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.
March 29, 2001
Julia Bowman Robinson is no longer with us, though her work survives through the efforts of many dedicated people. Julia was born on December 8, 1919 in St. Louis Missouri. As a child she faced a series of traumatic events, which affected her throughout her life. At the age of two, Bowman’s mother passed away. Julia and her sister, Constance, were sent with their nurse to Arizona to live with their grandmother. Their father owned a machine tool and equipment company, and soon after the death of his wife, lost interest in the business. Along the way he had saved up substantial amount of money and felt that he had enough income to support a family. Julia’s father remarried and closed down his business in order to move to Arizona and live with Julia and Constance.
At the age of nine, Julia came down with scarlet fever. It was because of this, that she was isolated from her family. During this time her father took care of her. Shortly after Julia recovered from scarlet fever, she came down with rheumatic fever and spent an entire year in bed. Because of these illnesses, Julia missed more than two years of school. Julia’s parents hired a tutor, a retired schoolteacher, to help her out with her schooling. In one year, through the help of her tutor, Julia spent three mornings a week completing the state requirements for the fifth, sixth, seventh, and eighth grades. In 1932, Julia entered Theodore Roosevelt Junior High, where she had difficult times because she had not been in the typical school setting for a while. She embarrassed herself quite often. Although Julia was shy in high school, it did not bother her that she was the only girl in her physics and advanced math classes. Julia began to do what was out of the norm of other people, that is she stayed in the math classes although she was the only girl. Despite that fact, she made the best grades in the class. While being educated, Julia was never encouraged by any of her teachers to pursue with advanced math, although she made the best grades. However, Julia went to San Diego State College majoring in math with the intent to become a teacher. Later on she altered her plans and transferred to the University of California at Berkley, still majoring in math, but now in research mathematics. No one believed that women could do math especially the advanced mathematics that is required in upper level classes in high school and college. Julia Bowman was a striking contradiction to this claim.
Bowman received a bachelor’s in arts in 1940 and began her graduate studies. In the colleges she attended, there were no women professors teaching mathematics, however there were women professors in Psychology and Biology. While Julia was taking math classes at Berkley, she met Ralph Robinson who not only was one of her professors, but also became her husband. Ralph had a tremendous impact on Julia during his one-on-one teaching. Julia had mentioned that she would not have become a mathematician if it were not for her husband Ralph and his great encouragement.
One of the impacts that Julia’s childhood diseases had on her was the inability to have children. This depressed Julia to no end because she could never have the thrill of helping something to live and grow. Ralph reminded her that she always had her mathematics, and this helped her to ease the pain along the way. The same year, 1948 Julia got her PhD from the University of California at Berkley and began working on Hilbert’s Tenth problem.
Even today, one cannot research Julia Robinson without running across David Hilbert’s Tenth Problem. This problem, along with others, was Julia’s passion in life. Although she had many mathematical successes, she is best known for her work on the tenth problem. Julia Robinson cannot be disconnected from David Hilbert’s Tenth Problem. The Tenth Problem is as follows:
DETERMINATION OF THE SOVABILITY OF A DIOPHANTINE EQUATION
Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined by a finite number of operations whether the equation is solvable in rational integers.
A Diophantine equation is an polynomial equation in which only integer solutions are allowed. They take their name from the ancient Greek mathematician Diophantus of Alexandria. A Diophantine equation may extend from a first-degree polynomial to a polynomial of an infinite degree. If the polynomial is of first degree, it is known as being linear. However, no matter what degree of the polynomial, the sum of the equation must equal zero. Fermat’s Last Theorem states that x^n + y^n = z^n. We can turn this into a Diophantine equation by subtracting z^n from both sides. Therefore forming x^n + y^n – z^n = 0. This is one form of a Diophantine equation. The general form of a Diophantine equation is x^n + y^m + z^t + ….. = 0. The terms may go on and on forever, with the same or different powers of exponents, and their sum will equal zero.
We will now discuss the linear Diophantine equations. Solutions are abundant for first-degree equations, however, each solution can be classified in one of three ways: no solution, a finite solution, or infinitely many solutions. There is an example proven for each of the three cases. The problem 2x –2y = 1 will never have a solution since the variables x and y are both multiplied by the even number two, therefore they will never equal an odd number. Subsequently, the number one is odd and we can verify that this will definitely have no solution in the integers. Another example that also has no solution would be 2x +2y –4z – 5 = 0. It follows that there exist certain Diophantine equations with no solutions. The second case is an equation with a finite solution. This is the most obvious of the three cases. An example 3x = 6 only has one solution, and that is where x is equal to two. Likewise, 2x + 4 – 16 = 0 can be solved with x only being equal to six. These are perfect example of Diophantine equations with a finite solution.
Finally, the third case for first-degree equations is an equation with infinitely many solutions. In order to solve these you may use the “coefficient reduction method”. This method involves reducing the equation to one unknown. This makes the more complicated problem solvable. For example, let us decipher the equation 7x – 17y = 1. First, solve the equation for x since the coefficient of x easily divides into the coefficient of y. Therefore, add 17y to both sides and you will get 7x = 17y + 1. Now divide everything by seven. This gives you that x = 2y + (3y + 1)/ 7. We know that all Diophantine equations have integer solutions, therefore, (3y + 1)/ 7 must be an integer. Denote this integer as z. Hence, 3y + 1 = 7z and x = 2y + z. Now we will again apply the “coefficient reduction method” and solve for y in the equation 3y + 1 = 7z. Subtract one from both sides of this equation and then follow by dividing through by three. This gives you that y = 2z + (z – 1)/ 3. Similar to the x equation where (3y + 1)/ 7 is an integer, (z – 1)/ 3 is also an integer in the y equation. Let us call (z – 1)/ 3 the variable t. Therefore we can state that 3t + 1 = z, also we can confirm that y = 2z + t. By substitution y = 2(3t + 1) + t, therefore y = 7t + 2. As stated above we know that x = 2y + z, which can be written as x = 2(7t + 2) + (3t + 1). This expression can be simplified to x = 17t + 5. Notice that x is now in terms of one variable instead of two. Now by using t equals, all integers you end up with an infinite number of solutions for the original Diophantine equation. This completely covers all three cases of the Linear Diophantine Equation.
Although Julia Robinson’s work did not center on Linear Diophantine Equations, these are the building blocks to what she proved. Julia along with others, including Yuri Matijasevich, proved that you cannot look at a Diophantine equation and be able to tell in a finite number of steps if you can solve the equation or not. The same can be said for how big is the universe or when is the beginning of time. No matter how sophisticated the technology, there is always some part to the answer, which we cannot know. This is true for the size of the universe and is true for the solution to David Hilbert’s tenth problem.
So, why is the answer to Hilbert’s tenth problem significant? The answer lies in coding. If you can come up with an equation that is hard to solve in a finite number of steps, you can use that equation to code in different commands and information that you do not want the general public to know. In order to decode the message you would have to have prior knowledge of what was put into the original equation.
Julia, just like many others, broke down many doors for women. In 1975, she was the first woman mathematician to be elected to the National Academy of Sciences. In 1982, she became president of the American Mathematical Society. A long way from being the only girl in her physics and math classes, Julia had a significant impact on a field that is generally controlled by males. She did not let this discourage her in anything that she did, but instead let this be a learning situation that made her stronger. Julia died on July 30, 1985.