Womens Contributions To Mathematics Essay, Research Paper

Abstract

Women in the world of mathematics is a subject that people rarely hear about. The only

time people do is if it?s a female math teacher. But what many do not know is that

women have made extremely important contributions to the world of mathematics.

Women have been documented to be involved in mathematics, since as early as the fifth

century A.D. Women such as Hypatia, Maria Gaetana Agnesi, Sophie Germain, Emmy

Noether, Ruth Moufang and Sun-Yung Alice Chang. These women have lived through

difficult times such as women?s oppression, the French Revolution, World War I and II,

which included Hitler?s administration over women?s schooling, and social prejudices.

This did not stop their yearning for math though. These women combined have earned

many different awards, specifically ones usually given to men. They have conquered the

biases people have had towards them and made what they do best count. Many of their

theorems and equations are still used today, and some are even being perfected by others.

It is important that the reader realizes that educating children about women in

mathematics is important. Many children think of mathematicians as men, and that is

totally untrue. That thought could possibly contribute to the fact that women are less

likely to enter the mathematics field compared to men. This is because they are not

educated properly on the subject, and are not given the opportunity to excel. There are

many more women in mathematics then mentioned above, but the ones named are very

important to the field and children need to know that. By taking these 6 women?s

contributions and focusing on how they apply to the middle school curriculum would be

very useful to any teacher. The children could each pick a female mathematician, and

make a poster and do a presentation about their findings. It could also be done as a

group project. As long as the topic gets discussed and that the girls come out feeling like

they could also get involved in mathematics.

Women?s Contributions to Mathematics

In the world of mathematics, you rarely hear anything about women

mathematicians. Although not much is said about women and math, there are many

women mathematicians who have made significant contributions to the field. From as

early as 370 AD, women have been contributing to the study of equations, theorems, and

even solving problems that have deemed themselves in the mathematical world as

impossible. Because of the time period that these women lived, many were not

recognized for their achievement; some were even banished or killed. Names such as

Hypatia, Maria Gaetana Agnesi, Sophie Germain, Emmy Noether, Ruth Moufang, and

Julia Bowman Robinson may not be common to the everyday person. But to

mathematicians around the world, especially women, they are a sign of achievement and

determination in a field dominated by men. In order to make women recognized in the

field of mathematics, educators need to spend time teaching their students that math is not

just for males. Because of the contributions of the women named above, math

exploration has been furthered and many questions have been answered, although some

are still to this day unresolved.

Hypatia

370?-415 AD

Hypatia is the first, truly documented woman mathematician. Her works have

given way to famous male mathematicians such as Newton, Descartes, and Leibniz.

Raised in ancient Egypt during the time that Christianity started to take over many other

religions, it was hard for Hypatia to study anything in an age where males dominated

many fields of study. Hypatia was looked at, though, as a woman of strong character, and

as a strong orator, astrologist, astronomist, and mathematician. Raised mostly by her

father, Theon, a known mathematician of the times, Hypathia gained a lot of knowledge

at a young age. She studied under her father?s supervision, which gave her the wanting to

know the unknown in mathematics.

Hyapthia made many contributions to the study of mathematics, her most famous

being her work on conic sections. A conic section is when a person divides cones into

different parts using planes. Because she edited a book written by Apollpnius so well, her

work survived all the way up until today. Her concepts later developed into what is today

called, hyperbolas, parabolas, and ellipses.

Hypatia died a very tragic death in 415 AD. Because she was a woman in the

field of mathematics and science, many rumors were spread about her. One of the

Christian leaders named Cyril heard of these rumors and because he did not like the civil

governor of Alexandria, where Hypatia lived, he made Hypatia a target. She was very

respected and he knew that killing her would definitely hurt the city. On her way home

one night, she was attacked by a mob and literally skinned with oyster shells. Some say

she died for the love of mathematics(Adair, 1995).

Maria Gaetana Agnesi

1718-1799

Maria Gaetana Agnesi was not really considered a mathematician in her time. But

now that some people look back, she made a very significant contribution to the world of

mathematics. She practiced mathematics during the Renaissance in Italy. During this

time, it was considered an honor to be an educated woman. So Maria was both looked up

to and considered a prodigy by the time she was very young. This could be attributed to

the fact that her father was an upstanding mathematician and professor in Milan, Italy.

