tag:blogger.com,1999:blog-26351126094072388282015-09-16T09:53:37.694-07:00Imagining ArchimedesGeoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.comBlogger38125tag:blogger.com,1999:blog-2635112609407238828.post-70494521999665493472012-09-03T16:18:00.002-07:002012-09-03T23:37:55.408-07:00Laplace Transforms on Wolfram Alpha<br /><div style="text-align: justify;">The use of Laplace Transforms in control theory and in signal analysis took off after WWII and has become a widely established tool of analysis. When I was a student in the dark ages (before the Internet and reality TV), we used look up tables to determine Laplace Transforms and their inverse. These tables are still widely used but online tools like Wolfram Alpha can also be readily used.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Like many aspects of Wolfram Alpha, the commands are largely intuitive and the program is forgiving with syntax:</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">For example, type in "Laplace Transform (t sin(2t))" into the dialogue box will produce the following answer and a graph of the function F(s). The graph of the transform is a nice bonus above the traditional look up tables, as it helps appreciate the new function that you formed through this transformation.<br /><br />The inverse operations are just as simple. For example, the program only blinked for a few moments on "Inverse Laplace Transform (3s /(s^2 + 6))" to produce "3 cos(SQRT(6) t)". Once again a useful graph is generated and can be downloaded.<br /><br />I encourage you to test how good Wolfram Alpha is at solving the inverse problems. Man (and Woman) against machine is always fun.<br /><br />I'm sure Marquis de Laplace would have been impressed !!<br /><br /></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com0tag:blogger.com,1999:blog-2635112609407238828.post-6802621109397697542012-09-03T15:55:00.001-07:002012-09-03T15:55:58.217-07:00Who was Laplace?<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-4D06OKqusJU/UEUyRzQej8I/AAAAAAAAAko/Y-t4VNaYzNw/s1600/180px-Pierre-Simon_Laplace.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" fea="true" src="http://1.bp.blogspot.com/-4D06OKqusJU/UEUyRzQej8I/AAAAAAAAAko/Y-t4VNaYzNw/s1600/180px-Pierre-Simon_Laplace.jpg" /></a></div><br /><br /><div style="text-align: justify;">Pierre Simon Laplace (1749-1827) is a giant figure in the history of mathematics and astronomy. His career coincided with great upheaval in his home country of France, in particular the overthrow of the Bourbon monarchy and the rise and fall of Napoleon Bonaparte. Laplace was the son of a farmer (as was Newton) and was brought up in Normandy before going to Paris and becoming a Professor of Mathematics at the Ecole Miltaire. Several great mathematicians, physicists and engineers were associated with the military schools in France at the time. For example, Fourier (1768-1830) and Carnot (1796-1832) followed a similar route and had a great impact around the time of Laplace. This golden era of French mathematics and physics is directly connected to the political and social upheavals of the time. Napoleon was a great supporter of mathematics and science in France, and himself closely associated with many of the leading mathematicians of the time. Carnot's father was a general in Napoleon's army and Fourier was a trusted associate of Napoleon, famously serving as a Governor of Lower Egypt. Laplace himself is famously reported to have quipped to Napoleon "I have no need of the hypothesis", when queried by the Bonaparte why his book on planetary motions didn't mention god.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Laplace's interests in understanding the motions of planets, particularly that of Saturn and Jupiter, are linked with his developments in Mathematics . Laplace also formed a famous theory on how the planets originated. He theorized that the solar system started as a massive cloud of dust that collapsed to form the sun with the remnants condensing to form planets. An updated versions of this model of planetary formation is now widely accepted and substantial evidence has now been gathered to support this theory. Laplace came up with many novel ideas about solving differential equations, particularly, through the use of potential functions. The famous second order differential equation named after him is widely used throughout physics and mathematics. Lapalce also made significant contributions to probability theory and numerical techniques for solving equations. The famous "Laplace Transform" used widely in control theory and signal analysis is actually a variation of the approach taken by Laplace himself, though his ideas made a significant contribution to the development of this type of analysis.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Laplace's worked during a time when there was less clear distinction made between mathematics and physics. His great success in making contributions to both fields suggest that maybe the modern tendency towards narrow specialisation is not helpful to developing new ideas. Certainly, his work is an inspiration to those who wish to work across fields.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">References</div><div style="text-align: justify;">Larousse, Dictionary of Scientists, 1994</div><div style="text-align: justify;">Wikipedia entry on Pierre Simon Laplace (accessed September 2012). </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com0tag:blogger.com,1999:blog-2635112609407238828.post-19758591951410860572012-08-17T23:48:00.000-07:002012-08-18T00:35:22.969-07:00Odd and Even Functions - Oddly Interesting<br /><div style="text-align: justify;">The idea that functions can be either "odd" or "even" is an initially a surprising idea. </div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Yes, a number can be <em>odd</em> or <em>even</em> and some people are <strong>odd </strong>but can a function be odd or even ?</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The formal definition of an even function is that f(-t) = f(t) for all of t, while for an odd function f(-t) = - f(t) for all of t.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">These formal definitions don't immediately expose the power of the concept. Examining graphs of odd and even functions quickly reveal why the concept is useful and interesting. For example, below is a graph of the function is x^3sin(x), which satisfies the condition of an even function.</div><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-0eD5-UMhMNE/UC8tFJDYD2I/AAAAAAAAAkE/WCYguNQoJQA/s1600/x%5E3sin(x).gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="145" src="http://1.bp.blogspot.com/-0eD5-UMhMNE/UC8tFJDYD2I/AAAAAAAAAkE/WCYguNQoJQA/s320/x%5E3sin(x).gif" width="320" /></a></div><br /> Figure: Plot of x^3 sin (x) (image from Wolfram Alpha)<br /><br /><div style="text-align: justify;">Please note the following that the function is fully symmetrical around the y-axis (which is is the same as saying f(-x) = f(x)) As a result, iintegrating the function from 0 to any positive value of x will have the identical answer as integrating from 0 to the same negative value. This also means that integrating f(x) from x= -b to x= b is the same as doubling the definite integral from x= 0 to x= b.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Some common even functions include, f(x) = x^2, f(x) = cos(x), f(x) = SQRT(1 + x^2), f(x) = 1/(4-x^2) and many more.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Odd functions also have an underlying symmetry but it is 180 degrees around the origin. For example, plotting f(x) = x^2 sin(x) quickly reveals a quite different symmetry from the x^3 sin(x).</div><div style="text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-LbkVLbZi5U0/UC80VtuzULI/AAAAAAAAAkU/PDL8yJD3k5Q/s1600/x2sinx.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="150" src="http://1.bp.blogspot.com/-LbkVLbZi5U0/UC80VtuzULI/AAAAAAAAAkU/PDL8yJD3k5Q/s320/x2sinx.gif" width="320" /></a></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Figure: Plot of x^2 sin(x) (image from Wolfram Alpha)</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">For odd functions, the y value for a particular positive value of x (say x= b) will by -y for x = -b. As a result, the definite integrals from 0 to b will all be the same absolute value but the opposite sign for 0 to -b. Thus, integrating any odd function from -b to b will always results in "0" as the answer (a nice relationship to exploit when simplifying definite integrals in Fourier Analysis). This is all obvious from looking at the graph above. As often the case in mathematics, symmetry is a wonderful thing !