system that most of the world uses today. Once zero was introduced, almost all of the mathematical mechanics would be in place to enable ancient Indians to study higher mathematics.

The concept of zero

Zero itself has a much longer history. The recently dated first recorded zeros, in what is known as the Bakhshali manuscript, were simple placeholders – a tool to distinguish 100 from 10. Similar marks had already been seen in the Babylonian and Mayan cultures in the early centuries AD and arguably in Sumerian mathematics as early as 3000-2000 BC.

But only in India did the placeholder symbol for nothing progress to become a number in its own right. The advent of the concept of zero allowed numbers to be written efficiently and reliably. In turn, this allowed for effective record-keeping that meant important financial calculations could be checked retroactively, ensuring the honest actions of all involved. Zero was a significant step on the route to the democratisation of mathematics.

These accessible mechanical tools for working with mathematical concepts, in combination with a strong and open scholastic and scientific culture, meant that, by around 600AD, all the ingredients were in place for an explosion of mathematical discoveries in India. In comparison, these sorts of tools were not popularised in the West until the early 13th century, though Fibonnacci's book liber abaci.

Solutions of quadratic equations

In the seventh century, the first written evidence of the rules for working with zero were formalised in the Brahmasputha Siddhanta. In his seminal text, the astronomer Brahmagupta introduced rules for solving quadratic equations (so beloved of secondary school mathematics students) and for computing square roots.

Rules for negative numbers

Brahmagupta also demonstrated rules for working with negative numbers. He referred to positive numbers as fortunes and negative numbers as debts. He wrote down rules such as: "A fortune subtracted from zero is a debt," and "a debt subtracted from zero is a fortune".

This latter statement is the same as the rule we learn in school, that if you subtract a negative number, it is the same as adding a positive number. Brahmagupta also knew that "The product of a debt and a fortune is a debt" – a positive number multiplied by a negative is a negative.

For the large part, European mathematicians were reluctant to accept negative numbers as meaningful. Many took the view that negative numbers were absurd. They reasoned that numbers were developed for counting and questioned what you could count with negative numbers. Indian and Chinese mathematicians recognised early on that one answer to this question was debts.

For example, in a primitive farming context, if one farmer owes another farmer 7 cows, then effectively the first farmer has -7 cows. If the first farmer goes out to buy some animals to repay his debt, he has to buy 7 cows and give them to the second farmer in order to bring his cow tally back to 0. From then on, every cow he buys goes to his positive total.

Basis for calculus

This reluctance to adopt negative numbers, and indeed zero, held European mathematics back for many years. Gottfried Wilhelm Leibniz was one of the first Europeans to use zero and the negatives in a systematic way in his development of calculus in the late 17th century. Calculus is used to measure rates of changes and is important in almost every branch of science, notably underpinning many key discoveries in modern physics.

But Indian mathematician Bhāskara had already discovered many of Leibniz's ideas over 500 years earlier. Bhāskara, also made major contributions to algebra, arithmetic, geometry and trigonometry. He provided many results, for example on the solutions of certain "Doiphantine" equations, that would not be rediscovered in Europe for centuries.

The Kerala school of astronomy and mathematics, founded by Madhava of Sangamagrama in the 1300s, was responsible for many firsts in mathematics, including the use of mathematical induction and some early calculus-related results. Although no systematic rules for calculus were developed by the Kerala school, its proponents first conceived of many of the results that would later be repeated in Europe including Taylor series expansions, infinitessimals and differentiation.

The leap, made in India, that transformed zero from a simple placeholder to a number in its own right indicates the mathematically enlightened culture that was flourishing on the subcontinent at a time when Europe was stuck in the dark ages. Although its reputation suffers from the Eurocentric bias, the subcontinent has a strong mathematical heritage, which it continues into the 21st century by providing key players at the forefront of every branch of mathematics.