He often had lectures and seminars at his house for people to come and hear about math.

She liked to listen to these lectures which may have sparked her interest in mathematics.

There are two accomplishments that Maria is accredited with. Her first is her

book that she got published called Analytical Institutions, which was about integral

calculus. Some say that it was originally written for her younger brothers, to aide them in

math. Now that the book has been translated, many mathematicians are using her work

and it is used as a textbook.

Her second accomplishment is a curve called the Witch of Agnesi. Maria came

up with the equation for this well known curve: y= a*sqrt(a*x-x*x)/x . The way to

generate the curve is xy2=a2(a-x)(Golden & Hanzsek-Brill, no date) The reason why it

is called the Witch of Agnesi is because the man who translated the name of the curve

may have mistranslated the Latin word versiera. It can either mean ?to turn? or ?the wife

of the devil.? This curve is very useful in the field of mathematics; even Fermat studied

this curve. Fermat also made the famous problem called Fermat?s Last Theorem, which

famous female mathematician Sophie Germain studied (Unlu, 1995).

Sophie Germain

April 1, 1776-June 27,1831

Sophie Germain, was born right before the French Revolution. She was born into

the middle class, and this meant that she had to hide her identity in order to practice math.

The middle class was not supportive of women studying math, therefor much of her work

is done under her pseudonym M. Leblanc. Because of the Revolution, Sophie had to

spend many days in her house, for fear of being killed in a revolt. She was intrigued by

the story of Archimedes and how he got killed because he would not respond to a soldier

while looking at a math problem. Some people think this is why Sophie choose to study

mathematics.

Sophie Germain studied under famous mathematician of the time, Carl Friedrich

Gauss. Gauss was really into number theory and Fermat?s Last Theorem. Fermat?s Last

Theorem is closely related to the Pythagorean theorem. Instead of using x2+y2=z2,

Pierre de Fermat used x,y,and z raised to powers of 3, 4, 5,etc. Many think that this

problem was unsolvable, but Fermat said that he had proof it could work. The mystery is

though, that Fermat never wrote down his solution. It was up to future mathematicians to

find the solution that Fermat claimed.

Sophie was up to the challenge, and in a letter to Gauss, written in 1808 she came

up with a calculation that said something about several solutions. Fermat?s theory says

there are no positive integers such that for n*2. But Sophie proved in her theorem that if

x, y, and z are to the fifth power than n has to be divisible by five. Sophie said that this

would work only with what are now called Germain primes. Germain primes are primes

such that when you take a prime, multiply it by two, and then add one, your answer will

be prime. Some Germain primes are 2, 3, 5, 11, 23 and 29 (Singh, no date). In 1825,

she proved, that for the first part of Format? Last theorem, these primes would work.

There are many other mathematicians that have followed up on Sophie?s work on

Fermat?s Last Theorem. Number theorist ,Euler and Legrange, proved that if p=3 is

prime, 2p+1 is also prime if and only if 2p+1 divides 2p-1. In 2000, famous number

theorist, Henri Lifchitz, found an easier way to determine a Germain prime. He says that

if p*=5 is prime, q=2p+1 is also prime if and only if q divides 3p-1. It turns out though in

1994, Andrew Wiles, a researcher at Princeton, claimed to have proof of the theorem.

His manuscripts have been reviewed and it is among the majority that he has proved it

(Swift, 1997)

Emmy Noether

March 23, 1882-April 14, 1935

Still in the late 1800s, it was not proper or allowed for a woman to go to college.

Emmy Noether became one of these women, when she was denied enrollment at the

University of Erlangen. They did allow her, though, to sit in on two years of math

classes and take the exam that would let her be a doctoral student in math. She passed the

test and after going for five more years, she was given a diploma. After graduation,

Emmy decided to take up teaching, but the university would not hire her because she was

a woman. So she decided to work along side her father, who at the time was a professor

at the university. Emmy Noether’s first piece of work was finished in 1915. It is work in

theoretical physics, sometimes called the Noether’s Theorem, which proves a relationship

between symmetries in physics and conservation principles. This basic result in the

general theory of relativity was praised by Einstein, where he commended Noether on her

achievement.

