Tuesday, January 13, 2009

What is "Engineering Mathematics" ?

I teach "Engineering Mathematics". What does "Engineering Mathematics" mean ? How is engineering mathematics different from any other sort of mathematics ? Does it has an underlying philosophy or approach ?

To address these questions, we need to firstly need acknowledge that most people who teach engineering mathematics don't generally see themselves as engineering mathematicians but normally identify themselves as either a mathematican who teaches engineers, an applied mathematician or an engineer who teaches mathematics. Not surprisingly, ideas about "Engineering Mathematics" reflect these different backgrounds and perspectives, so a generally agreed defintion and overall philisophy is unlikely. I think qustions about the nature of engineering mathematics point to a related question - what is engineering ?

On that question, many books, essays and papers have been written, and hundreds of defintions provided. Here are three:

"the profession in which the knowledge of the mathematical and physical sciences gained by study, experience and practice is applied with judgement to develop ways to utilise economics, materials and forces of nature for the progressive well being of human kind"
Engineering Council for Professional Development quoted in Johnston et al., "Engineering and Society", Prentice Hall, London, 2000, p. 533.

" the art of directing the great sources of power in nature for the use and convenience of man"
British Institute of Civil Engineering 1828 quoted in Ferguson, "Engineering and the Mind's Eye", MIT Press, Cambridge (USA) 2001.

"Engineers will translate the action the dreams of humanity, traditional knowledge and the concepts of science to achieve sustainable management of the planet through the creative application of technology"
Inst. of Professional Engineers New Zealand 1993 quoted in Johnston et al., "Engineering and Societry", Prentice Hall, London, 2000, p. 533.

The issues relating to the role of mathematics in engineering are quickly apparent in these definitions. The first defintion specifically mentions the role of mathematics, the second emphasizes "art" and third has "science" and "traditional knowledge" underpinning the actions of engineers. These differences are not just reflections of people's different preferences in defining their profession but reflect much deeper divisions in underlying philosophies about engineering. For example, Eugene Ferguson argues in his book "Engineering in the Mind's Eye" that a mathematical approach in engineering design at universities has been over emphsised at the expense of visualisation and drawing. Sharon Beder suggested in her book "The New Engineer" (MacMillian, Melbourne, 1998) that the amount of mathematics in traditional engineering courses reflected the desire of early engineering academics to impress other academics in their institutions of the intellectual rigor of their programs, rather than an analysis of what level of mathematics engineers really need. Of course, there are many more who would passionality argue that mathematics is a key aspect of engineering and central to its development.

For the purposed of this discussions, I will go with the defintion provided by the Engineering Council. The key terms in the definition for me are "knowledge", "study", "experience", "practice" and "judgement". Their use in the defintion suggest that mathematics and science, combined with practical experience, help the engineer to makes judgements and choices about how to utilise resources for the benefit of humanity. This implies to me that "Engineering Mathematics" must empasize the role of mathematics in making sound choices. As I labour toward some sort of coherent defintion (please be patient !!), a few aspects of the problem are becoming clear to me:

A) Engineers need to understand how mathematical principles can be applied to practical problems.

B) Engineers need to be confortable with using mathematics as a tool to inform judgements and choices.

C) Engineers need education in fundamental aspects of mathematics, in so far as a means of developing the skills associated the applications and forming judgements on technical matters.

I think the first two points are uncontroversial (though I am often surprised what some people would like to argue with !) but the third point very much reflects a judgement I have formed from personal experience. Some people argue that education in fundamental aspects of mathematics for engineers is more about "developing thinking and intellect", others would see as a simple educational necessity (i.e. don't run before you can walk). My view is somewhat different from both these positions, that is that educating engineers in fundamantal aspects of mathematics should always be done in the context of application and techical judgements. For example, when I teach techniques for solving differential equations, I emphasise right from the start the practical implications of these techniques and the strategies that engineers use to form and solve these types of problems. I would also agree with the proposition that pure intellectual appreciation of methematics should also encouraged among engineering students (the developing thinking argument) and in fact I think that both approaches (developing thinking/learning in context) can be complementary.
These deliberations don't lead me any closer to a clean defintion of "Engineering Mathematics" but rather simple emphasize how ideas about this area of knowledge are invarably interwinned with ideas about the nature of "Engineering" - a concept itself that is subject to debate and constantly evolving. I personally find this lack of defintion and evolutionary nature invigorating.

Saturday, January 10, 2009

Why does multiplying two negatives result in a positive ?

Last year, a student in my first year mathematics class asked me an apparently innocent question:

Why does -5 x -5 = 25 ?

