*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*= 4*k*
The number is one
more than a multiple of 4 :

*n*= 4*k*+ 1
The number is two
more than a multiple of 4 :

*n*= 4*k*+ 2
The number is three
more than a multiple of 4 :

*n*= 4*k*+ 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*= 4*k*, or*n*= 4*k*+ 2. That leaves the other two categories.
So, a prime number greater than 2 can be written as either

Over a hundred years ago Pierre de Fermat concluded that “odd numbers of the form

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.

*n*= 4*k*+ 1, or*n*= 4*k*+ 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*= 4*k*+ 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*= 4*k*+ 3 cannot be written as a sum of two perfect squares.” He asserted simply that*n*= 4*k*+ 3 ≠ a^{2}+ b^{2}. 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

*x*, I use the fact that^{n}*d/dx*(*x*) =^{n}*nx*^{n}^{−1}.^{ }So, if I have*f*(*x*) =*x*^{5}, its derivative should be 5*x*^{4}. This is easy. If I^{ }analyze trigonometric functions such as sin*x*and cos*x*, then I use the^{ }derivatives*d/dx*(sin*x*) = cos*x*, and*d/dx*(cos*x*) = − 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:

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*d*^{2}*y*/*dx*^{2}, and so forth. Thus,^{ }the previous linear equation would be written as (*x*^{2}+ 1)*y’*+ 3*xy*= 6*x*. 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

where the rate

*P*(*t*) represents the population change in time (*t)*, I writewhere 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*) =*P*_{0}*e*, where^{kt}*P*_{0}is the initial population, i.e.,*P*(*t*= 0) =*P*_{0}.
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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