PG and Research Department of Mathematics

Activities for the month of October

2017-2018

• ·        Started preliminary work for starting Math’s  Laboratory
• ·        Planning for National Level Seminar.
• ·        Class Committee meeting was conducted
• ·        Model Exam papers were issued to the students
• ·        Staff  Meeting was conducted
• ·        M.phil classes were started
• ·        Conducted Value Added Exam  for six batches
• ·        Portions were completed

Top 10 mathematical innovations

10. Logarithms (John Napier, JoostBürgi, Henry Briggs)

A great aid to anybody who multiplied or messed with powers and roots, logarithms made slide rules possible and clarified all sorts of mathematical relationships in various fields. Napier and Bürgi both had the basic idea in the late 16th century, but both spent a couple of decades calculating log tables before publishing them. Napier’s came first, in 1614. Briggs made them popular, though, by recasting Napier’s version into something closer to the modern base-10 form.

9. Matrix algebra (Arthur Cayley)

An ancient Chinese math text included matrix-like calculations, but their modern form was established in the mid-19th century by Cayley. (Several others, including Jacques Binet, had explored aspects of matrix multiplication before then.) Besides their many other applications, matrices became extremely useful for quantum mechanics. In fact, in 1925 Werner Heisenberg reinvented a system identical to matrix multiplication to do quantum calculations without even knowing that matrix algebra already existed.

8. Complex numbers (GirolamoCardano, Rafael Bombelli)

Before Cardano, square roots of negative numbers had shown up in various equations, but nobody took them very seriously, regarding them as meaningless. Cardano played around with them, but it was Bombelli in the mid-16th century who worked out the details of calculating with complex numbers, which combine ordinary numbers with roots of negative numbers. A century later John Wallis made the first serious case that the square roots of negative numbers were actually physically meaningful.

7. Non-Euclidean geometry (Carl Gauss, Nikolai Lobachevsky, JánosBolyai, Bernhard Riemann)

Gauss, in the early 19th century, was probably the first to figure out an alternative to Euclid’s traditional geometry, but Gauss was a perfectionist, and perfection is the enemy of publication. So Lobachevsky and Bolyai get the credit for originating one non-Euclidean approach to space, while Riemann, much later, produced the non-Euclidean geometry that was most helpful for Einstein in articulating general relativity. The best thing about non-Euclidean geometry was that it demolished the dumb idea that some knowledge is known to be true a priori, without any need to check it out by real-world observations and experiments. Immanuel Kant thought Euclidean space was the exemplar of a priori knowledge. But not only is it not a priori, it’s not even right.

6. Binary logic (George Boole)

Boole was interested in developing a mathematical representation of the “laws of thought,” which led to using symbols (such as x) to stand for concepts (such as Irish mathematicians). He hit a snag when he realized that his system required x times x to be equal to x. That requirement pretty much rules out most of mathematics, but Boole noticed that x squared does equal x for two numbers: 0 and 1. In 1854 he wrote a whole book based on doing logic with 0s and 1s — a book that was well-known to the founders of modern computer languages.

5. Decimal fractions (Simon Stevin, Abu’lHasan Al-Uqlidisi)

Stevin introduced the idea of decimal fractions to a European audience in a pamphlet published in 1585, promising to teach “how all Computations that are met in Business may be performed by Integers alone without the aid of Fractions.” He thought his decimal fraction approach would be of value not only to merchants but also to astrologers, surveyors and measurers of tapestry. But long before Stevin, the basic idea of decimals had been applied in limited contexts. In the mid-10th century, al-Uqlidisi, in Damascus, wrote a treatise on Arabic (Hindu) numerals in which he dealt with decimal fractions, although historians differ on whether he understood them thoroughly or not.

4. Zero and 3. Negative numbers (Brahmagupta)

Brahmagupta, a seventh-century Hindu astronomer, was not the first to discuss negative numbers, but he was the first to make sense of them. It’s not a coincidence that he also had to figure out the concept of zero to make negative numbers make sense. Zero was not just nothingness, but a meaningful number, the number you get by subtracting a number from itself. “Zero was not just a placeholder,” writes Joseph Mazur in his new book Enlightening Symbols. “For what may have been the first time ever, there was a number to represent nothing.”

2. Calculus (Isaac Newton, Gottfried Leibniz)

You know the story — Newton gets all the credit, even though Leibniz invented calculus at about the same time, and with more convenient notation (still used today). In any event, calculus made all sorts of science possible that couldn’t have happened without its calculational powers. Today everything from architecture and astronomy to neuroscience and thermodynamics depends on calculus.