After some uncomfortable silence and a couple of minutes of moving by head around from hand to the other and scratching my chin (in a vain attempt to appear intelligent and considered)I resorted to the standard rescue line used by the intellectually challenged:

"Good question ! I'll have to think about that and get back to you."

My intellectual limitations aside, this is a good question. The other major rules of arithmetic seem obvious and intuitive, for example, if I have five dollars and you give me another five dollars, I indeed now have ten dollars (5 + 5 = 10). If you ask me to divide that amount evenly among five people, that will result in five piles of two dollars ( 10/5 = 2, 5 x 2 = 10). If from this ten dollars you take eight, I clearly have two left (10 - 8 = 2). If I owe you a further ten dollars, than it also follows that I am now eight dollars in debt (2 - 10 = -8). If I keep acumulating a debt of five dollars for five days, than I would owe twenty five dollars after five days (-5 x 5 = -25). Furthermore, if I now split this debt among twenty five people, they would each owe one dollar (-25/25 = -1).

Up to this point, these basic procedures of adding, subtracting, dividing and multilplying seem entirely consistent with our experience of the world.

However, the idea of -5 x -5 is hard to express in terms of a simple business transaction or an everyday example of counting. In fact, the idea of multiplying two negtives is quite abstract.

Do we accept that -5 x -5 = 25 by convention or is there a deeper reason ?

In fact, there are very good logical grounds why we accept that multiplying two negatives results in a positive. The argument goes as follows.

Multiplying numbers that are added (or subtracted) is the same as multiplying the numbers separately and than adding (or subtracting) the terms together (e.g. (2+5) x 6 = (2 x 6) + (5 x 6)).

Therefore, we can construct the following argument about multiplying any unknown (signified by y) with a negative:

(-1) x y = (-1) x y + y - y (yes, a bit of algebraic fiddling but true !)
= (-1 + 1)y - y (using the principle above)
= - y

Therefore, multiplying any number by a negative results in that nunmber changing signs and it follows:

-5 x -5 must produce a positive.

Logic has provided an answer where intuition and common experience have failed !

This matching of mathematical ideas with "intuition" and common experience becomes difficult as more advanced ideas of mathematics are introduced to students. The inherent abstraction associated with mathematics is one of its great attritubutes, as these abstractions lead to wonderful insights and ideas, but also can provide a barrier in learning. I think to a certain degree struggling with these ideas is a sign that you are really thinking about the issues, or at least that is my excuse !
Note: It can be argued that the distributive law of muliplication (the centre of the argument I presented) can be viewed as a convention of a particular system or rather than as a "law" and that it is possible to construct different systems built around quite different conventions. Barry Mazur explains this interesting view in his excellent book "Imagining Numbers" (Penguin, London, 2003).

Monday, January 5, 2009

Why do Engineering students study so much calculus ?

Why do Engineers students study so much calculus ?
There are many answers to this one !
1. "We had to suffer now its your turn !" (grizzled old Professor)
2. "As part of a government plan to keep underdesirables (e.g. mathematics lecturers) off the street" (cynical student)
3. "So much !!!! In my day, we were doing triple integration by the time we left primary school" (mathematics Professor in early stages of dementia)
Here is my answer. Engineering students should study calculus because , firstly, it provides them with a powerful tool to analyse the physical world and , secondly, because is a beautiful topic in itself, a triumph of imagination and analysis. Differential calculus, which I would describe as the mathematics of change, is incredibly useful for analysing the movement of fluids, the flow of heat, the rates of chemical reactions, the effect of changing electrical/magnestic fields, the movements of machines and numerous other engineering examples. Integral calculus, which I would describe as the mathematics of accumulation, allows to calculate area and volumes of complex shapes, centroids and moments of structures, accumulated energy and heat, and generally the overall effect of any varying process.

The fact that both forms of calculus are directly related to each other (i.e. one is the inverse operation of the other) is also one of the most profound and useful intellectual insights of all time. For example, this means that gradient of a tangent line to parabola (i.e the instanteous rise of the curve) is 2x, given y = x2. If we graph this gradient function (g = 2x), the area under that curve is the anitderivative (x2). In other words, the mathematics of change can also be used to describe the mathematics of accumulation. This is not only true for this simple example but for any function that you care to think off. Accumulation is the mirror image of change. Bravo Mr Newton and Mr Leibniz - who both worked this out in the 17th century, though there is still much debate about who was actually first to make the breakthrough (see "A Tour of the Calculus" by Berlinski for an entertaining overview of this topic).
The advent of symbolic computer programs and, more recently, CAS calculators has taken much of the pain from University calculus. This technological advance has shifted the emphasis from memorizing algebraic routines towards using calculus as a tool to analyse a problem.