During the 1920s Noether did foundational work on abstract algebra, working in

group theory, ring theory, group representations, and number theory. During the time that

she was a teacher, Germany was involved in WWI and WWII. Because of the war, and

since Noether was a Jew, she was forced out of Germany and went to live in the United

States ( ?Emmy Noether?, no date).

While in the United States, Noether taught at an all girls college. Her students

loved her and many followed her teachings. Some say that they way she taught was

phenomenal. She was clear and used many different methods of teaching so that her

students could understand math easier. She was praised by Einstein constantly on her

theory of relativity. Albert Einstein paid her a great tribute in 1935: “In the judgement of

the most competent living mathematicians, (Emmy) Noether was the most significant

creative mathematical genius thus far produced since the higher education of women

began.” Throughout her career she worked with many mathematicians such as Emanuel

Lasker, Bartel van der Waerden, Helmut Hasse and Richard Brauer. Twice Noether was

invited to address the International Mathematical Congress (1928, 1932). In 1932 she

received the Alfred Ackermann-Teubner Memorial Prize for the Advancement of

Mathematical Knowledge. It is said that her greatest work was that of abstract algebra

(Taylor, 1995).

Ruth Moufang

January 10, 1905-November 26, 1977

Like the Nazis refused Emmy Noether the right to teach, Ruth Moufang was also

denied the right. Because of this, Ruth Moufang decided to enter the field of industrial

mathematics, and work on the elasticity theory. She was the first German woman to have

a doctorate in this field. Ruth Moufang published one famous paper on group theory.

This paper was first written based on the writings of Hilbert. Ruth?s most famous

teachings were on number theory, knot theory, and the foundations of geometry. She also

is famous for what we call today, Moufang planes and Moufang loops. Moufang loops

are a class of loops which arise naturally in many other fields such as finite group theory

and algebraic geometry (O?Conner & Robertson, 1996).

Sun-Yung Alice Chang

March 24, 1948-present

Sun-Yung was born in Ci-an, China. During research, no information was found

on the time period when she was born. What was found though is an abundance of

information on her college life and what her contributions to mathematics were.

Sun_Yung Chang received her doctorate in mathematics from University of

California. She then went to teach college math at UCLA. Currently, she still teaches at

UCLA, but since she started many things have happened to her.

Her greatest accomplishment is when she received the Ruth Lyttle Satter prize for

her contributions to mathematics over the last five years. She was awarded the prize for

her contributions to partial differential equations and on Riemannian manifolds. The

study of manifolds having a complete Reimannian Metric is called Reimannian geometry

(Weinsstein, 1996-2000). This is a topic that Sun-Yung studied a lot. Sun-Yang says, in

her speech at the American Mathematical Society, ?Following the early work of J Moser

and influenced by the work of T Aubin and R Schoen on the Yamabe problem, P. Yang

and I have solved the partial differential equation of Gaussian/scalar curvatures on the

sphere by studying the extremal functions for certain variation functionals. We have also

applied this approach in conformal geometry to the isospectral compactness problem on

3-manifolds when the metrics are restricted in any given conformal class. More recently

we have been studying the extremal metrics for these functionals. We are working to

derive further geometric consequences. This latter piece of work is a natural extension of

the earlier work by Osgood-Phillips-Sarnak on the log-determinant functional on compact

surfaces.?(O?Conner & Robertson,1998, p. 2) Sung-Yung is already considered to be a

great mathematician, even though she says there is still work to be done.

Women in Mathematics connected to the Middle School Curriculum

In Sun-Yung?s speech, given at the acceptance of her award in 1995 she states,

?Since the Satter Prize is an award for women mathematicians, one cannot help but to

reflect on the status of women in our profession now. Compared to the situation when I

was a student, it is clear that there are now many more active women research

mathematicians. I can personally testify to the importance of having role models and the

companionship of other women colleagues. However, I think we need even more women

mathematicians to prove good theorems and to contribute to the profession.? (O?Conner

& Robertson, 1998, p. 2)

This is exactly why this topic needs to be discussed in the middle grades. Girls

need to know that mathematics is not only for men. Young girls may be less apt to go

into the field of mathematics based on the biases that have been going for years.