1. Arabic numerals

Did you ever wonder why the Romans didn’t do much creative quantitative science? Try doing a complicated calculation with their numerals. Great advances in Western European science followed the introduction of Arabic numerals by the Italian mathematician Fibonacci in the early 13th century. He learned them from conducting business in Africa and the Middle East. Of course, they should really be called Hindu numerals because the Arabs got them from the Hindus. In any case, mathematics would be stuck in the dark ages without such versatile numerals. And nobody would want to click on a Top X list. (Wait — maybe they would. But you won’t see any list like that on this blog.)

Abel Prize 2017: Yves Meyer wins 'maths Nobel' for work on wavelets

Frenchman wins prestigious prize for theory that links maths, information technology and computer science

French mathematician Yves Meyer was today awarded the 2017 Abel Prize for his work on wavelets, a mathematical theory with applications in data compression, medical imaging and the detection of gravitational waves.

Meyer, 77, will receive 6 million Norwegian krone (about £600,000) for the prize, which recognises outstanding contributions to mathematics and is awarded by the Norwegian Academy of Science and Letters.

The Abel Prize has been awarded annually since 2003 and was last year won by Briton Andrew Wiles for his work on solving Fermat’s Last Theorem. It is considered by many to be a maths equivalent of the Nobel Prize, which has no prize for mathematics.

 Mathematics FOR Life 

Five Historic Female Mathematicians You Should Know

1 of 5

Hypatia (ca. 350 or 370 – 415 or 416)

(© Bettmann/CORBIS)

No one can know who was the first female mathematician, but Hypatia was certainly one of the earliest. She was the daughter of Theon, the last known member of the famed library of Alexandria, and followed his footsteps in the study of math and astronomy. She collaborated with her father on commentaries of classical mathematical works, translating them and incorporating explanatory notes, as well as creating commentaries of her own and teaching a succession of students from her home. Hypatia was also a philosopher, a follower of Neoplatonism, a belief system in which everything emanates from the One, and crowds listened to her public lectures about Plato and Aristotle. Her popularity was her downfall, however. She became a convenient scapegoat in a political battle between her friend Orestes, the governor of Alexandria, and the city’s archbishop, Cyril, and was killed by a mob of Christian zealots.

Sophie Germain (1776 – 1831)

When Paris exploded with revolution, young Sophie Germain retreated to her father’s study and began reading. After learning about the death of Archimedes, she began a lifelong study of mathematics and geometry, even teaching herself Latin and Greek so that she could read classic works. Unable to study at the École Polytechnique because she was female, Germain obtained lecture notes and submitted papers to Joseph Lagrange, a faculty member, under a false name. When he learned she was a woman, he became a mentor and Germain soon began corresponding with other prominent mathematicians at the time. Her work was hampered by her lack of formal training and access to resources that male mathematicians had at the time. But she became the first woman to win a prize from the French Academy of Sciences, for work on a theory of elasticity, and her proof of Fermat’s Last Theorem, though unsuccessful, was used as a foundation for work on the subject well into the twentieth century.

Ada Lovelace (1815 – 1852)

(© Heritage Images/Corbis)

Augusta Ada Byron (later Countess of Lovelace) never knew her father, the poet Lord Byron, who left England due to a scandal shortly after her birth. Her overprotective mother, wanting to daughter to grown up as unemotional—and unlike her father—as possible, encouraged her study of science and mathematics. As an adult, Lovelace began to correspond with the inventor and mathematician Charles Babbage, who asked her to translate an Italian mathematician’s memoir analyzing his Analytical Engine (a machine that would perform simple mathematical calculations and be programmed with punchcards and is considered one of the first computers). Lovelace went beyond completing a simple translation, however, and wrote her own set of notes about the machine and even included a method for calculating a sequence of Bernoulli numbers; this is now acknowledged as the world’s first computer program.

