These are actual student papers that were not designed to be web pages. They may contain historical, grammatical, mathematical, or formatting errors. These papers were graded using the criterion mentioned in the paper directions, and the writing checklist. The test review sheets and the WebCT tests are good indicators of the mathematics that was discussed in class during and/or after each presentation.
Grace Brewster Murray Hopper
By Samantha Mathews and Melissa Mogensen
Grace Brewster Murray Hopper was one of the most influential women in the world of computer science. An admirable patriot to her country, Grace spent the larger part of her life dedicated to the United States Navy becoming the oldest person to retire from active service. Hopper developed the first computer compiler and developed a computer language that helped the computer world become what it is today. Her contributions were also in the world of mathematics, where her work was done on irreducibility criteria. Hopper spent half of a century dedicated to keeping the United States on the edge of high technology.
Grace Hopper was born on December 9, 1906 in New York City. Hopper was great granddaughter to Alexander Russell who was a rear admiral in the United States Navy. This was Hopper’s role model and personnel hero. Its no wonder Grace spent the majority of her life dedicated to the Navy. She was the granddaughter of a civil engineer, John Van Horne, who gave Hopper got her first experiences with angles, curves, and angles. Her father, Walter Fletcher Murray, was an insurance broker. Her mother, Mary Campbell Horne Murray, actually had special arrangements made so that she could study geometry. At that point in time, women were not allowed to study algebra or trigonometry. It was society’s opinion that women only needed to know basic mathematical skills in order to use it on household accounts and family finances. Grace had a particular love for gadgets; she was known to take apart household clocks to find out how they worked. Her parents instilled the ambition and drive that made Grace the women she was. Her father wanted his daughters to have the same opportunities that her brother did.
Throughout Hopper’s student career, she received a superior education. Grace attended Graham School and Scroonmakers School in New York City, both were private schools for girls. When it came to time for Grace to attend college, she attempted a Latin exam and failed. She decided to attend Hartridge School in Plainfield, New Jersey in the fall of 1923. She stayed there for a year before enrolling in Vassar College at the age of 17, where she majored in mathematics and physics. In 1928 she attended Yale University in New Haven, Connecticut. Shortly after graduating in 1930, Grace married her former English professor. She was awarded her Ph.D. in mathematics in 1934 by Yale University for her thesis, “New Types of Irreducibility Criteria.”
After her student career, Hopper began teaching at Vassar College where her first year salary was only eight hundred dollars. In 1936, she published a paper on “The ungenerated seven as an index to Pythagorean number theory.” After the bombing of Pearl Harbor, Grace decided she would join the Navy, and follow in her great grandfather’s footsteps. However, she had several obstacles in her way. Because she was underweight, only 105 pounds, and was considered overage, she was not eligible for military enlistment. After getting a leave of absence from Vassar College, a waiver for the weight requirement, and special government permission, she was sworn into the Reserve in December of 1943 where she was commissioned to Lieutenant Junior Grade. Then in 1944, she started to work on the Bureau of Ordinance Computation Project at the Cruft Laboratories at Harvard University. Here Hopper worked on the Mark I with Howard Aiken and was the third person to program the Mark I. The Mark I was the world’s first large-scale automatically digital computer, and was very large, 51 feet long, 8 feet wide, and 8 feet high. It was made of more than 760,000 pieces and could perform 3 additions per second and store 72 words. This computer was used by the Navy for gunnery and ballistic calculations. During her work on the Mark I, Hopper was given credit for coining the term “bug”, which is a reference to a glitch in the computer. She actually found a moth inside the computer, which was causing the problems.
Grace got divorced in 1945 from Vincent with no children. In 1946, Hopper was too old to stay in active service and retired. Soon after this, she began to work for Eckert-Maunchly Computer Corporation as a senior mathematician where she worked with John Eckert and John Maunchly on the UNIVAC computer. The UNIVAC used vacuum tubes instead of electromechanical relay switches like the Mark I did. It was also up to twenty times faster. Also in 1946, she published a book, “A Manual Of Operations for the Automatic Sequence Controlled Calculator.” While she was at Harvard, she designed an improved complier and helped develop Flow-Matic, the first English-language data-processing compiler. A complier is a special program that processes statements that are written in a programming language, and turns them into a “code” that a computer’s processor uses. Flow-Matic became a model for a new program COBOL (Common Business Oriented Language), which eventually came out in 1959. This was the first user-friendly business software program. Her aim in compliers was that there needed to be standardization. This made it possible for computers to respond to words rather than numbers. Programmers, previous to COBOL, would write programs in binary code, strings of one’s and zero’s. This left room for mistakes and errors to programmers and was extremely time consuming.
Hopper’s age forced her to retire from the Navy in 1966 at the rank of Commander. However, the Navy recalled her in less than seven months down the road, because they were unable to develop a working payroll, not even after 823 attempts. This reinstatement made her the first women to return to active duty. In 1986, she retired for good from the Navy at rank of Rear Admiral where she was the oldest person to retire from active duty. After her final retirement, she worked with Digital Equipment Corporation as a senior consultant. She worked there until she died in her sleep on January 1, 1992 at the age of eighty-five. She received a full military funeral at Arlington National Cemetery, Virginia.
