Mathematics is the language used to quantify anything in science.  Still, some aspect of the tool we call math is utilized in nearly every field from business, to sales, and of course, engineering.

The field of Mathematics may be divided into several branches like calculus, linear algebra, and statistics.  While all of these subjects are fascinating, let us focus on calculus, which can be summarized as "the study of change."  If no facet of the universe ever changed, we would have no use for calculus, but then again, such a universe could not support life altogether (life cannot exist without chemical reactions).

At its heart, calculus focuses on functions, which are equations describing how variables are related to one another.  The simplest kind of function consists of two variables; one variable is dependent, the other is independent.  If a taxi driver charges a customer $1.50 per minute, then the cost function for riding in the taxi would be C = 1.5t, where C is in dollars and t is in minutes.  Here, time is an independent variable, and cost is a dependent variable (as the cost depends on the duration of the cab ride).  In other words, cost is a function of time, or C = f (t).

The simple taxi function given above relates cost to time, but functions can describe the relationships between other dependent variables and time.  We could, for example, consider the temperature of a hot cup of coffee.  One can imagine that the coffee's temperature value would decrease as time goes on until it reaches the room's air temperature.  The key difference between the taxi function and the coffee function, is that the function for the ride in the taxi was specified by the taxi driver.  No one specified the temperature function of the coffee.  The coffee's temperature function is the result of a differential equation, which is also known as a governing equation - a law of nature, and in the case of the coffee, thermodynamics.

An ordinary differential equation relates a dependent variable to an independent variable and to its own derivatives of various orders.  A derivative of a dependent variable describes its rate of change with respect to an independent variable - this is the first derivative (order one).  The second derivative of a dependent variable is the rate of change of its rate of change with respect to an independent variable (order two).

The origin of all differential equations in science come from the laws of nature.  The law of nature describing the motion of a body is Newton's Second Law, Fnet = ma.  When this relationship is first introduced to high school physics students, it is not described as a differential equation, although in reality, it is one.  The reason that this is a differential equation, is because acceleration, a, is the second derivative of position (velocity is the first derivative of position).  The solution to Newton's equation of motion is a function: the position of a body as a function of time, x = f (t).  This function is of course situation-dependent: it depends on the forces that act on the mass, and the value of the mass itself.

Let us try to find the vertical position function (altitude) for a ball dropped off of a rooftop, y = f (t).  If we ignore the aerodynamic force that acts on the ball, its differential equation of motion becomes very simple: a = - 9.8 m/s², or simply, y'' = - 9.8, where each prime denotes a derivative with respect to time.  Integrating both sides once, we get y' = -9.8t + c1.  Integrating both sides once more, we get the general solution we are in search of: y = -4.9t²+c1t+c2, where c1 and c2 are unknown constants. 

In order to solve for those unknown constants, we need to know the initial conditions of the body; that is, we need its initial vertical position and velocity.  If the building has a height H, then f(0) = H.  As the ball is dropped in this problem, its initial velocity is zero, so f'(0) = 0.  Using these two initial condition, we can get the particular solution for the vertical position of the ball as a function of time: y = -4.9t² + H.  We could not have arrived at our function without the initial conditions, and that makes perfect sense: you cannot know where you are going if you don't know where you are coming from.  And, it is no coincidence that we needed two initial conditions to arrive at our particular solution: the governing equation was a second-order differential equation.

The position function derived above tells us the height in meters of a ball dropped  from a building that is H (meters) tall at any time t (seconds) after it is dropped (ignoring aerodynamic effects).  This result illustrates the power of differential equations.  A law of nature defines how a dependent variable must behave.  If we solve the differential equation that defines this behaviour, we arrive at a solution.  Using known laws of nature, one can do all of this without doing any experimentation. 

If we can solve a governing differential equation for any particular situation where the initial conditions are known, we can predict its future before it occurs.  As such, fortune-telling may be done with a pen and paper rather than a crystal ball.

Isaac Asimov is on a short list of my favourite science authors.  The list has two names on it: Arthur C. Clarke and Isaac Asimov.  Both write excellent science fiction (Asimov's "I Robot" and Clarke's "Fountain's of Paradise" are my personal favourites) and both write excellent topical science articles and essays.  While Orson Scott Card writes some compelling sci-fi (Ender's Game is probably my favourite novel of all time), he is not a "popularizer" of science - it is rare to find an author that is skilled in both fiction and non-fiction.

Asimov may have been the most prolific author ever, having published upwards of four hundred pieces.  His direct style in story-telling and, at times, redundant style of communicating in non-fiction does not appeal to everyone, but it does appeal to me.

Asimov's largest volume of work involves the communication of science, but physics in particular.  I just finished reading "Asimov on Numbers," which involves mathematics, but really focuses on, well, numbers.  It is a collection of essays written over a period of many years, beginning in 1959.  Although all articles are between four and five decades old, they have aged well.  The only instances where the book feels old is when population and financial figures are discussed, as the absolute values of both have inflated significantly in the ensuing years.

I learned a lot in reading these essays, mostly about the history of mathematics (very fascinating) and the Earth's geography (which, as it turns out, can be described so well numerically).

Asimov challenges the reader to consider the very act of counting, and note that the ten-based arabic counting system used around the world today is in fact quite arbitrary.  The base of ten is convenient as we have ten fingers, and groups of ten appeal to people (take the ten commandments for example).  In fact, we could have had a different base...perhaps a base of twelve.  In such a case, a number like 18 (in the base ten) would actually be written as 16, since the first digit would represent 12, and 12 + 6 = 18.  This may seem absurd, but computers are perfectly happy using the binary counting system, which uses a base of two.

