There is a common belief that females are less mathematically capable than males. This belief is fairly constant across populations.This belief is not entirely unfounded. Although evidence from the many studies performed on gender differences in mathematics is inconsistent, small but statistically significant differences are the norm. There are no significant differences between boys and girls math achievement in elementary school and few differences at any age, although these differences are small, parents and teachers often expect large discrepancies between boys and girls performance in math class. Because, others expectations can have a strong influence on one's attitudes and behavior. Even parents unfamiliar with this study tend to rate daughters as less mathematically able than sons’, even if these daughters and sons are performing at the same level. Many parents accept their son’s mathematical successes as evidence of innate ability, while they think of their daughters' successes as hard work compensating for an innate lack of ability.

Over the years, there have been many improvements for women in the mathematics field. There is still some discrimination, and men still outnumber women in this field, but that is beginning to change. The discussions that are now occurring related to this topic have been a huge contribution to the start of these changes. People now talk about the way things are for women in this field, why they are this way, and how we can strive to improve them. Times have changed, and more women are entering the upper echelons of the once, exclusively masculine professions of mathematics and science, when there, they are thriving[6].

Mathematicians in general are not a well known group. However, women mathematicians are even less well known. It is possible to read most histories of mathematics and find little or no mention of women mathematicians, even if few there were. This article is therefore intended to trace the impact some of these women have had on the development of mathematics. It appears that the reason for this was that in almost any age women faced many barriers, which men of far less ability did not have to face. For women, talents alone were not sufficient criteria for success in mathematics. The woman also needed to have drive and determination, not only to ignore role stereo types, but also to overcome the restrictions placed on their education. It was not until these barriers were crossed that women were able to develop and enjoy their talents[2].

The other women discussed in this article are daughters of mathematicians.. Women in mathematics and science need to be visible, to openly display their passion for their chosen profession. They need to carry their understandings of mathematics into their communities and society at large. They should also be role models and sources of encouragement for young girls and women. Women have made important advances in education over the last few decades, and the gender gap[3].

Beginning from the early centuries, women have struggled, not only to prove themselves, but also to be accepted by the general society, as someone who is equally smart and competent as men. This article is about the contribution of women in mathematics, the following section gives you some of the many renowned women, who have inspired and paved way for many female aspirants who strive to contribute in this field. Some of the famous women mathematicians who proved their mathematical genius and undoubtedly, will always find a place in the pages of history [4],[6].

2.1 Hypatia of Alexandria (AD 350 – 415)

Hypatia was one of the earliest known female mathematicians in the world, known not only for her excellence in mathematics and astronomy. Hypatia was born in Alexandria, Egypt, around 350AD and was a recognized scholar. Besides being a mathematician, she was an astute astronomer and a philosopher. She believed in the theories of Plato and Aristotle.

Her immortal contributions to the mathematical world, paved the way for inexhaustible research on many topics. She wrote  a commentary on the 13^th volume of the famous Greek mathematics textbook, 'Artihmetica'. She edited Ptolemy's famous version of the 'Almagest'. She edited her father's commentary on 'Euclid's Elements'. She also made a commentary on a famous work on Conics by 'Apollonius'.

2.2 Maria Gaetana Agnesi (1718 -1799)

She was considered to be the first important female mathematician after Hypathia. As mentioned already, this Italian mathematician and philosopher was a child prodigy, who was known as the "Seven-Tongued Orator" at the age of eleven. She could fluently speak Italian and French by the age of 5, and by her eleventh birthday, she mastered Greek, Hebrew, Spanish, German, and Latin. By the time she was 14, she was studying ballistics and geometry. She is known for her detailed explanation of differential and integral calculus in her book 'Instituzioni analitiche ad uso della gioventù italiana', translated as analytical institutions for the use of Italian Youth in English, which was published in the year 1748 in Milan.She also gained fame for the curve called the "Witch of Agnesi", named after her. She wrote the equation of this curve, 'versiera' as it was previously known, which was originally studied and constructed by Pierre de Fermat and Guido Grandi. Her work was translated into English, where, the word versiera, which seemed very similar to the word 'avversiera' (meaning 'the wife of the devil') got translated as the word 'witch'. Hence, the curve came to be known as the Witch of Agnesi.

