Wednesday, May 23, 2012

Excerpts from Sophie's Diary: A Mathematical Novel

Today we enjoy a bit of math, as told in Sophie’s Diary: A Mathematical Novel. Written by Dr. Dora Musilek, this novel was inspired by the French mathematician Sophie Germain, an important contributor to number theory and mathematical physics. Her correspondence with some of history’s great mathematicians such as Lagrange, Legendre, and Gauss are known while her life prior to that is shrouded in the unknown. Dr. Musilek explored Germain’s early life and uses the concept of an adolescent’s diary to discuss how Germain may have taught herself math while dealing with the social upheaval of the French Revolution, which occurred at this time in her life. Read on for an engaging lesson in math.

Monday | January 2, 1792

I begin the new year with more determination and a renewed resolve to study prime numbers. One of my goals is to acquire the necessary mathematical background to prove theorems.

Prime numbers are exquisite. They are whole pure numbers, and I can manipulate them in myriad ways, as pieces on the chessboard. Not all moves are correct but the right ones make you win. Take, for example, the process to uncover primes from whole numbers. Starting with the realization that any whole number n belongs to one of four different categories:
The number is an exact multiple of 4 : n = 4k
The number is one more than a multiple of 4 : n = 4k + 1
The number is two more than a multiple of 4 : n = 4k + 2
The number is three more than a multiple of 4 : n = 4k + 3

It is easy to verify that the first and third categories yield only even numbers greater than 4. For example for any number such as k = 3, 5, 6, and 7, I write: n = 4(3) = 12, and n = 4(6) = 24; or n = 4(5) + 2 = 22, and n = 4(7) + 2 = 30. The resulting numbers clearly are not primes. Thus, I can categorically say that prime numbers cannot be written as n = 4k, or n = 4k + 2. That leaves the other two categories.

So, a prime number greater than 2 can be written as either n = 4k + 1, or n = 4k + 3. For example, for k = 1 it yields n = 4(1) + 1 = 5, and n = 4(1) + 3 = 7, both are indeed primes. Does this apply for any k? Can I find primes by using this relation? Take another value such as k = 11, so n = 4(11) + 1 = 45, and n = 4(11) + 3 = 47. Are 45 and 47 prime numbers? Well, I know 47 is a prime number, but 45 is not because it is a whole number that can be written as the product of 9 and 5. So, the relation n = 4k + 1 will not produce prime numbers all the time.


Over a hundred years ago Pierre de Fermat concluded that “odd numbers of the form n = 4k + 3 cannot be written as a sum of two perfect squares.” He asserted simply that n = 4+ 3 a2 + b2. For example, for k = 6, n = 4(6) + 3 = 27, and clearly 27 cannot be written as the sum of two perfect squares. I can verify this with any other value of k. But that would not be necessary.

And now we skip ahead to another excerpt where differential calculus is described.

Wednesday | February 27, 1793

I feel strong enough to resume my studies. My mind is clear again to meet the challenges of a new topic that at first seemed insurmountable. I resumed my studies of differential calculus.

There is something magical about Infiniment petits. I went back to the basic definition: “a derivative of a function represents an infinitesimal change in the function with respect to whatever parameters it may have.” The simple derivative of a function f with respect to x is denoted by f’(x), which is the same as df/dx. Newton used fluxions notation dz/dt = ż, but it means the same, so I will use f’ or df/dx from now on. Well, I can now take the derivative of certain classes of functions because I just follow certain rules.

If my function is of the type xn, I use the fact that d/dx(xn) = nxn−1. So, if I have f(x) = x5, its derivative should be 5x4. This is easy. If I analyze trigonometric functions such as sin x and cos x, then I use the derivatives d/dx(sinx) = cos x, and d/dx(cosx) = − sin x.

Taking derivatives is so easy! I could spend hours deriving more complicated functions. However, I wish to learn also how to see the world through mathematics. I must find the connection between differential equations and physics. I am eager to explore this applied aspect of mathematics.

Let’s start with a differential equation, an equation involving an unknown function and its derivatives. It can be relatively easy such as
or a bit more complicated such as the linear differential equation:
or even a nonlinear equation such as this: 

A differential equation is linear if the unknown function and its derivatives appear to the power 1 (products of these are not allowed) and nonlinear otherwise. The variables and their derivatives must always appear as a simple first power. Nonlinear equations are difficult to solve and some are impossible.

First I need to master linear equations. Some mathematicians use the notation y’ for the dy/dx derivative, or y’’ for d2y/dx2, and so forth. Thus, the previous linear equation would be written as (x2 + 1)y’ + 3xy = 6x. I need to keep these differences of notation in mind, since I am studying from five different books.

I studied the properties of differential equations and learned to solve them. Now, I must learn how to apply differential equations. But how do I translate a physical phenomenon into a set of equations to describe it? It is impossible to depict nature in its totality, so one usually strives for a set of equations that describes the physical system approximately and adequately.

Say that I want to predict the growth of population in Paris. To do it, I can use an exponential model, that is, an equation that represents the rate of change of the population that is proportional to the existing population. If P(t) represents the population change in time (t), I write
where the rate k is constant. I observe that if k > 0, the equation describes growth, and if k < 0, it models decay. The exponential equation is linear with a solution P(t) = P0ekt, where P0 is the initial population, i.e., P(t = 0) = P0.

Mathematically, if k > 0, then the population grows and continues to expand to infinity. On the other hand, if k < 0, then the population will shrink and tend to 0. Clearly, the first case, k > 0, is not realistic. Population growth is eventually limited by some factor, like war or disease. When a population is far from its limits of expansion, it can grow exponentially. However, when nearing its limits, the population size can fluctuate. Well, I think that the equation I use to predict the rate of change of population can be modified to include these factors to obtain a result closer to reality.

Aristotle thought that nature could not be expected to follow precise mathematical rules. But Galileo argued against this point of view. He envisioned the experimental mathematical analysis of nature to be used to understand it. Newton was inspired by Galileo and later developed the laws of motion and universal gravitation. Newton, Leibniz, Euler, and other great people then created the mathematics that help us converse with the universe.

Oh, how glorious it is to speak such a language and understand the whispers from the heavens and the world around me.

Dr. Dora Musilek is a research scientist and also lectures on the role and contributions of women scientists and mathematicians. She holds a Ph.D. in aerospace engineering. You can learn more about Dr. Musilek and her novel at sophiesdiary.net You can learn more about Dr. Musilek’s writing process at MAAA Books Blog.

These views are the opinion of the author and do not necessarily either reflect or disagree with those of the DXS editorial team. 

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