Friday, January 2, 2009

Counting in eights

Imagine a parallel universe where everything is the same as our world except the people occupying this alternate existence have four digits on each hand and foot, and (unsurprisingly) count in groups of eights .... how does that work ?
Lets start counting 1, 2, 3, 4 (first hand), 5, 6, 7, 10 (second hand), 11, 12, 13, 14 (first foot), 15, 16, 17, 20 (every digit has now been used).
From this little example, we can see that "16" in our base 10 world is "20" in this strange place.
How would we make sense of this world ? Lets imagine we are driving down a road in this parallel universe and we see a sign saying "Maximum Speed 65 km/hr". What does this translate to in our universe ?
Lets start with the "65" part of the problem.
65 in octal (counting in eights) = (6 x 8 ) + (5 x 1 ) = 53 in base 10
Note: 655 in octal = (6 x 64) + (5 x 8) + (5 x 1) = 429 in base 10. Converting from a another base number to base ten always follows this simple procedure. Going the other way (e.g. from a base 10 number to another base) is a little trickier but basically involves dividing the base 10 number by the base of the new number and breaking down the number into multiples of the new base.
e.g. 1 53/8 = (6 x 8) + 5 remainder = 65 in octal
e.g. 2 429/8 = (53 x 8 ) + 5 remainder = (((53/8) x 8) x 8)+ 5 remainder
= ((6) 8 + 5 remainder) x 8) + 5 remainder = (6 x 64) + (5 x 8) + (5 x 1)
= 655 in octal
Easy so far (don't ask about fractions !). What about the unit "km" ? In our system this refers to a 1000 m. Lets assume that a metre in our eight toed counter universe is the same distance as our comfortable sane base 10 world. Therefore, 1000 m in octal = (1 x 512) + (0 x 64) + (0 x 8) + (0 x 1) = 512 m in base 10.
What about the unit of hours ? Lets us assume (a common expression for mathematicians !) that an hour is the same duration of time in both universe. This means that 65 km/hr is the equivalent as 53 x 0.512 km/hr = 27 km/hr in our neck of the woods. Our parrallel universe is not a fast place; I suggest not getting out of first gear and leaving the handbrake on !
Please notice that in an octal system, numbers will tend to be "larger" i.e. need more columns of numbers. This explains why computers sometimes store very large numbers using a hexadecimal system (base 16). On the other hand, the octal system needs less symbols to express numbers ("8" and "9" are redundant in our octal universe), which taken to an extreme in the binary system ("0" and "1") reduces counting to the manipulation of on/off signals. This reduction of symbols, first envisaged by Leibniz in the 17th cenury, greatly facilitates the mechanisation of counting and, indeed, binary is used in all modern computers for the manipulation and storage of numbers. The fact that Leibniz saw this advantage some 280 years before the first computer started crunching numbers in binary, tells you something about his powers of imagination and insight. Touche Mr Leibniz (who also invented calculus in his spare time).

Thursday, January 1, 2009

Imagining Archimedes

Welcome to Imagining Archimedes ! This blog has two main purposes, firstly, to celebrate the beauty and power of mathematics and, secondly, to provide an online service to first year engineering students studying mathematics (the subjects I teach at Swinburne University of Technology).

The site is named in honour of Archimedes (287-211 BC), one of the most brilliant mathematicians of all time, a talented inventor and engineer, a great physicist and an inspiration to anybody who has looked at thier surroundings and imagined another world beyond mere appearances. For me, mathematics is a language of the imagination, in the same way that the best poetry, art and music try to express what we feel, see and imagine. Amazingly, mathematics is also deeply practical and has underpinned finance and technology for thousands of years. In this way, mathematics is both beautiful and powerful, and for me, Archimedes is the embodiment of this ideal.
(Please see http://www.cs.drexel.edu/~crorres/Archimedes/contents.html for more about Archimedes)

Who am I ? My name is Geoff Brooks and I am a Professor of Engineering Mathematics at Swinburne University of Technology in Melbourne (Australia). My background is in Chemical Engineering and Process Metallurgy. I started my career as a very practical engineer, designing and building all sorts of equipment in the 1980s, before catching the research bug and doing my PhD at the University of Melbourne in the field of process metallurgy. Most of my current research is focused on using mathematical models to understand and optimise the processes for making metals. The ultimate goal of this work is to minimize the impact of metal production on the environment - the original motivation for starting my PhD. My knowledge of mathematics comes from both a deep love of the topic and from the experience of using mathematics to address practical issues.
(Please see http://www.swin.edu.au/feis/mathematics/staff/gbrooks.html for more about my research and background)