</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The relationships that follow on from these simple definitions are even also satisfying, namely:</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">a) An Odd function multiplied by an Odd function results in an Even function ( O x O = E)</div><div style="text-align: justify;">b) An Odd function multiplied by an Even function results in an Odd function (O x E = O)</div><div style="text-align: justify;">c) An Even function multiplied by an Even function results in an Even function (E x E = E)</div><div style="text-align: justify;">d) The reciprocal of a Odd function is an Odd function ( 1/O = O)</div><div style="text-align: justify;">e) The reciprocal of an Even function is an Even function (1/E = E)</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Somehow, these relationships seem intuitive and are in keeping with our numerical sense of "odd" and "even". They can also be easily confirmed through plotting and/or simple substitution.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">In fact, I strongly encourage you to prove these relationships to yourself. That way, you will really see what is oddly interesting about symmetry.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div><br /><br /><br /><br /><br /><br /><br /><br /><br />Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com0tag:blogger.com,1999:blog-2635112609407238828.post-40159066147314340292012-08-11T00:25:00.000-07:002012-08-13T18:19:49.890-07:00Why Engineering Mathematics is not a "service" subject<br /><div style="text-align: justify;">We all have sensitive points !</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">I find that discussion of the 1976 Olympic Men's Hockey final distressing (Yes, Australia lost to New Zealand but please don't talk about it). A statistical analysis of the Bulldogs performance in AFL is another matter that is best left alone with me (OK, we only have one Premiership).</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">At work, the description of Mathematics as a "service" subject in Engineering is likely to raise the blood pressure. The term service implies that mathematics is some kind of ssecondary topic to Engineers, a kind of background material before they get to the meat of their degree. RRRRRRR</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Of course, some Engineers don't directly use much mathematics in their daily jobs, this is a particularly true in areas of management, sales/marketing and production. Even in the "hard" technical areas of Engineering such as design and research only a few are regularly performing mathematical operations in their daily jobs. Quite correctly, alot of Engineering involves "soft" skills associated with teamwork, communication and generic management skills. I feel no need to denigrate these skills compared to mathematics, physics and the core sciences associated with engineering, as it clear to me that great engineering is much as a triumph of organisation and human co-operation, as it is a celebration of powerful mathematics and science.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">The film "Apollo 13" (and book by Jim Lovell and Jeffrey Kluger) is an excellent portrayal of the need for strong leadership and teamwork, as well as deep technical knowledge, in solving challenging technical problems - in this case, finding a way to safely return astronauts from a damaged spaceship. "Huston, we have a problem" is the famous catch phrase from the film. This classic understatement from Gene Kranz (the legendary NASA flight director) who muttered these words in real life, emphasised the need for calm analysis in the face of imminent disaster. As the films shows in some detail, what follows. is a story of determination, teamwork, careful sciscientific analysis of data and systematic evaluation of the options. The hero's of the films are as much the scientists, engineers and technicians on the ground as the three men in the damaged ship. We see the various players carefully checking calculations, modifying equations and running algorithms, as the drama unfolds. Human joy is unleashed as the Astronauts voices are heard after splash down, even the rock like Kranz sheds a tear.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">What a wonderful celebration of Engineers and Scientists !</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">And here is my point ..... all of this is underpinned and linked by mathematical skills and the language of mathematics. It is rigorous training in arithmetic, trigonometry, algebra and advanced mathematics that allows the engineers to make sound choices under extreme pressure. As they rush to find the right path home, it is confidence in the core mathematics and physics behind their calculations that allows them to make life and death decisions.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;">Of course, not many engineering projects are as dramatic as "Apollo 13" but the point remain the same, even when engineers are not directly carrying out mathematical operations and analysis, it is their training and confidence in mathematics and fundamental sciences that empowers them to make wise choices. Mathematics is not only the language of technology but also one it its corner stones. Mathematics is not "servicing" Engineering, it is a core topic, a central part of its nervous system and present in all its vital organs.<br /><br />In summary, my advice to any young engineer is to pay close attention to your mathematics, develop your analytical skills and avoid supporting sportings teams that have only spasmodic success.</div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div><div style="text-align: justify;"><br /></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com4tag:blogger.com,1999:blog-2635112609407238828.post-59286907375094658942012-08-04T15:40:00.000-07:002012-08-28T18:21:53.024-07:00Integration Using Wolfram Alpha<div class="separator" style="clear: both; text-align: justify;">Wolfram Alpha is a web based tool that allows you carry out quite sophisicated mathematical operations, producing both analytical and graphical answers. The operations of Wolfram Alpha are based on the software "Mathematica" that was developed in the late 1980's by Stephen Wolfram and his team. The web tool is much more forgiving than the off line software in terms of syntax, though Mathematica is much more powerful and can be used as a sophisicated programming language.</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: justify;">However, Wolfram Alpha is simple to use, performs most mathematical opeations relevant to undergraduate and High School students and is FREE !!! </div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;">To illustrate this great utility, head to <a href="http://www.wolframalpha.com/">www.wolframalpha.com</a>, think of a mathematical operation and type in your question.</div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;">I typed in "Integrate cos(x) from x = -pi to pi" into the input box and pushed enter. If you don't feel so game as me, you can go to the example page where there are numerous examples of mathematical operations that can be performed and the input forms preferred by the program. If you guess the form, like I often do, the program will do its best to make sense of your crude mathematical jottings,</div><div class="separator" style="clear: both; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: justify;">A few micro seconds later, the following output came back:</div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><a href="http://1.bp.blogspot.com/-lve5sPec2zQ/UB2aOp0N9SI/AAAAAAAAAjQ/AxQKyHNcHCA/s1600/integrating+cosx.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" eda="true" src="http://1.bp.blogspot.com/-lve5sPec2zQ/UB2aOp0N9SI/AAAAAAAAAjQ/AxQKyHNcHCA/s1600/integrating+cosx.gif" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><a href="http://1.bp.blogspot.com/-Nd9edsaU_4w/UB2agZfOOZI/AAAAAAAAAjY/HuiDEV3CeFE/s1600/graph+int+cos(x).gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" eda="true" src="http://1.bp.blogspot.com/-Nd9edsaU_4w/UB2agZfOOZI/AAAAAAAAAjY/HuiDEV3CeFE/s1600/graph+int+cos(x).gif" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;">(Images from www. wolframalpha.com)</div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><br /></div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none; text-align: justify;">The graphs generated by Wolfram Alpha are good quality and can be saved as PDFs. I also think the graphical nature of the solution is most helpful in visualising the mathematical operation being performed. For example, the symmetry of cos(x) integral around the y axis is quite evident in the solution above. This property means that we classify cos(x) as an "even function". A parabola is another example of an even function. "y=x" and "y=sin(x)" are simple examples of functions that don't have this symmetry, which I will discuss in more detail in a later entry.