Teachers need to tell about the importance of mathematical skills for both boys and girls,

and also need to plan activities centered around women in mathematics. By talking to

young girls in middle school about female mathematicians, educators could possibly

ignite a flame, under possibly, another great female mathematician.

Although many do not think of women as mathematicians, there are many women

who have proved themselves in the mathematical world. Through their theorems and

problem solving, these women have furthered the world of mathematics, for others to

someday conquer.

References

Adair, G. (1995). Hypatia. Agnes Scott College [Online]. Available:

http://www.agnesscott.edu/lriddle/women/agnesi.htm [1 March 2000].

Emmy Noether (no date). [Online].

Available:http://www.coastal.edu/academics/science/jump/biography/enoether.ht

ml [5 March,2000].

Golden & Hanzsek-Brill. (no date). Investigation of the Witch Curve. [Online].

Available: http://jwilson.coe.uga.edu/Texts.Folder/Agnesi/witch.html [1 March,

2000].

O?Conner, J.J., & Robertson, E.F. (1996). Ruth Moufang. [Online]. Available:

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Moufang.html [24

February 2000].

O?Conner, J.J., & Robertson, E.F. (1998). Sun-Yung Alice Chang. [Online]. Available:

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Chang.html [6

March 2000].

Singh, Simon. (no date). Math?s Hidden Women. [Online]. Available:

http://www.pbs.org/wgbh/nova/proof/germain.html [1 March 2000].

Swift, Amanda. (revised in 1997). Sophie Germain. Agnes Scott College [Online].

Available: http://www.agnesscott.edu/lriddle/women/germain.htm [1 March

2000].

Taylor, Mandie. (1995). Emmy Noether. Agnes Scott College [Online]. Available:

http://www.agnesscott.edu/lriddle/women/noether.htm [2 February 2000].

Unlu, Elif. (1995). Maria Gaetana Agnesi. Agnes Scott College [Online]. Available:

http://www.agnesscott.edu/lriddle/women/agnesi.htm [1 March 2000].

Weisstein, Eric. (1996-2000). Riemannian Geometry. Wolfram Research Inc. [Online].

Available: http://www.mathworld.wolfram.com/RiemannianGeometry/html [7

March 2000].

Bibliography

References

Adair, G. (1995). Hypatia. Agnes Scott College [Online]. Available:

http://www.agnesscott.edu/lriddle/women/agnesi.htm [1 March 2000].

Emmy Noether (no date). [Online].

Available:http://www.coastal.edu/academics/science/jump/biography/enoether.ht

ml [5 March,2000].

Golden & Hanzsek-Brill. (no date). Investigation of the Witch Curve. [Online].

Available: http://jwilson.coe.uga.edu/Texts.Folder/Agnesi/witch.html [1 March,

2000].

O?Conner, J.J., & Robertson, E.F. (1996). Ruth Moufang. [Online]. Available:

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Moufang.html [24

February 2000].

O?Conner, J.J., & Robertson, E.F. (1998). Sun-Yung Alice Chang. [Online]. Available:

http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Chang.html [6

March 2000].

Singh, Simon. (no date). Math?s Hidden Women. [Online]. Available:

http://www.pbs.org/wgbh/nova/proof/germain.html [1 March 2000].

Swift, Amanda. (revised in 1997). Sophie Germain. Agnes Scott College [Online].

Available: http://www.agnesscott.edu/lriddle/women/germain.htm [1 March

2000].

Taylor, Mandie. (1995). Emmy Noether. Agnes Scott College [Online]. Available:

http://www.agnesscott.edu/lriddle/women/noether.htm [2 February 2000].

Unlu, Elif. (1995). Maria Gaetana Agnesi. Agnes Scott College [Online]. Available:

http://www.agnesscott.edu/lriddle/women/agnesi.htm [1 March 2000].

Weisstein, Eric. (1996-2000). Riemannian Geometry. Wolfram Research Inc. [Online].

Available: http://www.mathworld.wolfram.com/RiemannianGeometry/html [7

March 2000].