Sofia Kovalevskaya (1850 – 1891)

(© Michael Nicholson/Corbis)

Because Russian women could not attend university, Sofia Vasilyevna contracted a marriage with a young paleontologist, Vladimir Kovalevsky, and they moved to Germany. There she could not attend university lectures, but she was tutored privately and eventually received a doctorate after writing treatises on partial differential equations, Abelian integrals and Saturn’s rings. Following her husband’s death, Kovalevskaya was appointed lecturer in mathematics at the University of Stockholm and later became the first woman in that region of Europe to receive a full professorship. She continued to make great strides in mathematics, winning the Prix Bordin from the French Academy of Sciences in 1888 for an essay on the rotation of a solid body as well as a prize from the Swedish Academy of Sciences the next year.

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Emmy Noether (1882 – 1935)

(Public Domain)

In 1935, Albert Einstein wrote a letter to the New York Times, lauding the recently deceased Emmy Noether as “the most significant creative mathematical genius thus far produced since the higher education of women began.” Noether had overcome many hurdles before she could collaborate with the famed physicist. She grew up in Germany and had her mathematics education delayed because of rules against women matriculating at universities. After she received her PhD, for a dissertation on a branch of abstract algebra, she was unable to obtain a university position for many years, eventually receiving the title of “unofficial associate professor” at the University of Göttingen, only to lose that in 1933 because she was Jewish. And so she moved to America and became a lecturer and researcher at Bryn Mawr College and the Institute for Advanced Study in Princeton, New Jersey. There she developed many of the mathematical foundations for Einstein’s general theory of relativity and made significant advances in the field of algebra.

George Boole: Five things you need to know about the man behind today's Google Doodle

The mathematician became one of the founding fathers of modern computer science and engineering – despite never finishing school.

Google has chosen to honour the 200th birthday of the British mathematician George Boole, whose work on logic and algebra has been credited with laying the foundations for the computer age. 

Here are five things you may not know about him:

1. His system of Boolean Logic paved the way for modern electrical engineering and computer science 

What exactly today’s Google Doodle means

Boole created a system, known as Boolean Logic, where all mathematical variables could only boil down to two variables - “true” or “false” or “on” and “off”. 

These ideas were put to use more than 70 years after his death when Victor Shestakov at Moscow State University in Russia proposed using the system to design electrical switches, according to the Scientific American. 

This simple “on-off” system later went on to form the bedrock of all computer code.

2. He was self-taught and had little formal education

Boole was born in Lincoln in 1815 to a shoemaker and had little formal schooling beyond primary school. 

At the age of 16 he was forced to become the main breadwinner for his parents and three younger siblings after his father’s business collasped. 

He became a teacher in Doncaster and Liverpool before returning to Lincoln where he got involved in the local Lincoln Mechanics’ Institution when it was founded in 1833. 

From there he began to learn mathematics but it took him several years to master calculus as he had no tutor. 

3. He was a polymath 

In addition to mathematics, he also taught himself French, German and Latin. 

Once he had mastered those languages, he went on to teach himself Greek and translated a poem by the ancient Greek poet Meleager which his father had published in 1828 when he was just 14.

This led to a local controversy as a Lincoln schoolmaster claimed a 14 year old boy with little formal schooling could not possibly have done it by himself. 

4. He founded a school when he was just 19

While Boole was learning mathematics and foreign languages, he also found the time to found a small school in his hometown in 1834. 

Four years later in 1838 another schoolmaster in Waddington, Lincolnshire died and Boole was invited to run his boarding school. 

In 1840, he opened a boarding school in Lincoln again and was beginning to have his mathematical work published. 0:3

5. He became the first professor of mathematics at the newly founded Queen’s College, Cork (now University College Cork) in Ireland in 1849. 

Despite having no university degree, Boole’s increasing fame in mathematical circles lead to other mathematicians recommending him for the professorship. 

It was there that he met his wife Mary Everest in 1850 and would go on to have five children before his death in 1864. 

To celebrate the bicentenary of his birth, University College Cork, has set up georgeboole.com to mark his life and works.

It has set up the Boole2School programme which takes a “George Boole” into classrooms across Ireland to learn more about him.

