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:

X^2-1=0 x^2+1=0

(x+1)(x-1) (x+i)(x-i)=0

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

coefficients.

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

number. This

concludes
that you

cannot
use

Eisenstein’s criterion on this

equation.

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’s irreducible.

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

decreasing pattern:

X^n + An-1 X^n-1…+Ao=0

In this case it’s

X^n + Ao = 0, so it’s not

applicable.

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

(x+1)*(x+1)*(x+1)*(x+1)+1=0

X^4+4X^3+6X^2+4X+2=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

(x+1)*(x+1)*(x+1)+1=0

x^3+3x^2+3x+2=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

Eisenstein’s
criterion.

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!

http://mathpages.co/hoe/kath406.htm