2.3 Marie-Sophie Germain (1776-1831)

A truly inspirational French woman who overcame all odds to achieve fame/recognition as a true mathematician She was self-taught, introduced to this subject by the story of Archimedes' death that she read in her father's library. She was intrigued by the thought that how could geometry be so interesting that archimedes ignored his death but, not the subject?. This sparked her curiosity, and even when her own family did everything to prevent her from studying, through her dedication and passion, she compelled them to allow her to learn. Because she was a woman, she would submit her analysis papers under a man's name, M. LeBlanc, but when her teacher impressed by the paper insisted on meeting her, it was then that her true identity came forth. She continued to learn under the tutelage of renowned mathematicians of the time including Josephi Louis Lagrange, Adrien-Marie Legendre, and Carl Friedrich Gauss.

Sophie was born in a wealthy upper class French family, in 1776, the year of the American Revolution. Any sort of 'brainwork', was regarded as unhealthy and dangerous for women in those times. Sophie faced a lot of problems in getting her education, due to the social taboos that existed in her society. Still, she learned mathematics and carved a niche for herself in this field. She is often called 'The Revolutionary Mathematician'. She worked on number theories and gave many an interesting theorems on prime numbers. She even discovered new identities. Many such numbers are now called "Sophie Germain primes”. Her work on 'Fermat's Last theorem' was a path-breaking one. She was the first woman to attend both 'Academie des Science' and an 'Institut de France' session.

2.4 Augusta Ada King, Countess of Lovelace (1815-1852)

Born as Augusta Ada Byron, the only legit daughter of Lord Byron, this English mathematician achieved quite a lot of fame and recognition for her analytical and computing abilities. In fact, she is widely known as the world's first computer programmer. Her mother discouraged her to learn literature as she feared that Ada too, may turn out to be reckless and emotionally unstable as her father, with whom she separated soon after Ada's birth. However, Ada showed great interest in becoming an analyst and a metaphysician, and was highly intelligent and farsighted for her time. In fact, when she heard of Charles Babbage's idea of inventing a calculating engine, she was the only one who believed that computers could do more than just calculating. Babbage called her ‘The Enchantress of Numbers’.

She suggested Babbage to write a plan where the 'Analytical Engine' could calculate Bernoulli numbers, which later on developed as the very first 'computer program'. This is the reason, why the U.S. Department of Defense named a software language of theirs, as 'Ada' in the year 1979.She translated the memoir of Italian mathematician Luigi Menabrea on Babbage's machine, the Analytical Engine. She also included her own notes explaining the functioning of the Analytical Engine. Her notes were lengthier than the original translation.

2.5 Sofia Vasilyevna Kovalevskaya (1850-1891)

She was born on 15th January, 1850, in Moscow, Russia. She was the first Russian woman to be recognized as a mathematician. Sofia's interest in mathematics began when she observed her room's wallpaper that had calculus notes of her father. These papers were put on the wall, for there was a shortage of wallpaper. Her passion for studies was such that she even agreed to marry. For a young and unmarried woman, in those times, was not allowed to travel alone to outside place, the nearest university being in Switzerland. She had to leave her homeland, Russia, to fulfill her dreams of becoming a teacher. Women were not allowed to become lecturers in universities. Yet, after going through many hardships, she managed to attain a Ph.D. from the University of Gottingen, became a lecturer at the University of Stockholm, and won the Prix Bordin from the French Academy of Sciences. She opposed her elders to pursue higher studies. She researched on differential equations known as 'Kovalevskaya Top'. She worked on the 'Cauchy-Kovalevskaya theorem', a very basic theorem that helps understand differential equations.

2.6 Hertha Marks Ayrton (1854-1923)

Born as 'Phoebe Sarah Marks', she changed her name to Hertha during her teens, after an eponymous heroine of a Swinburne poem that profoundly criticized organized religious beliefs, as Phoebe herself was an agnostic. This English mathematician paved her way from being the daughter of a seamstress to becoming the recipient of the Royal Society's prestigious Hughes Medal for her discovery in the physical sciences. After passing the Mathematical Tripos in 1880, and receiving her B.Sc. degree from the University of London, she taught mathematics to children. In her career, she devised and solved mathematical problems, a majority of which were published in the Mathematical Questions and Their Solutions from the "Educational Times". She also was a renowned inventor and physicist.