</div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><br /></div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;">Of course, this problem is too simple, now I type in a problem that is a little more challenging:</div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><br /></div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;">"Integrate x^2 cos(x)"</div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><br /></div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none; text-align: justify;">Wolfram Alpha eats up problems like this (this one would take me several minutes using pen and paper and even a a few seconds using my well worn integral tables) and even provides the solution steps, which is an invaluable to any student (or even a rusty Professor) trying to get on top of the mysteries of integration.</div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none; text-align: justify;"><br /></div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none; text-align: justify;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-YcEcT97A458/UB2h-lKoovI/AAAAAAAAAjo/_iLibR9I2AE/s1600/Integrate+x2cosx+steps.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" eda="true" height="256" src="http://3.bp.blogspot.com/-YcEcT97A458/UB2h-lKoovI/AAAAAAAAAjo/_iLibR9I2AE/s320/Integrate+x2cosx+steps.gif" width="320" /></a></div>and another useful graph<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-J0SSbvXJPEo/UB2iNgYNExI/AAAAAAAAAjw/iVNHR7f8h4Q/s1600/Integrate+x2cosx+graph.gif" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" eda="true" src="http://1.bp.blogspot.com/-J0SSbvXJPEo/UB2iNgYNExI/AAAAAAAAAjw/iVNHR7f8h4Q/s1600/Integrate+x2cosx+graph.gif" /></a></div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><br /></div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;">(Images from <a href="http://www.wolfamalpha.com/">www.wolfamalpha.com</a>)</div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><br /></div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none; text-align: justify;">Once again, note how the graphical representation of the function makes it immediately clear that the function has a particular symmetry.</div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none; text-align: justify;"><br /></div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none; text-align: justify;">Wolfam Alpha has it critics and Stephen Wolfram himself is a controversial figure but personally I am most grateful to have access to such a obviously useful tool.</div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><br /></div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><br /></div><div style="border-bottom: medium none; border-left: medium none; border-right: medium none; border-top: medium none;"><br /></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com0tag:blogger.com,1999:blog-2635112609407238828.post-58787642223657163432010-08-08T17:04:00.000-07:002010-08-08T17:22:41.977-07:00Out of the Flatlands !<div align="justify">Today, I commence lecturing in HMS 112 (the second session subject mathematics subject for first year students at Swinburne), which is exciting for me and hopefully also for the students ! Of course, I appreciate that students returning from the break may still be building up enthusiasm for the subject ... I suggest a shot of <span class="blsp-spelling-error" id="SPELLING_ERROR_0">caffeine</span> and a gentle start.<br /><br /></div><div align="justify"></div><div align="justify">Why is it exciting ? Because I get to escape the "<span class="blsp-spelling-error" id="SPELLING_ERROR_1">flatlands</span>" of high school mathematics for once and all; I leave behind x y plots and simple one variable problems and head towards the valleys and byways of 3-D land, where unexpected dips and rises test your powers of visualisation and calculus. Where a slope is not a simple slope but needs to be defined relative to the land around it and where the mathematical symbols get curly and more cryptic !<br /><br />Even at this stage of the adventure, our powers of imagination are being tested (was is the difference between a circular paraboloid and a two sheet hyperboloid ?) and we haven't even entered complex number land yet !!!!<br /><br /></div><div align="justify"></div><div align="justify"></div><div align="justify">My advice to students entering this new land .... hold on, enjoy the ride and keep a sense of humour.</div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com10tag:blogger.com,1999:blog-2635112609407238828.post-41336044645098671342010-03-06T01:12:00.000-08:002010-03-07T15:56:48.192-08:00Fourier Series and other amazing feats on WolframAlpha<div align="justify"><br />As occasional visitors of my blog would know, I am a great enthusiast for <a href="http://www.wolframalpha.com/">http://www.wolframalpha.com/</a>. For those <span class="blsp-spelling-corrected" id="SPELLING_ERROR_0">unfamiliar</span> with the <span class="blsp-spelling-error" id="SPELLING_ERROR_1">Wolframalpha</span> website, it is a powerful online tool that allows you <span class="blsp-spelling-corrected" id="SPELLING_ERROR_2">perform</span> quite <span class="blsp-spelling-corrected" id="SPELLING_ERROR_3">sophisticated</span> algebraic feats with excellent graphical solutions provided with details of the algebra. Fortunately, the commands for the software are quite intuitive and easy to learn. I suggest going to the website and start playing (try "plot x^2<span class="blsp-spelling-error" id="SPELLING_ERROR_4">sinx</span>", "Integrate <span class="blsp-spelling-error" id="SPELLING_ERROR_5">xcos</span>(x^2) from x =1 to 3", "Differentiate x^2In(x)" and "Solve x^3-2x^2 + 6x - 10 =0" for starters - there is an example page to help you with syntax and common commands).</div><br /><div align="justify"></div><div align="justify">For students studying Fourier Series, the site is particularly useful. Some of the <span class="blsp-spelling-corrected" id="SPELLING_ERROR_6">exercises</span> I can recommends for students of the Fourier Series:</div><br /><div align="justify"></div><div align="justify">A. Visualising the periodicity of <span class="blsp-spelling-corrected" id="SPELLING_ERROR_7">function</span> with multiple terms</div><div align="justify"></div><br /><div align="justify">e.g. Compare a "Plot Sin(x/2) + Sin(x) + Sin(3x)" with "Plot Sin(5x) + Sin(x) + Sin(3x)"<br /></div><div align="justify"></div><div align="justify"><br />B. Integrating terms in evaluating the coefficients of the Fourier Series<br /></div><div align="justify"></div><div align="justify"><br />e.g. If you are <span class="blsp-spelling-corrected" id="SPELLING_ERROR_8">determining</span> the "a1" coefficients for f(x) = 2x + 3 over the period 2pi, "Integrate (2x+3) cos(x) from x = -pi to pi"<br /></div><div align="justify"></div><div align="justify"><br />C. Carrying out a full Fourier expansion of f(x) over a period of 2pi for n terms</div><div align="justify"></div><div align="justify"><br />e.g. "<span class="blsp-spelling-error" id="SPELLING_ERROR_9">FourierTrigSeries</span> 2x+3, x, 6"</div><div align="justify"></div><div align="justify"><br />D. Checking whether a certain function is odd or even</div><div align="justify"></div><div align="justify"><br />e.g. "Plot x^2, <span class="blsp-spelling-error" id="SPELLING_ERROR_10">sinx</span>" to compare an even with an odd function, and "Plot x^2 <span class="blsp-spelling-error" id="SPELLING_ERROR_11">sinx</span>" to see what happens when you multiply an odd and even <span class="blsp-spelling-corrected" id="SPELLING_ERROR_12">function</span>.</div><div align="justify"></div><div align="justify"><br />E. Performing a half range cosine expansion of a function with a period of 2pi</div><div align="justify"></div><div align="justify"><br />e.g. "<span class="blsp-spelling-error" id="SPELLING_ERROR_13">FourierCosSeries</span> 2x+3, x, 6"</div><div align="justify"></div><div align="justify"><br />F. Performing a half range sine expansion of a function with a period of 2pi</div><div align="justify"></div><div align="justify"><br />e.g. "<span class="blsp-spelling-error" id="SPELLING_ERROR_14">FourierSinSeries</span> 2x+3, x, 6"</div><div align="justify"></div><div align="justify"><br />G. Carry our a Fourier series expansion in complex form</div><div align="justify"><br />e.g. "<span class="blsp-spelling-error" id="SPELLING_ERROR_15">FourierSeries</span> 2x+3, x, 6"<br /><br /></div><div align="justify"></div><div align="justify">In all cases, the software can be used to aid learning and also check the answers you are calculating or deriving, AND IT IS ABSOLUTELY FREE !