Math Jokes:

       ``Three employees of NOSC (an engineer, a physicist and a mathematician) are staying in a hotel while attending a technical seminar. The engineer wakes up and smells smoke. He goes out into the hallway and sees a fire, so he fills a trashcan from his room with water and douses the fire. He goes back to bed.
Later, the physicist wakes up and smells smoke. He opens his door and sees a fire in the hallway. He walks down the hall to a fire hose and after calculating the flame velocity, distance, water pressure, trajectory, etc. extinguishes the fire with the minimum amount of water and energy needed.
Later, the mathematician wakes up and smells smoke. He goes to the hall, sees the fire and then the fire hose. He thinks for a moment and then exclaims, "Ah, a solution exists!" and then goes back to bed.

Many of the toughest decisions faced by cancer patients involve knowing how to use numbers -- calculating risks, evaluating treatment options and figuring odds of medication side effects.

But for patients who aren't good at math, decision science research can offer evidence-based advice on how to assess numeric information and ask the right questions to make informed choices.

"The ability to understand numbers is associated with all kinds of positive health outcomes, including for cancer patients," said Ellen Peters, professor of psychology at The Ohio State University.

"The problem is that too many people aren't good with numbers or are afraid of math. But we're starting to figure out the best ways to help these patients so they aren't at a disadvantage when it comes to their treatment."

Peters, who is director of the Decision Sciences Collaborative at Ohio State, presented research on cancer patients' health and numeracy -- the ability to understand and use numbers -- Feb. 20 in Boston at the annual meeting of the American Association for the Advancement of Science.

Numerous studies have shown that people who are less numerate experience worse health outcomes. Peters says these are examples of the "tyranny of numbers." For example, diabetics with lower numeracy scores have higher blood sugar levels. And children with diabetes have higher blood sugar levels if their parents are less numerate.

A 2010 study by Peters shows how skill with numbers can affect breast cancer patients. In this research, women who had surgery for breast cancer were presented with options for further treatment, including hormonal treatment, chemotherapy, combined treatment or no treatment. The patients were given information, based on their characteristics, on how likely they were to survive 10 years for each possible treatment plan.

The patients were then asked to estimate, based on this information, what their own chances of survival were for 10 years with each treatment.

The patients who scored higher in numeracy were more pessimistic than the data suggested they should be. But their estimates of their own survival did vary based on the numbers they were given.

"For those who were less numerate, their survival estimates were pessimistic, but remained the same no matter what numbers they were presented. It was as if they didn't read the numbers at all," Peters said.

"This is critical. We were giving them information that should help them choose the best treatment, but they were ignoring it."

Other research shows that less numerate people "rely more on their emotions" to make health-related decisions. They are also more swayed by how information is presented to them rather than by the information itself, she said.

If a patient recognizes that he or she is not good with numbers, how can he or she cope? Peters said research suggests four strategies:

Ask for the numbers. This may seem counter-intuitive, but research backs it up. In one study, less numerate people were asked to estimate their risk of side effects from a medication. Some were given numeric information about the risks of a particular side effect, while others were told only that there was a risk. When they weren't given the numbers, 70 percent of less numerate people overestimated their risk, but only 17 percent did when given the numbers. They didn't do as well at evaluating risk as more numerate people when given the numbers, but they still did much better than when they didn't have them at all.

Ask what the numbers mean. Along with the numbers, doctors should be able to tell you what the numbers mean in practical terms. "If 80 percent of people are helped by this particular drug, is that good or bad? Ask your doctor to say if this is above or below average, if it is a fair, good or excellent treatment compared to other options," she said.

Ask for absolute risk. Saying that a particular drug doubles your risk of a dangerous side effect sounds scary. But this is what is called a relative risk. The absolute risk is more important. "If you're doubling your risk from 0.01 percent to 0.02 percent, that is much less threatening than if you are doubling from 10 percent to 20 percent," Peters said.

Cut down the choices. If you're given a bewildering list of choices for treatment, ask your doctor to choose the best two options to consider. "It is absolutely OK to tell the doctor that this is too complicated. You don't need to have doctors make a treatment decision for you, but they should be able to identify the most critical information for you to consider."

Health care providers should do a better job in presenting critical information to patients, Peters said. But when they don't, patients should ask for help.

"Numbers are important, whether you like them or not. And nowhere are they more important than when it comes to your health," she said.