There is even an essay on the subject of large numbers.  Here, we are asked to ponder the question, "What is the biggest number one might ever need to describe any measurement within our universe?"  I already knew that a googol referred to 10 to the power one hundred.  However, the fact that a twelve-year-old had proposed the number was news to me.  But there are numbers bigger than a googol...I won't mention them though, because that would spoil the fun of reading about them from Asimov (I actually laughed out loud reading this particular essay).

I particularly enjoyed learning about the ancient greeks, such purists when it came to mathematics, that they refused to recognize any geometrical activity that required more than a ruler and compass.  And yes, there are multiple articles about the irrational number known as pi.

Articles centering around geography use numbers to describe large things, like the volume of water on Earth, or the height of our highest mountain peaks.  We are also invited to think about how many electrons can fit in the volume of the universe.  This is a pretty big number (though not quite a googolplex).

I read and write so much about science, but seldom recognize the language with which we communicate it quantitatively: mathematics.  With this in mind, my next posting will touch on my favourite math topic: differential equations.

Typically, my semesters are spent teaching Physics courses.  The past two semesters have been a little bit different for me, with some Mathematics courses thrown into the mix.  A few weeks ago, in my Calculus course, one student, we will call “Jimmy”, asked if math could be used for something useful.

I explained that math was the language of science, and science affects us every day.  Also, business requires math to predict where it will go based on where it has been and other factors.  The social sciences require a deep understanding of statistics to make sense of our lives.

Jimmy said that business and science are good, but not all that important to him at this stage of his life.  “Can math help us find love?” he asked.  What an excellent question.

The students from my Calculus course would like

to wish you a happy Valentine's Day!

Together, the classroom of 18-year-old students and I began to investigate if we could use functions and calculus as a tool to help us find love.  The process took some time, and in the end, we did not answer this question to its full extent, but we did answer a different one: “What is the likelihood of finding love?”

With some chalk, a blackboard, and our brains, we determined what the class endearingly named “The Love Function”.  A function is a relationship between variables.  The love function outputs the probability, as a percentage, of a person to have fallen in love at least once at some point in their past given their current age in years.  The kind of love we are talking about here is the intimate kind (every child loves their parents; they are biologically programmed to do so).

For the love function to work, all one must do is input their age in years into the function, and out comes the likelihood that they are in love or have been at some point in the past.  The probability is dependent on time, the independent variable.

Several steps were taken to come up with an appropriate function to describe this probability.  First, a model, or function type needed to be determined.

The odds of finding someone to love vary as we go through life.  A baby cannot fall for someone intimately, and even a young adolescent is unlikely to do so in a meaningful way.  The twenties are definitely the high point for falling in love, as most first marriages occur when people reach their late twenties.  We decided that if someone has never fallen in love by the time they are 30 years old, it is still possible for it to happen, though less likely.

The probability for cupid to strike at least once increases for the entire domain of time that is a person’s life.  Those that know calculus realize that the function must therefore have a positive derivative at all times.  The function model we elected to use for the love function took the form: L(t) = A/(1+BC^t-D).  A function of this type increases slowly for some time, then increases dramatically, and then increases slowly again, as long as C is a value between zero and one.  To fully define the function, we now needed to determine the values of constants A, B, C, D.

We made three assumptions that enabled us to complete the love function.  The first one was that 90% of people who live beyond 80 years of age fall in love at least once.  This allowed us to seek the end behaviour of the function, which showed that A = 90.

The second assumption is that just 4% of 16-year-olds have found love at least once.  This may seem low, but we agreed as a class that it is hard to find true love until you know yourself fairly well, and few teenagers can make that claim.  This data point allowed us to solve for two unknowns: D = 16 and B = 21.5.

The final constant took some thinking to solve for.  The third assumption we made was that the maximum opportunity to fall in love occurred when one is towards the end of school or at the start of their first real job.  This moment in time, we assumed, occurs when someone is, on average, 24 years old.  The moment of maximum opportunity coincides with the maximum slope of the love function.  The students in the class, with some guidance from me, realized that the maximum slope occurs when the concavity of the function is zero.

So, we took two derivatives of the function.  We then set that function equal to zero, and input the time of maximum opportunity, t = 24.  This enabled us to solve for the final constant, C = 0.625.  With the constants all solved for, we could present the completed love function as follows:

L(t) = 90/[1+21.5(0.625)^t-16] [%]

If you replace the t in the equation above with your age in years, the calculation will output the likelihood that you are in love now, or have been in love at least once in your life, as a percentage.  The graph below is a plot of the love function.  What is the likelihood that you have found love?

The accuracy of this probability function is related to the correctness of our three assumptions as well as the validity of the function model we chose.  The use of some statistical data would allow the love function to be a bit more accurate.  As a first guess, I think this is a reasonably accurate result.

Of course, the true definition of love sits in the category of philosophy – a place where math has difficulty to swim.  The meaning of love is on the minds of many of you as I post this article today, the 14^th of February, Valentine’s Day.

If nothing else, the exercise was useful for the math class as it provided a meaningful backdrop for mathematical modeling and calculus.  Still, I think that the results of the love function are worth considering. 

The love function tells us that Valentine’s Day is or has been an important day to the vast majority of us fortunate enough to live a full life.  Also, the fact that the function is always increasing indicates that it is never too late to fall in love for the first time.

I present to you, "The Love Function"

 

Finally, I think that the steep slope of L(t) that we see in our twenties is representative of the turbulent period of time that this tends to be: a time when we are both stressed and swept away by the all-encompassing powerful entity that is love.