</div><div align="justify"></div><div align="justify"></div><div align="justify"></div><div align="justify"></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com29tag:blogger.com,1999:blog-2635112609407238828.post-20658693064445978722010-03-03T02:18:00.000-08:002010-03-03T03:01:01.688-08:00Going from words to symbols<div align="justify"><br />One of the important skills that we develop in Engineering mathematics, is the ability to develop mathematical relationships from written (or verbal) descriptions. This translation (or perhaps interpretation ?) from "ideas" into equation form is challenging because its requires both mental dexterity and familiarisation with mathematical language.<br /><br /></div><div align="justify"></div><div align="justify">It does take some confidence to translate "If you have a room of volume 60 cubic meters, that is one meter higher than it is wide and one meter longer than it is high, what are the dimensions of the room ?"<br /><br /></div><div align="justify"></div><div align="justify">into<br /><br /></div><div align="justify"></div><div align="justify">(x+1)(x+ 2) x = 60; w=x, h= (x+1) and L=(x+2), solve for x<br /></div><div align="justify"></div><div align="justify"><br />This is particularly apparent when first year engineering mathematics students tackle vector problems that start with descriptions like "A ferry is crossing a river ....". I think this dis-comfort reflects a background of solving problems that either already defined in mathematical terms or has a ready made picture representation provided with the problem. Unfortunately, the problems presented to engineers and applied mathematicians are rarely presented so neatly.<br /><br />My advice to developing this skill can be broken down into the following steps:<br /><br />a) Try to represent the problem as a picture through a freehand sketch (labelling lines and symbols from the written description of the problem).</div><div align="justify"></div><div align="justify"><br />b) Try to visualise the problem from this picture representation, forming an image in our mind, identifying what specific problems you are trying to solve.</div><div align="justify"></div><div align="justify"><br />c) Express the problem in symbolic form, writing down definitions of the symbols you are using or any assumptions you need to make.</div><div align="justify"></div><div align="justify"><br />d) Solve the equation (or equations) you have formed, showing each step systematically.<br /><br /></div><div align="justify"></div><div align="justify">e) Look at your answer and your original picture of the problem and ask yourself two important questions:<br /></div><div align="justify"></div><div align="justify"><br />(i) Have you answered the original question ?<br /></div><div align="justify"></div><div align="justify"><br />(ii) Does your answer make sense ? (Is it believable ?)<br /><br /></div><div align="justify"></div><div align="justify">Of course, like any skill, practice will develop your abilities. It must also be admitted that there is an element of "art" to the processes described above that is beyond words or description .... which makes it fun and challenging !</div><div align="justify"></div><div align="justify"></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com1tag:blogger.com,1999:blog-2635112609407238828.post-14973992122843030192010-03-02T19:04:00.000-08:002012-08-11T21:30:25.042-07:00Vectors made easy !<div align="justify"><br />The unit vector notation used in Engineering mathematics is wonderfully simple and powerful.<br /><br />Imagine we have a position in the x/y plane, lets call it P, and we want to form a vector from the origin to this point. We call that position vector <strong>OP</strong>.<br /><br />Lets say that P is located at x=2 and y=3, now we can draw a line from the origin to P and put an arrow head along it. We have a vector.<br /><br />Now lets define the unit vector (i.e. one unit in a particular direction) in the x direction as <strong><em>i</em></strong> and the unit vector in the y direction as <strong><em>j</em></strong>.<br /><br />We can now say OP = 2<em><strong>i</strong></em> + <em>3<strong>j.</strong></em><br /><br />Lets try a few things ....<br /><br />What is the magnitude of this vector ?</div><div align="justify"></div><div align="justify"><br />If we draw a right angled triangle from <strong>OP</strong>, we can quickly see that the magnitude of OP must be equal to the square root of (2 squared + 3 squared) = sqrt(13).</div><div align="justify"></div><div align="justify"><br />What is the unit vector of <strong>OP</strong> ?</div><div align="justify"></div><div align="justify"><br />It must be <strong>OP</strong>/sqrt(13) i.e. one unit in the direction of <strong>OP</strong>.</div><div align="justify"></div><div align="justify"><br />What about if we want to add another vector (<strong>OQ</strong> = 4<strong><em>i</em> </strong>+ 1<strong><em>j</em></strong>) to <strong>OP</strong> ?</div><div align="justify"></div><div align="justify"><strong><br />OP</strong> + <strong>OQ</strong> = (4 + 2)<strong><em>i</em> </strong>+ (3 + 1)<strong><em>j</em></strong> = 6<strong><em>i</em></strong> + <strong>4<em>j</em></strong>.</div><div align="justify"></div><div align="justify"><br />What if I want to find the vector <strong>QO</strong> ?</div><div align="justify"></div><div align="justify"><strong><br />QO</strong> = -<strong>OQ</strong> = -(4<em><strong>i</strong></em> + <em>1<strong>j</strong></em>) = -<strong>4<em>i</em></strong>-1<strong><em>j</em></strong></div><div align="justify"></div><div align="justify"><br />What about if we want to find the vector <strong>PQ</strong> ?</div><div align="justify"></div><div align="justify"><br />We use the head to tail rule and say <strong>PO</strong> + <strong>OQ</strong> = <strong>PQ</strong> = -(2<strong><em>i</em></strong> + 3<strong><em>j</em></strong>) + (4<strong><em>i</em></strong> + 1<strong><em>j</em></strong>) = 2<strong><em>i</em></strong> - 2<strong><em>j.</em></strong></div><div align="justify">This approach makes the whole problem of manipulating vectors easy and more powerful than constantly referring to angles and scalar qauntities.</div><div align="justify"></div><div align="justify"></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com1tag:blogger.com,1999:blog-2635112609407238828.post-41234657840800234062010-03-01T13:47:00.000-08:002010-03-01T14:20:03.824-08:00Who was Fourier ?<div align="justify"><br />Joseph Fourier (1768-1830) like many brilliant scientists and mathematicians before the 20th century and modern tendency towards narrow specialisation, excelled in many fields and combined theoretical brilliance with practical ability. An orphan at the age of ten, he was educated in a military school and an abbey, showing outstanding mathematical ability from a young age. Joseph was a man of his times and played his own role in the French Revolution, subsequently serving in Napoleons armies before taking up a position at the Governor of Lower Egypt (imagine, your mathematics lecturer being the governor of lower Egypt !). He loyalty to Napoleon continued, as he also served in his armies during Napoleon's brief return to power in 1815. Certainly, the revolution had been important in providing a person like Fourier of humble birth ( he was the son of a tailor) opportunities to excel and make a mark in French society.<br /><br /></div><div align="justify"></div><div align="justify">After several adventures in Egypt, he returned to France and mixed his ability in administration with experimental science and mathematics. He was made a Baron in 1808 and served in senior roles in the Academy of Sciences. Fourier was particularly interested in finding mathematical methods for describing heat flow. It was in this context that Fourier developed the idea that any continuous or dis-continuous function could be expressed as a infinite series of trigonometric functions. He wasn't able to prove this to be correct or general but he did develop techniques that proved invaluable since for many wide ranging mathematical problems. It is Fourier who showed that any complex wave form could be broken down into a combination of simpler wave forms. This remains a brilliant insight that guarantees his place as one of the great figure of mathematics. Merci Monsieur Fourier !