In 1884, she invented a line-divider, a draftsman's device that could divide a line equally in different parts, and also enlarge or reduce the figures. Including this, she registered 26 patents in her lifetime; five on mathematical dividers, 13 on arc lamps and electrodes, and the remaining on the propulsion of air. In 1899, she became the first woman to read her own paper, titled "The Hissing of the Electric Arc", before the Institution of Electrical Engineers (IEE). Soon, she was elected as the first female member of the IEE. She was also the first woman to read a paper before the Royal Society. In 1904, she read her paper titled "The Origin and Growth of Ripple Marks" which was later published in the Proceedings of the Royal Society. For her work and research on electric arc and sand ripples. She was bestowed the Hughes Medal by the Royal Society in 1906. She is one of the only two women who have been given this honor so far[2].

2.7 Mary Fairfax Somerville (1780-1872)

Mary Somerville was born at a time when it was considered that women did not possess the intellect or the capacity required to understand subjects such as, math and science. This Scottish mathematician and science writer was hardly given a chance to receive an education due to her gender, but she did manage to teach herself in the long run, with some unofficial help from her uncle and her brother's tutor. After the death of her first husband, who didn't consider much of her intellectual and academic interests, she got the freedom to pursue her studies, which continued after her second marriage to Dr. William Somerville, a supportive and understanding partner who admired her for her academic abilities. With the freedom to learn, she achieved great honors during her lifetime for her work in mathematics and science. Even till her old age (she died at the age of 92), she continued to "read books on the higher algebra for four or five hours in the morning, and even to solve problems", as stated in her autobiography.

She gained fame after the successful translation of Laplace's Mécanique Céleste. titled 'The Mechanism of the Heavens’. This work is considered as her most famous mathematical writing ever. She was the first female to be nominated as the member of the Royal Astronomical Society, jointly along with Caroline Herschel.It was her writing involving the discussion of a hypothetical planet perturbing Uranus, in the 6th edition of On the Connexion of the Physical Sciences (1842) that influenced John Couch Adams and led him to discover Neptune. In 1848, she published Physical Geography, when she was 69 years old. This was her most successful book till that time, and was widely used in schools and universities for half a century. She was the recipient of the 'Victoria Medal' of the Royal Geographical Society in the year 1869. In fact, William Whewell coined the term "scientist" when he was reviewing her work, 'On the Connexion of the Sciences', in 1834[3].

2.8 Amalie Emmy Noether (1882-1935)

Amalieg Emmy Voether was born on 23^rd March, 1882 in Germany. Amalie was a mathematician who is remembered for her revolutionary work in many fields. Albert Einstein described her as"the most important woman in mathematical history, since, the higher education of women began".

She is known for her revolutionary work in theoretical physics and abstract algebra (with special focus to rings, groups, and fields). The theorem named after her, Noether's Theorem successfully explains the relationship between conservation laws and symmetry. The term Noetherian Ring was coined in her honor, arising from her publication, Idealtheorie in Ringbereichen, which was called "revolutionary" by renowned algebraist Irving Kaplansk. Her findings changed the approach of abstract algebra. Her contributions in mathematics were highly significant. She worked on theories of algebraic invariants and number fields, calculus of variations, non commutative algebras, hyper complex numbers, and algebraic topology. She also developed the theory of ideals in commutative rings. She won the Ackermann-Teubner Memorial Award, along with Emil Artin for their contributions to mathematics. She also taught many renowned future mathematicians including Grete Hermann, Ernst Witt, and Max Deuring.

2.9 Dame Mary Lucy Cartwright

She is one of the most successful British mathematicians, who, through her mathematical findings and publishing, gained varied accolades of the highest degrees. She was the first woman to obtain a final degree in mathematics from Oxford. After graduating, she taught mathematics for four years to school children and then pursued her studies further in the subject at Oxford, under the guidance of famous mathematicians, G. H. Hardy and E. C. Titchmarsh. She became the Director of Studies in Mathematics at Girton College in 1936 after holding the post of a lecturer in this college. In her career, she published more than 100 papers on topics such as classical analysis, differential equations, and related topological problems.