During her lifetime, Grace Hopper received numerous awards. She was named the first computer science Man of the Year in 1969 by the Data Processing Management Association. On September 16, 1991, President George Bush awarded Hopper the National Medal of Technology. She was the first woman to ever receive this award. In addition to these awards, Grace was awarded 36 honorary doctorates from such colleges and universities as Newark College of Engineering, University of Pennsylvania, Pratt Institute, and Long Island University, just to name a few.
Grace only had a few gender obstacles to overcome. Her only significant gender conflict was when she tried to enter the Navy. The reason she was even considered for the Navy was because she was a mathematics professor. The fact that she was one of only a few women in the field of computer science never bothered her, nor seemed to affect her. Hopper did not seem to experience any problems in her education either. Grace was an extremely lucky woman and her accomplishments and contributions will always be remembered.
One of the most well known methods of finding irreducibility of polynomials with integer coefficients is demonstrated by Eisenstein’s criterion. When something is irreducible, it can’t be factored into smaller polynomials with rational coefficients. For example:
This is reducible because this is irreducible
it breaks down into a because when it’s
smaller polynomial with broken down, it leaves
rational coefficients. irrational
Eisenstein’s criterion states that if all the coefficients, except possibly the first one, are divisible by a prime “p”, and the constant coefficient is not divisible by p^2, then the polynomial is irreducible. His equation is the following:
X^n + An-1 X^n-1…+Ao=0
When you come across a complicated polynomial you can try using this method; however, this doesn’t always work. It only works if the polynomial follows the rules stated above. For example:
X^2 + 10X + 5 = 0 X^2 – 8X + 4 = 0
When you look as this In this equation
equation you notice your coefficients
the prime the have the prime
coefficients have in number 2 in common;
common is 5. Therefore, therefore, your
p=5. The next p=2. The next step
thing you notice is that is dividing it’s
(5/p^2) does not give square by the
you a rational number constant
when it is divided; coefficient. In
therefore, this equation this case (4/p^2)
is irreducible. gives us a rational
concludes that you
Eisenstein’s criterion on this
X^2 – 4X + 2 = 0
The prime number that the
coefficients have in common
is 2 and when you divide
(2/2^2) it comes out as an
irrational number; therefore,
by Eisenstein’s criterion,
It sometimes happens that the criterion is not applicable to the polynomial because it does not follow the criteria. For example:
X^4 + 1=0
In Eisenstein’s criterion
the X’s follow a
X^n + An-1 X^n-1…+Ao=0
In this case it’s
X^n + Ao = 0, so it’s not
For every great equation there is always a trick if something doesn’t work out. In this case, since X^4 + 1= 0 does not follow Eisenstein’s criteria, we can transform it into something that’ll work, for example:
F(x)=X^4 + 1 = 0 à g(x) = f(x+1) =
(x+1)^4 + 1 = 0
Now this polynomial satisfies the conditions of the Eisenstein’s criterion. We find that p=2 and since (2/2^2) leaves us with an irrational number, we conclude that this polynomial is irreducible.
This trick works because any factor of f(x) would be a factor of g(x) by substituting “x” by (x+1) in each factor.
However, this trick doesn’t always work:
f(x)=X^3+1=0 à g(x)=(x+1)^3-1=0
In this case, transforming
The function into g(x) still
didn’t help us solve it
because you this equation
still doesn’t have the
criteria needed to use
Eisentein’s criterion basically reduces the problem of factoring a difficult polynomial to a problem of factoring integers by using the coefficients of the former polynomial to see if they have a common prime divisor.
Grace Murray Hopper, instead of the open Dumas polygon, introduced the closed convex polygon, which is applied to the deduction of irreducibility criteria. This was dependent on the size and the divisibility properties of the coefficients. For the closed convex polygon, an approximate multiplication theorem holds and may be used to deduce irreducibility criteria depending on the size of the coefficients. A convex polygon is a closed figure in a plane whose angles are less than 180 degrees. For example:
This is a convex polygon because
the angles are less than 180 degrees
Grace Murray Hopper found a way to convert a polynomial into a convex polygon. With this conversion she found a way to decompose the polygon the way that Eisenstein broke down the polynomials.
This is an example of an icosehedron, which is going to be decomposed into a tetrahedron.
This shows the decomposition the icosahedron and how it is broken up.
This is the final step and the one that Hopper used in order to solve for irreducibility. This is all that I know about Hopper because all I had to work with was an abstract of her paper. The basis of her work was irreducibility and the process of turning then into closed convex polygons and determining their reducibility.
“Abstracts of papers”, American Mathematical Society Bulletin, Vol 40, pg 216, New York, 1934. This source stated the summarization of her thesis, which was helpful because her thesis was unavailable.
Dickason, Elizabeth. “Remembering Grace Murray Hopper: A Legend in Her Own Time,” CHIPS On-line, http://www.norfolk.navy.mil/chips/grace_hopper/file2.htm. This gave a great biography on Hopper.
Dickason, Elizabeth. “Looking Back:Grace Murray Hopper’s Younger Years,” CHIPS On-line, http://www.norfolk.navy.mil/chips/grace_hopper/young.htm. This source gave a description of Hopper’s life when she was younger.
Distinguished Women of Past and Present. “Grace Murray Hopper” http://www.distinguishedwomen.com/biographies/hopper/html A description of Hopper’s life was given in this source.
“Irreducibility Criteria”, This explained Eisenstein’s criterion and the tricks about finding irreducibility. It was very helpful in writing this paper!