<br /><br /></div><div align="justify"></div><div align="justify">PS. In recent times, people have argued that Fourier was also the first to correctly identify the mchanism of global warming (see <a href="http://www.aip.org/history/climate/co2.htm">http://www.aip.org/history/climate/co2.htm</a>)</div><div align="justify"></div><div align="justify"><br />Sources:</div><div align="justify">Larousse, Dictionary of Scientists, 1994, New York</div><div align="justify">Delvin, The Language of Mathematics, 1998, New York</div><div align="justify">Wikipedia</div><div align="justify"></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com0tag:blogger.com,1999:blog-2635112609407238828.post-34024228287587436202010-03-01T00:42:00.000-08:002010-03-01T14:24:14.720-08:00Fourier's great idea<div align="justify"><br />As students enter second year mathematics, they will be introduced to a famous mathematical concept called the "Fourier Series", which (unsurprisingly) developed by Fourier in the early part of the 19<span class="blsp-spelling-error" id="SPELLING_ERROR_0"><span class="blsp-spelling-error" id="SPELLING_ERROR_0">th</span></span> century. The Fourier series is based on a very elegant idea that has proven to be very useful in solving equations described the motion of waves, the flow of heat and almost any function or physical behaviour that has a bit of "up and down" (which mathematicians call "periodic").<br /><br /></div><div align="justify"></div><div align="justify">The basic idea is that any periodic function can be approximated by combining sine and cos functions in an infinite series:</div><div align="justify"></div><div align="justify"><br />e.g. f(x) = constant + a1<span class="blsp-spelling-error" id="SPELLING_ERROR_1"><span class="blsp-spelling-error" id="SPELLING_ERROR_1">cosx</span></span> + b1<span class="blsp-spelling-error" id="SPELLING_ERROR_2"><span class="blsp-spelling-error" id="SPELLING_ERROR_2">sinx</span></span> + a2cos2x + b2sin2x ......<br /><br /></div><div align="justify"></div><div align="justify">In this form, the overall period of this function is 360 degrees (2 pi) - you can easily prove to yourself that when you combine <span class="blsp-spelling-corrected" id="SPELLING_ERROR_3">trigonometric</span> functions of different periods, the longest period dominates the overall periodic behaviour of the series. Like the Taylor series (which uses an infinite combination of <span class="blsp-spelling-corrected" id="SPELLING_ERROR_3">polynomial</span> terms), the more terms included in the series, the greater convergence between the series and original function.<br /><br /></div><div align="justify"></div><div align="justify">This idea is, in fact, correct for many continuous and discontinuous functions though Fourier's <span class="blsp-spelling-corrected" id="SPELLING_ERROR_4">original</span> development of the series (in 1822) did not elucidate the limits of this theorem. Fourier did develop a very clever way of <span class="blsp-spelling-corrected" id="SPELLING_ERROR_6">evaluating</span> the constants in the equation through integrating combinations of f(x) and sine and cosine functions over one period of the function. This procedure, which now can be easily performed by computers ("<span class="blsp-spelling-error" id="SPELLING_ERROR_7"><span class="blsp-spelling-error" id="SPELLING_ERROR_4">Mathematica</span></span>" or my favourite website <a href="http://www.alphawolfram.com/">http://www.alphawolfram.com/</a>) or by a hard working second year engineering student armed with a table of standard integrals.<br /><br /></div><div align="justify"></div><div align="justify">The Fourier series, long with Taylor's series, is one of the most important mathematical tools available to engineers and scientists for analysing wave functions (e.g. radio waves, music, surf. etc.), solving differential equations and even as a means for compressing and storing data.</div><div align="justify"></div><div align="justify"></div><div align="justify"></div><div align="justify"></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com0tag:blogger.com,1999:blog-2635112609407238828.post-76238070794618003152009-08-21T21:27:00.000-07:002009-08-21T22:23:31.325-07:00The Multi-dimensional Universe<div align="justify"><br />In first year engineering mathematics course at Swinburne we try to lead students from the boring 2-D universe of high school (uniforms, canteen lunches and x vrs y graphs) to the much more exciting and vivid multi-dimensional world of advanced mathematics ! This progression - more of a leap - requires some imagination and determination. The first step is the visualisation of 3-D space, than a progression to a more general idea of "dimensions". When I teach partial differentiation, I get the students to think about how the slope of a hill changes when you keep one dimension constant (i.e. don't move sideways and walk upwards), as compared to the slope if you swap the dimensions you keep constant (i.e., don't move upwards but only sideways). These relationships become clearer when you draw graphs of these relationships for different physical environments (e.g. slopes in a valley as opposed to a ridge or a steep point hill). This type of approach can give you a sense of what the mathematical symbols means. These hill walking mental games are not just metaphors for the mathematical operations we are studying but direct physical examples of the mathematical ideas we are exploring.<br /><br />The imagination is also required when making the next intellectual journey ....that is, seeing the concept of dimension in a more general way. For example, realising that when studying how heat is flowing through the wall of a house, you can visualise the "valleys" and "hill tops" that the temperature profile will take in the three spatial dimensions of the wall, in the same way that you extended your view of the world by moving beyond x and y graphs. If you can visualise "temperature" as an extra "dimension" in the wall than you will start de-mystifying the mathematical operations taught to you (partial derivatives, cross products, etc.). After all, we are studying these operations for largely practical reasons, such as, calculating the temperature profiles of walls, the velocity profile of gas flowing in a duct and a myriad of other engineering problems, so visualising the mathematics in physical terms provides a direct intellectual route to performing the engineering calculations that any decent engineer would like to make. Some determination is required to master the mechanics of these operations - my head still spins a little when taking the partial derivative of a partial derivative - but I would argue that the imagination/visualisation part of this trip is the most difficult and most rewarding aspect of first year mathematics.<br /></div><div align="justify">Welcome to the multi-dimensional universe !</div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com3tag:blogger.com,1999:blog-2635112609407238828.post-44660421372848407452009-06-27T01:49:00.000-07:002009-06-27T03:44:46.316-07:00The Mathematics of Measurement<div align="justify"><br />We are surrounded by measurement devices. The modern world is abound with instruments providing values for temperature, humidity, weight, time, speed, force, pH, radioactivity, power, height, voltage, current and even our attractiveness to the opposite sex ! It is a naive person indeed who accepts a measurement on face value. Accurate and reliable <span id="SPELLING_ERROR_0" class="blsp-spelling-corrected">measurement</span> of any quantity is difficult and errors, whether they be random or <span id="SPELLING_ERROR_1" class="blsp-spelling-corrected">systematic</span>, are normal.</div><div align="justify"></div><div align="justify"><br />For example, if you are told that the temperature of your house is 26.56632 C, a thinking person would ask:</div><div align="justify"><br />How do you know the value to such accuracy ?</div><div align="justify"></div><div align="justify"><br />At what time did you take this value and does it vary with time ?<br /></div><div align="justify"><br />Is the value an average of many values taken from many positions in the house or is taken from a set position in the house ?<br /></div><div align="justify"><br />If taken from one position, is this value representative of the "house" as a whole ?</div><div align="justify"></div><div align="justify"><br />If it is an "average" value, how precisely is this average calculated ?<br /><br />Are there any corrections made for the way the thermometers are distributed in the house ?<br /><br />For example, if there are ten thermometers in the basement and only one in the front lounge, wouldn't a straight averaging of these values give a distorted figure ?<br /><br />How much variation is there in the values "averaged" ?<br /><br />Does this variation in the case of multiple values relate to the position of the measurement or is apparently random ?</div><div align="justify"></div><div align="justify"><br />Have the thermometers been calibrated against a standard ?</div><div align="justify"></div><div align="justify"><br />These questions all converge onto two main points: What does the measurement tell us about the system we are studying and how <span id="SPELLING_ERROR_2" class="blsp-spelling-corrected">accurate</span> is the measurement ?