She is best known for Cartwright's theorem, named after her, which "gave an estimate for the maximum modulus of an analytic function which takes the same value no more than p times in the unit disc." In the year 1945, she simplified Charles Hermite's elementary proof of the irrationality of π. Her collaboration with famous mathematician J. E. Littlewood, in the 1940s, helped find the solution to the Van der Pol equation and eventually paved the way for the discovery of many of the phenomena that later came to be known as "chaos”. She became the President of the London Mathematical Society in the year 1951, the first woman holding this title. She was the first female mathematician to be elected as a “Fellow of the Royal Society (FRS)”, in the year 1947. She was also the first female to be awarded the Sylvester Medal of the Royal Society in 1964. In 1968, she received the De Morgan Medal of the London Mathematical Society, and in 1969, she was honored by the Queen and became 'Dame Mary Cartwright', Dame Commander of the Order of the British Empire.

2.10 Euphemia Lofton Haynes (1890-1980)

Born as Martha Euphemia Lofton, she was a brilliant student with a love for mathematics. This African-American mathematician pursued her studies even after her marriage, as a result of which she earned a Ph.D. from the Catholic University of America, her dissertation being on "The Determination of Sets of Independent Conditions Characterizing Certain Special Cases of Symmetric Correspondences". She gave 47 years of her life to the field of education, teaching mathematics and English in schools and colleges, and playing an evident role in the integration of the public schools in D.C. She was the first African-American woman to have earned a Ph.D. in mathematics. She was also the first female to chair the District of Columbia School Board. After her death, she left USD 700,000 to the Catholic University of America. These funds were used to establish and support the Euphemia Lofton Haynes Chair in the Department of Education and to support a loan fund for needy students in the School of Education [6].

2.11 Marjorie Lee Browne (1914-1979)

Yet, another brilliant mathematician of the African-American descent, Marjorie Lee Browne dedicated her knowledge and her personal earnings to help other gifted and talented African-American students to thrive and learn. Similarly, how her father, who was a railway postal clerk but, was gifted in math, encouraged her to study the subject. After completing her high-school studies, she majored in mathematics and graduated cum laude, in the year 1935 from Howard University. She also earned her Ph.D. in the year 1949 from the University of Michigan; this was one of the very few universities that accepted African-American students in the United States at the time. After her Ph.D., she joined the North Carolina College (now North Carolina Central University (NCCU)), and contributed 30 years of her life teaching and researching in this institution. She also served as the head of the department from 1951 to 1970. Her work mainly focused on topology, linear and matrix algebra. Her work also, showcased simple evidences of topological properties significance and relations between classical groups.

After Euphemia Haynes, Browne and Evelyn Boyd Granville were the next African-American women to have earned a Ph.D. in mathematics. Dr. Browne became the first to receive the W.W. Rankin Memorial Award for Excellence in Mathematics Education, in the year 1975. The award states the following, "She pioneered in the Mathematics Section of the North Carolina Teachers Association, helping to pave the way for integrated organizations."She gauged the crucial need of computer science early on, which is why she wrote a USD 60,000 grant to IBM, in the year 1960, to bring a computer to NCCU? It was one of the very first computers in academic facilities, and most likely the first at a historically black school.

2.12 Julia Bowman Robinson (1919-1985)

A truly admirable and inspirational American mathematician, who irrespective of her health issues and insecurities, managed to achieve honors that no other American woman had achieved. She contracted scarlet fever and rheumatic fever during childhood, as a result of which, most of her childhood was spent in isolation. She graduated from San Diego High in 1936, with honors in mathematics and science courses. She also won the Bausch-Lomb Medal for all around excellence in science. She received her B.A. degree in the year 1940, and her Ph.D. in 1948 from the University of California at Berkeley. Mathematics was not only a passion for her, but a treatment that helped her come out of depression when she was unable to conceive due to health issues. Her husband, renowned mathematician Raphael Mitchell Robinson, helped her regain her interest in mathematics to overcome her grief.

She was the first woman mathematician to be elected in the United States National Academy of Sciences, in 1975.She is best known for her work on Hilbert's Tenth Problem that comes under Diophantine analysis, and Decidability. She also worked for the office of Naval Research on a problem in hydrodynamics. She was also the first female officer in the American Mathematical Society, and after four years, she ended up becoming the first woman president of the society. She was also the recipient of the MacArthur Foundation Prize Fellowship of USD 60,000 for five years, in the year 1983.

