Presentation for my Rotary Club
Mathematics – a presentation for my Rotary Club on 10 November 2003
Rotaryclub Rijnwoude
“Wiskunde” is a unique Dutch word, make up by the Flemish mathematician Simon Stevin (Brugge 1548 – Leiden⁄The Hague, between 20-2 and 18-4-1620). He used “wisconst” for ars mathematica and between 1605 and 1608 “he wrote Wisconstighe ghedachtenissen”. "Wis" stands for "certain" or "sure" as opposed to "uncertain", and "kunde"for "art". So for Dutch speaking people wiskunde is the art of thoughts we are certain of. It does not depend on observations on natural phenomena, like science, nor on revelations, like religion.
See this mathematical chronology.
See this mathematical chronology.
I will restrict myself in two ways:
I will only talk on applied mathematics, not on theoretical, or pure mathematics;
I will only present examples from my own professional work.
We start with counting. First, as a todler, we count real-life things ( like one apple, two apples,…), later abstract : when I was four years old, the sisters of my playing mate - neighbours of my grand parents, thaught me counting above thousand (million, billion, trillion) as we sat on their garden fence.
When we count, we use natural numbers: 1, 2, 3, …. . . They are integer and positive = larger than zero. The allow us to add. We can find out how many objects we own.
Later we want to solve equations like x+10=3. We have to invent negative integer numbers. They allow us to find x=-7 as the solution of x+10=3 We now have integer numbers, both positive (larger than zero) as negative (smaller than zero). They allow us to subtract. We now can know have many things we have lend and borrowed.
With integers we can multiply. But we also want to solve equations like x*10=3. Once again, we have to invent another type of numbers: they allow us to find x=3⁄10 as the solution of x*10=3. Thanks to Simon Stevin (De Thiende, 1585) we recognise this as the decimal number 0.3. We now can divide objects like three apples among ten children.
Next, we want to solve equations like x2=2, where x2 is shorthand for x times x. But its solution is no rational number, so once again we invent numbers: the irrational numbers. A simple proof that x is not rational. Assume it is rational, so it can be written as a⁄b, where a, or b, but not both, can be divisible by 2. So we have x= a⁄b. Or x2=a2⁄b2=2, or a2 =2* b2. The right-hand side 2* b2 of this equality is divisible by 2, and so the left-hand a2 must be.
So a is divisible by 2, say a=2*c. Then a2 =22 *c2 = 2* b2 . Both the left side 22 *c2 and the right side 2* b2 are divisible by 2. Factor that common factor out and we are left with: 2*c2 = b2.
The left-hand side 2*c is divisible by 2, so the right-hand side b2 must be too. So b is divisible by 2. So both a and b are divisible by 2. And this contradicts our assumption that a and b are not both divisible by 2. We can only conclude that the solution of x2=2 is not rational. We now can find the length of a square piece of land if we know its area. Any number, be it rational or irrational, we call a real number.
So a is divisible by 2, say a=2*c. Then a2 =22 *c2 = 2* b2 . Both the left side 22 *c2 and the right side 2* b2 are divisible by 2. Factor that common factor out and we are left with: 2*c2 = b2.
The left-hand side 2*c is divisible by 2, so the right-hand side b2 must be too. So b is divisible by 2. So both a and b are divisible by 2. And this contradicts our assumption that a and b are not both divisible by 2. We can only conclude that the solution of x2=2 is not rational. We now can find the length of a square piece of land if we know its area. Any number, be it rational or irrational, we call a real number.
Next we want to solve equations like x2=- 2. But every real number, be it positive or negative, multiplied by itself, will yield a positive number “minus times minus is positive”). How do we solve this dilemma? By defining unreal (imaginairy) numbers as numbers whose square is negative. The unit of a pute imaiginary number mathematicians use the symbol i. In electronics the symbol j is used, because i is used for current. We can form a so-called complex number as the sum of a real and an imaginairy number. With complex numbers we can solve equations like Ax2 + Bx + C = 0. Or even equations like Anxn + An-1xn-1 + An-2xn-2+ ... + A1x + A0 = 0. So now we can compute the internal rate of return of any investment.
The above shows a mathematician is driven by an innate desire to solve problems. If possible, he uses concepts and tools he knows. If these are not sufficient, he developes new ones. The applied mathematician is satisfied with that. The pure mathematician not so much wants to solve practical problems, as to generalise. That can take some time, sometime even centuries.The Pythagorean theorem is much more recent than the experience of the pyramid builders, who made right angles with a triangle with sides 5, 12 and 13. In the past, the average math teacher taught often only abstract concepts, not their application in practise.
An example from my own practice. At Akzo we used a complex calculation based on the theories of Russian Pontryagin and the American Richard Bellman to calculate the optimal temperature profile of a chemical reactor to make nylon. It is one thing to compute the optimal profile once, taking to various technical constraints such as capacity burners. But we need more answers: does it make sense to install a larger burner? The brute force approach is to repeat the calculations, with a larger burner. But from the theory of static optimisation I knew the optimal solution inherently can provide the sensitivity to all relevant parameters. Now dynamic optimisation is far more complex than static optimisation. But using my guts feeling I wrote down the sensitivity equations for dynamic optimisation. A test calculation showed my formulas produced the correct results. Enthusiastically, I wrote a paper. But it was rejected: if it was so simple, the answer surely could be found in standard textbooks. I asked: in which book? On what page? No answer ever came forward. Later a smart student from Twente University wrote his graduate thesis under my guidance. Later on he even got his Ph.D. on the same subject. My formulas proved to be correct in all but exceptional cases.
With the same theory I could indicate within five minutes how to switch over from the production of clear nylon to white nylon in minimum time. White nylon is made by adding titanium dioxide to clear nylon. It was common practise to use a pump with the proper dose. The production of the first few hours, and trow that away: it no longer sufficiently clear, but also not yet not white enough. The solution is to use two pumps simultaneously for a short time, and then switch them both off at the right moment, and after some time switch on the original pump again. The right moments are computed as those moments that result in the titanium dioxide concentration just reaching the allowable maximum, and then dropping as quickly as possible to its allowable minimum, and then slowly rising to the nominal concentration.
Colleagues buying houses I could tell what their optimal morgage payments would be. As long as the nett interest (after tax deduction) is less than what savings produce nett, you should only pay interest, and do not pay off anything. At the latest moment possible, start paying off your morgage debt maximally. See also Pay off the morgage optimally.
Most intriguing was the question of a collegue of a different department at Akzo Research. He needed help to solve an ordinary non-linear differential equation. The answer indicated the time needed before the dependent variable would reach zero. I did not need much time to solve his equation, because a standard approach proved succesful. The answer was 6.5 hours. He was enthousiastic. “Do you know what you have calculated?”. I looked at him closely and had a hunch: “The sinking of the Titanic”, I answered (my encyclopedia writes three hours). He became more enthousiastic, even agitated. “How did you know that? I was there with my mother and brother and I can remember we were in the water and I had to let her go, and drowned”, he said.
That evening the library bus was parked in front of our house. At the back of the bus I found a book on the first and fatal trip of the Titanic. In the appendix a passenger list. There was only one woman with two little boys. So I could tell him his former name the following morning. But he knew that already.
That evening the library bus was parked in front of our house. At the back of the bus I found a book on the first and fatal trip of the Titanic. In the appendix a passenger list. There was only one woman with two little boys. So I could tell him his former name the following morning. But he knew that already.