</div><div align="justify"></div><div align="justify"><br />Mathematics is highly useful in evaluating many of these issues. For example, statistics can be used to evaluate <span id="SPELLING_ERROR_3" class="blsp-spelling-corrected">variation</span> in measurements and calculus can be used to "average" values and quantify variation. Above all, mathematics can ease the <span id="SPELLING_ERROR_4" class="blsp-spelling-corrected">hand waving</span> and provide <span id="SPELLING_ERROR_5" class="blsp-spelling-corrected">quantifiable</span> answers to these questions.</div><div align="justify"></div><div align="justify"><br />For example, imagine you are calculating the distance travelled by a trolley moving at constant velocity, using the very simple formulae:</div><div align="justify"></div><div align="justify"><br />distance (m) = velocity (m) x time (m) or D = V x t</div><div align="justify"></div><div align="justify"><br />The velocity has been measured as 22.35 m/s and the time has been measured as 10.00 s. What is the error associated with this <span id="SPELLING_ERROR_6" class="blsp-spelling-corrected">calculation</span> ?</div><div align="justify"></div><div align="justify"><br />Given there crude measurement, we can assume that there the random error of the measurement is one half of the last graduation of the device. Simply put, if you are using a mm graduated ruler, we can assume that the error associated with the rule is +/- 0.5 mm. This may not be correct, for example, if my sight is poor the error may become larger or if the graduations on the ruler have been badly printed, this assumption may also be too low. Another possibility is that I incorrectly placed the ruler and introduced a large systematic error (as opposed to <span id="SPELLING_ERROR_7" class="blsp-spelling-error">the</span> "random" errors I have been discussing) However, without any more information the "half the smallest graduation" principle is a reasonable starting point for our deliberations.<br /></div><div align="justify"><br />Therefore, in this calculation, we can assume the that the velocity is 22.35 +/- 0.005 m/s and the time was 10.00 +/- 0.005 s.<br /></div><div align="justify"></div><div align="justify"><br />Method 1. </div><div align="justify"></div><div align="justify"><br />We know from the fundamental derivation of <span id="SPELLING_ERROR_8" class="blsp-spelling-corrected">calculus</span> that <span id="SPELLING_ERROR_9" class="blsp-spelling-error">dD</span>/<span id="SPELLING_ERROR_10" class="blsp-spelling-error">dt</span> is approximately equal to (small <span id="SPELLING_ERROR_11" class="blsp-spelling-corrected">change</span> in D/small change in t) or more simply put the gradient of the curve at a particular point is approximately equal to the ratio of a small change in the resultant variable to a small change in the independent variable. This principle can be used to approximate the error using the following formulae:</div><div align="justify"></div><div align="justify"><br />error in D = <span id="SPELLING_ERROR_12" class="blsp-spelling-error">dD</span>/<span id="SPELLING_ERROR_13" class="blsp-spelling-error">dt</span> x error in t = V x error in t = 22.35 x 0.005 = 0.11175 m.</div><div align="justify"></div><div align="justify"><br />Therefore, we have the result of 223.5 +/- 0.1 m. This approach ignores any error in the V value, as it treats the problem as being D = f(t). This would be fine if V was <span id="SPELLING_ERROR_14" class="blsp-spelling-corrected">truly</span> constant or the error associated with V was very small compared to that associated with t. This approach is particularly useful when the function is complex (e.g. D = <span id="SPELLING_ERROR_15" class="blsp-spelling-error">Vcos</span> (t^2)) and other methods are difficult to use.</div><div align="justify"></div><div align="justify"><br />Method 2.</div><div align="justify"></div><div align="justify"><br />We estimate the error by calculating the answer using the most pessimistic values and take this answer away from the value calculated without considering the error. In this case:</div><div align="justify"></div><div align="justify"><br />(22.355 x 10.005) - (22.35-10.00) = 223.66175 - 223.5 = 0.16175m</div><div align="justify"></div><div align="justify"><br />Therefore, the answer is 223.5 +/- 0.16 m.</div><div align="justify"></div><div align="justify"><br />Method 3.<br /></div><div align="justify"></div><div align="justify"><br />It can be shown by a simple proof, that when the errors associated with measurements are relatively small, that when two values are multiplied together, the relative error (absolute error/value) of the new value is the sum of the relative errors of the original values. In our example, this results in:<br /><br /></div><div align="justify"></div><div align="justify">error in D = D ((error in V/V)+(error in t/t)) = 223.5 ((0.005/22.35)+(0.005/10)) = 0.16175 m<br /></div><div align="justify"></div><div align="justify"><br />Therefore, the answer is 223.5 +/- 0.16 m.<br /><br /></div><div align="justify"></div><div align="justify">Clearly, the first method underestimated the error and the results from the final two techniques should be used in this case. This simple example illustrates some of the complexity in determining what a measurement really means and how mathematical approaches are useful and dealing with the complex issues associated with measurement.</div><div align="justify"></div><div align="justify"></div><div align="justify"></div><div align="justify"></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com1tag:blogger.com,1999:blog-2635112609407238828.post-16283968903918986552009-06-07T23:21:00.001-07:002009-06-12T11:05:13.660-07:00The Business Mathematics Connection<div align="justify"><br />Niall Ferguson's "The Ascent of Money" is a highly entertaining history of business that seeks to explain how business practice has had a profound effect on human history. The title of documentary series is a deliberate pun on the influential BBC TV series "The Ascent of Man" from the 1970s. This series presented a grand overview of the history of human civilisation, in which commerce was barely mentioned, where as Greek mathematics and Galileo's trail by the church were described in great detail. Apparently, a young Niall felt that something was missing and decided that once he had become a world famous economic historian he would have his revenge ! I, for one, enjoyed the pun !<br /><br /></div><div align="justify"></div><div align="justify">In one episode, Ferguson traced the history of lending, arguing that the fortune generated by the business innovations of the Medici family and other Italian businessman effectively funded the Renaissance. This claim may somewhat under estimate the importance of artistic and scientific ideas but is certainly an effective counterbalance to the traditional dis-taste and dis-interest that many historians have shown towards the influence of commerce on human affairs.<br /><br /></div><div align="justify"></div><div align="justify">Of particular interest to me, was Ferguson's emphasis on the impact of the introduction of "Arabic" numerals to Europe (which we now know came from India) on the ability for traders to effectively barter and exchange currency and goods. As Ferguson explained, Roman numerals, was practically useless for large commercial transactions and that Southern European traders found the counting systems used by their counterparts from the Muslim world to be far more practical. In this way, business lead a revolution in mathematics.<br /><br /></div><div align="justify"></div><div align="justify">This link between business and mathematical innovation is profound. The very business of counting in groups of numbers (binary, decimal or duodecimal) is almost certainly linked to the growth trade in the ancient world. The concept of exponential functions is similarly linked to the development of interest calculations and banking practices in the late middle ages. It is also well established that basic concepts of probability and statistics were developed in a business context, in particular, around the complicated calculations of insurance and risk assessment in the 19th century. This interaction between business and mathematical innovation continued in the 20th century with the development of game theory and other techniques of discrete mathematics.<br /><br /></div><div align="justify"></div><div align="justify">I'm personally not surprised by this profound link. In my own experience in small business, the back and forward of everyday commerce is a fertile ground for innovation and new ideas. The atmosphere is very different from academia, where often new ideas can be squashed by petty snobbery's, ideological positions, intellectual fashions and just plain conservatism. In business, the attitude often is, if it works, than lets use it ! This, of course, means that lots of mediocre ideas also fly but that's part of territory.