2.13 Alexandra Bellow (1935 - Present)

Alexandra Bellow is a famous mathematician from Romania, who was working as a full-time professor at the Northwestern University until her retirement in the year 1998. Her career was highly illustrious and she is well-known for her mathematical work, mainly in ergodic theory. She graduated in the year 1957, from the University of Bucharest, and received her Ph.D. from Yale in 1959. She is the wife of the famous mathematician Alberto Calderon, who is no longer with us. Her ex-husband was the famous Nobel Prize recipient Saul Bellow who is known for his literary works. Throughout her career, she has contributed immensely to the fields of ergodic theory, probability, and analysis. She also contributed to the editorial boards of the Transactions of the American Mathematical Society, the Annals of Probability, and Advances in Mathematics. She published various papers throughout her career and is the recipient of awards such as, Fairchild Distinguished Scholar Award and Humboldt Prize[4].

2.14 Shafi Goldwasser (1958 - Present)

Shafrira Goldwasser is an Israeli-American computer scientist, who is currently a Professor at the Massachusetts Institute of Technology, teaching electrical engineering and computer science. She is also the Professor of Mathematical Sciences at the Weizmann Institute of Science, Israel. She has worked magnificently in areas including complexity theory, computation number theory, cryptography, and her work with zero-knowledge proofs is highly acknowledged. Her husband, computer scientist Nir Shavit, is also a Gödel Prize winner like herself.

She is the first recipient of the RSA (Research Scholar Award) Professorship, which she received in the year 1997.She has won the Gödel Prize twice in theoretical computer science. First in 1993, for The knowledge complexity of interactive proof systems, and in 2001, for Interactive Proofs and the Hardness of Approximating Cliques. She has won many other awards, a few among them are: ACM (Association Computing Machinery) Grace Murray Hopper Award, RSA Award in Mathematics for outstanding mathematical contributions to cryptography, Benjamin Franklin Medal in Computer and Cognitive Science, and the Turing Award for her work in the field of cryptography, along with Silvio Micali [6].

Mathematics in India at that time was family-based. Mathematical education was largely restricted within the family and there was not much scope of innovation. Father passed on the commentaries to his son, who pursued it. The role of women in mathematics during those times was highly restricted or even probably non-existent. Religion also played a key role. Mathematical beliefs were tantamount to religious beliefs and changing religious beliefs were not acceptable. The next significant figure was Bhaskara I (7th century A.D.). He was a contemporary of Brahmagupta at the Ujjain center and led the Asmaka School. This school would have the study of the works of Aryabhata as their main concern and certainly Bhaskara was a commentator on the mathematics of Aryabhata. More than 100 years after Bhaskara, the astronomer Lalla, lived another commentator on Aryabhata. Bhaskara II (1114 - 1185 A.D.) also known as Bhaskharacharya was born in Maharashtra. He is considered as an outstanding poet-mathematician of his times. He was the head of the astronomical observatory at Ujjain, where, other famous Indian mathematicians including Brahmagupta had studied and worked previously. He worked on algebra, number systems, and astronomy. He wrote beautiful texts illustrated with mathematical problems and he provided the best summary of the mathematics and astronomy of the classical period. He made fundamental contributions to the development of number theory, the theory of indeterminate infinite series expressions for sine, cosine and tangent, computational mathematics, etc. 200 years after Bhaskara did any significant work happened in Indian Mathematics. Bhaskara produced six works during his lifetime: Lilavati, Bijaganita, Siddhantasiromani, Vasanabhasya of Mitaksara, Brahmatulya, and Vivarana. .

Lilavati is the first part of Bhaskaracharya's work Siddhantashiromani, which he wrote at the age of 36. Siddhantashiromani consists of four parts namely1) Lilavati 2) Algebra 3) Planetary motions and 4) Astronomy [7].

Lilavati has an interesting story associated with how it got its name. Bhaskaracharya created a horoscope for his daughter Lilavati, stating exactly when she needed to get married. He placed a cup with a small hole in it in a tub of water, and the time at which the cup sank was the optimum time Lilavati was to get married. Unfortunately, a pearl fell into the cup, blocking the hole and keeping it from sinking. Lilavati was then doomed never to wed, and her father Bhaskara wrote her a manual on mathematics in order to console her, and named it Lilavati. This appears to be a myth associated with this classical work. Lilavati was used as a textbook in India in Sanskrit schools for many centuries. Even now, it is used in some Sanskrit schools [8].