<br /></div><div align="justify"></div><div align="justify"><br />I look forward to the next episode of Ferguson's "The Ascent of Money" and learning more about the link between "dirty money" and mathematics !</div><div align="justify"></div><div align="justify"></div><div align="justify"></div><div align="justify"></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com0tag:blogger.com,1999:blog-2635112609407238828.post-9867795079723408272009-05-29T22:07:00.000-07:002009-05-29T22:51:55.416-07:00The Ascent of Freeware<div align="justify"><br />In the last month, a new website created by a team lead by Stephen Wolfram (<a href="http://www.alphawolfram.com/">http://www.alphawolfram.com/</a>) has generated considerable interest among mathematicians, scientists, engineers and the wider community. In the popular media, the site is characterised as an attempt to challenge the supremacy of "google" but a visit to Alpha Wolfram will quickly reveal that the site offers a very different service. For example, one can type "Integrate x^2cosx" and get a full analytical answer to the integral (including the steps), an alternate solution, a graphical representation of the integral, a definite integral solution and a series expansion of the solution, within seconds. Impressive indeed ! Type in "Solve x^3 + 2x^2 + x - 6 = 0", and the full solution of the cubic with steps and graphical interpretation appear moments later. Certainly, I have been able to think of analytical problems that the software can't deal with and the on line service is not really appropriate for dealing with large data sets (see <a href="http://www.scilab.org/">http://www.scilab.org/</a> for powerful freeware for manipulating matrixes and high level scientific programming), but this is nit picking - Alpha Wolfram is a triumph.<br /><br /></div><div align="justify">Alpha Wolfram places much of the analytical mathematical power of Mathematica and Maple in the hands of anybody with access to the web. AND IT IS FREE ! It will cause mathematics teachers at all levels to re-think what kind of homework questions are worth asking, in particular, it should push assessment towards "setting up the problem" and "analysing the answers", and away from the application of largely mechanical procedures for solving various standard equations. It maybe to early to say the traditional idea of getting a 1st year Engineering student to go through hundreds of standard integrals is now dead but certainly, this approach is in danger of becoming irrelevant and going the way of "log tables" and using Euclid's "Elements" as a textbook.<br /><br /></div><div align="justify">Viva La Freeware !!</div><div align="justify"></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com0tag:blogger.com,1999:blog-2635112609407238828.post-68284546191906327172009-05-17T04:42:00.000-07:002009-05-29T22:54:17.274-07:00In Praise of Newton-Raphson<div align="justify">The Newton-Raphson technique for finding roots of equation via an iteration process is one of the first numerical techniques taught to students of mathematics. As a technique, it illustrates important features common to many numerical techniques used in mathematics, namely:<br /><br /></div><div align="justify">A) it is based on a very simple mathematical idea, that is, that extrapolating a value from a curve back to the x axis, by assuming a linear relationship, is a good way to form a more accurate guess for the intercept of the curve with the x axis,<br /><br /></div><div align="justify">B) after a few manual calculation using the technique, you are eternally grateful to the inventors of the computer (Hail Babbage, Turing, Zuse and friends !)<br /><br /></div><div align="justify">C) it is very simple to turn the procedure into an automated program,<br /><br /></div><div align="justify">D) the better the initial guess, the quicker you will arive at the solution and save computational time,<br /><br /></div><div align="justify">E) the more accurate the solution you desire, the greater the number of iterations,<br /><br /></div><div align="justify">F) finding a strategy for dealing with rounding errors and storing numbers with the appropriate level of precision between iterations are not trivial problems,<br /><br /></div><div align="justify">G) without care, it is possible to diverge of the wrong solution or (even worse) even to send the computer off to an unending loop of diverging solutions (i.e. "wrong" over and over and over again), and<br /><br /></div><div align="justify"></div><div align="justify">H) it really works - there are few curves that it can't deal with but these are relative oddities compared to the great number of curves that the technique solves readily.<br /><br />As a young engineer, I wrote several programs that used the Newton-Raphson technique to find solutions to the various equations I had formed in my models. Invariably, once I had found a good method for avoiding divergent solutions, the Newton-Raphson routine would find a solution. Like many before me, I found the technique surprisingly powerful , verstaile and useful. Now, students can "play" with the technique using graphical calculators or spreadsheet programs on a lap top. In essense, once you have a "curve", whether it be formed by data or through a known equation, the technique can be used to find solution for particular intercepts (e.g. y = 0) without having an analytical solution - that may not be possible or indeed just beyond your algebraic ability. </div><div align="justify"></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com0tag:blogger.com,1999:blog-2635112609407238828.post-33922259742228255572009-05-02T02:09:00.000-07:002009-05-02T03:40:30.215-07:00The Box Problem<div align="justify">A common problem used to illustrate how differential calculus can be used for optimisation is "the box problem". The box problem goes as follows; imagine you manufacture boxes (W metres wide, D metres deep and H metres high) and you wish to minimise the amount of cardboard used to produce your standard box with volume V (V= W.D.H cubic metres).<br /><br />The first step is to set up an area equation, which is the quantity that we are trying to minimize:<br /><br /></div><div align="justify">A = (area of the two sides defined by the width) + (area of the two sides defined by the depth) + (area of the top and bottom sides)</div><div align="justify">=2W.H + 2D.H + 2W.D<br /><br /></div><div align="justify"></div><div align="justify">Now we have three unknowns and two equations. One option is to form solution based on an assumed ratio (C) of the width to the height, which we can use to simplify our area equation to:<br /><br /></div><div align="justify"></div><div align="justify">A = 2W.H. + 2(V/W) + 2(V/H) by using the the volume equation to substitute for D and using C= W/H, we get :<br /><br />A= 2C.H^2 + 2 (V/C.H) + 2(V/H) = 2C.H^2 + (2/H)((V/C) + V)<br /><br /></div><div align="justify"></div><div align="justify">If we graph this function (A vrs H) and forget negative values of both A and H, we can see a clear vertical asymptote along the A = 0 and a minimum near the origin that is a function of our choices for V and C. Of course, this equation is ripe for differentiation:<br /><br /></div><div align="justify"></div><div align="justify">dA/dH = 4C.H - (2/H^2)((V/C) + V)<br /><br /></div><div align="justify"></div><div align="justify">At the minimum, it must follow:<br /><br /></div><div align="justify"></div><div align="justify">dA/dH = 0 = 4.C.H - (2/H^2)((V/C) + V)<br /><br /></div><div align="justify"></div><div align="justify">therefore,<br /><br /></div><div align="justify"></div><div align="justify">H = ((V + VC)/(2 C^2))^1/3<br /><br /></div><div align="justify"></div><div align="justify">Now, we have a ready way of optimising the quantity of cardboard for any given volume and ratio of height for depth. What solutions do we get if we assume a certain ratio to the width to the breadth ? Which is the true minimum (i.e. independent of our assumptions of ratios of dimensions) ? Excellent questions ! Start analysing and optimising ..... welcome to Applied Mathematics !</div><div align="justify"></div><div align="justify"><br /></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com0tag:blogger.com,1999:blog-2635112609407238828.post-72410008300399102232009-04-24T23:30:00.000-07:002009-04-25T01:52:04.664-07:00Practical Implications of Calculus<div align="justify">Calculus is widely used by engineers and scientists to analyse practical problems. One common approach is to analyse a particular system using fundamental physics for a particular geometry (e.g. a force balance around a <span class="blsp-spelling-error" id="SPELLING_ERROR_0">spherical</span> particle falling in a liquid) to form equations. These equations are than either integrated or differentiated to produce useful relationships for a given set of boundary conditions (e.g. settling time of a particle as a function of size and density for a given initial particle velocity). The success of this approach normally depends on the nature of the phenomena being studied (some very chaotic and/or highly non-linear phenomena are difficult to model), the assumptions made in setting up the problems and the difficulty in solving the equations formed. Often, numerical techniques are used to find solutions to these equations and any good engineering mathematics course teaches a range of relevant numerical techniques to differentiate and/or integrate equations that are either difficult or impossible to solve directly.