The last verse of Lilavathi demonstrates the poetic brilliance of Bhaskara: “(Lass) Lilavati is born in a respectable family, stands out in any group of enlightened persons, and has mastered idioms and proverbs. Whomsoever, she embraces will be happy and prosperous" [1],[8].

However, the same verse could also be interpreted as:”This Lilavati clearly explains fractions, simple fractions, multiplication etc. It also, beautifully describes problems in day-to-day transactions. Rules are transparent and examples are beautifully worded. Those who master this Lilavati will be happy and prosperous" [9].

Bhaskaracharya wrote this work by selecting good parts, from Sridharacharya's Trishatika and Mahaviracharya's Ganitasarasamgraha and adding material of his own. Lilavati became quite popular in India during the time it was first composed [10].

Lilavati mainly deals with what we call as `Arithmetic' in today's mathematical parlance. It consists of 279 verses written in Sanskrit in poetic form. There are certain verses which deal with Mensuration, Volume of pyramid, cylinders, heaps of grains etc., wood cutting, shadows, and trigonometric relations and also on certain elements of Algebra such as finding an unknown quantity subject to certain constraints using the method of supposition. We have divided the contents of Lilavati into 4 chapters namely: (i). Arithmetic (ii). Algebra (iii). Trigonometry and Geometry (iv) Discrete Mathematics. Arithmetic is directly related to commerce which is evident. An important thing to observe is that the verse claims that both addition and subtraction could be performed place-wise either from right to left or vice-versa. Bhaskaracharya provides a method for finding the solution which makes use of the Euclidean Algorithm for finding the Greatest Common Divisor (G.C.D). The Kuttaka method is said to be an important contribution of Bhaskaracharya [7].

Lilavati, Bhaskaracharya's monumental work is not only a literary treatise, but also occupies a distinguished and honored place in the history of Mathematics. It is a testimony to the extra- ordinary mathematical acumen of Bhaskaracharya, who is regarded as one of the most innovative mathematicians of India of his era. He was also an excellent teacher as indicated by the teasing and pleasing verses through which he tests his disciple's abilities to solve mathematical problems. Bhaskaracharya may not know anything about what a Proof is. However, his terse verses which contain the algorithmic rules and the generalizations which he recommends indicate that there may have been a certain rationale or reasoning in the minds of the mathematicians of those times. Moreover, his methods are not vague by any means and are quite precise. They probably relied more on intuition and did not feel the necessity of providing the rationale they had, if any. This is best example by the genius, Srinivasa Ramanujan of our times whose intuitive leaps of imagination were closer to Bhaskaracharya in spirit [5],[7].

However, this was also the limitation of Indian Mathematics. It did not nourish to its full potential. There are several reasons for this. There were no symbols or notation invented to handle mathematical objects and this was a huge handicap. This is a very important process which leads to the unreasonable effectiveness of mathematics. The language in which Sanskrit was difficult and only the very well learned scholars could decipher the poetic verses. The other reason for its loss of popularity was the lack of a proof or elaboration of the rationale behind the methods. Western mathematics and science very much demanded not just the result but why the result was true, if it was true at all. This may seem too much to ask for, but it was useful, because it guaranteed the determinism of Mathematics [11],[12].

All mathematical knowledge became sacrosanct, thanks to the rigorous demand of proof. It became sacred and deterministic to the extent that many people would take to it because they could not face the realism of the world and other sciences. Paul Erodes, the Hungarian mathematical genius of our times was an example of this kind. He took to mathematics because it was the only thing in this world that was guaranteed to be true, whatever truth meant to him. Nothing else appealed to him because of their innate uncertainty barring mathematics. Another huge handicap of Indian mathematics was the lack of illustrations or diagrams. As a result of this, Indian treatment of geometry was only very elementary. It is possible that they used some diagrams while solving problems, but this was never communicated in their works. What these ancient Indian mathematicians seem to excel was in their intuitive way of thinking and the algorithmic approach to mathematics. This is quite useful, especially in the computer age where such algorithmic approach would have received a huge support and would have been an advantage. Another advantage of Indian mathematics was the conciseness of the documentation.