<br /><br />Another interesting application of calculus is to analyse data. Consider a set of data collected in an experiment ..... imagine we are measuring X and Y simultaneously. When we plot X against Y, the curve generated may clearly show a relationship exists but the relationship is not simple or immediately apparent. A very simple method to start analysing this mysterious relationship, is to differentiate the X Y plot numerically (i.e. calculate the slope at points along the curve) and form a new plot of <span class="blsp-spelling-error" id="SPELLING_ERROR_1">dX</span>/<span class="blsp-spelling-error" id="SPELLING_ERROR_2">dY</span> <span class="blsp-spelling-error" id="SPELLING_ERROR_3">vrs</span> X. Now remember that we differentiate particular functions, new very specific relationships are formed. For example, differentiating a trigonometric function will generate another trigonometric function, and in the case of simple trigonometric functions like sine and <span class="blsp-spelling-error" id="SPELLING_ERROR_4">cosine</span>, functions are formed that have very specific geometric relationships to the original functions (e.g. cosine has the same shape and periodic form of sine but is "out of phase" with that relationship). In the case of <span class="blsp-spelling-error" id="SPELLING_ERROR_5">polynomials</span>, differentiating produces a function of lower order; the slope of a cubic follows a parabolic relationship, the differential of a parabolic functions produces a linear function and so on. This means that by differentiating a curve (i.e. measuring the slope of the curve at each point) some of these underlying relationships in the data maybe revealed.<br /><br />This approach can be extended to differentiating the <span class="blsp-spelling-error" id="SPELLING_ERROR_6">dX</span>/<span class="blsp-spelling-error" id="SPELLING_ERROR_7">dY</span> curve formed, as double differentiation also can unlock some underlying relationship For example, differentiating <span class="blsp-spelling-error" id="SPELLING_ERROR_8">sinX</span> will form <span class="blsp-spelling-error" id="SPELLING_ERROR_9">cosX</span> and differentiating that relationship will produce a <span class="blsp-spelling-error" id="SPELLING_ERROR_10">negative</span> version of the original relationship. Double differentiation of a cubic function will generate a linear function (try it !). Thus, the "slope of the slope" can potentially tell <span class="blsp-spelling-error" id="SPELLING_ERROR_11">alot</span> about the original relationship. This line of attack can be extended to integration, through measuring the area under the Y curve and plotting this relationship against X. Of course, both taking the slope and measuring the area can be used in combination to tackle the problem.<br /><br />The beauty of this methodology is that the procedure is very simple (e.g. measuring a slope of a line) and easily automated. You can try it yourself .... I suggest asking a mathematically inclined friend to dream up a complex function that is the combination of well known simple functions (e.g. <span class="blsp-spelling-error" id="SPELLING_ERROR_12">cosx</span> + x^3 + exp(x)), get him or her to form an x y table of values from this relationship and than ask you to derive the underlying relationship from this data set. The detective job in front of you is made simple by modern graphical/<span class="blsp-spelling-error" id="SPELLING_ERROR_13">CAS</span> calculators that allow ready numerical differentiation and integration of curves. Sometimes, a combination of intuition, luck and insight is required to identify the underlying relationship but the journey is normally fun. Try it !!!</div><div align="justify"></div><div align="justify"></div><div align="justify"></div><div align="justify"></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com0tag:blogger.com,1999:blog-2635112609407238828.post-9678840906335928822009-04-23T11:57:00.000-07:002009-04-23T12:58:44.837-07:00Thinking about the foundations of calculus<div align="justify">Just recently, I went through the standard derivation of the fundamental theorem of calculus with my students ..... forming tangent lines to a curve, calculating the gradient of that line using an increment, taking the increment towards infinity than repeating similar arguments for the area under a curve before forming the wonderful conclusion that the mathematics of calculating an area under a curve is the reverse of the process for calculating the gradient of a curve. In short, if you understand the mathematics of change, you also understand the mathematics of accumulation and vice versa. This was the brilliant insight that both Newton and Leibniz claimed as their own in the 17th century and formed the basis of the field we know as "Calculus".<br /><br />This derivation is rightly considered one of the great mathematical breakthroughs of all time and its conclusions are indeed far reaching. During the lecture, I presented the orthodox view that Newton and Leibniz are the great intellectual heros of this breakthrough with a nod of appreciation to ancient Greeks like Archimedes who developed integral calculus via the method of exhaustion. As I was going through these arguments, I found myself questioning this idea of Newtons and Liebniz's pivotal role in the development of calculus. Wasn't the real breakthrough the idea that if you take an increment and imagine it decreasing towards infinity, you can drive useful geometrical relationships ? Isn't that idea, which I think we can accredit to Archimedes, the real intellectual breakthrough ? If you know that idea and have the tools of Cartesian co-ordinates (thank you Descartes !), than won't the relationships that Newton and Leibniz formed eventually fall out ?</div><div align="justify"><br />Even as I write these heretical ideas down I feel my inner critic saying "No, these ideas only seem obvious because of the brilliant insights of Newton and Leibniz !" That may be true but historians of mathematics writing on calculus have shown that calculus quickly formed as a field after the developments in algebra instigated by Descartes and other mathematicis just proceeding Newton and Descartes. It is also acknowledged that Barrow (Newton's teacher at Cambridge) had an early form of differential calculus before Newton (see <a href="http://www.maths.uwa.edu.au/~schultz/3M3/L18Barrow.html">http://www.maths.uwa.edu.au/~schultz/3M3/L18Barrow.html</a> for an excellent overview of his ideas). After consulting my inner critic, I think the view I am forming can be expressed as follows: understanding the importance of taking increments towards zero was a great intellectual breakthrough that allowed the development of calculus, simplifying algebra through the Cartersian co-ordinates provided wonderful tools by which to understand the mathematics of change and accumulation and the derivation of calculus by Newton and Leibniz represent the accumulation of this intellectual development. In short, their intellectual insights owe a great deal to Archimedes, Descartes and Barrow.<br /><br /></div><div align="justify"></div><div align="justify">One of the interesting observation one can make from these discussions is that the way calculus is taught follows a very different route from its historical development. At high schools, we indoctrinate students in algebra, than introduce differential calculus and limits, and than form integral calculus. In history, calculus was formed in almost the opposite order. I suppose, as long as you understand the key intellectual points underpinning calculus, it doesn't really matter in which order you have learn't them. </div><div align="justify"></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com0tag:blogger.com,1999:blog-2635112609407238828.post-28984383538564051132009-04-09T23:16:00.000-07:002009-05-29T22:55:59.690-07:00A very brief history of calculators or how my brother amazed my school<div align="justify"><br />I started high school in 1973, three years after the end of the Beatles and a generation before the end of the cold war. Everybody wore their hair long, ludriciously wide ties were considered fashionable, most engineers (like my father) owned a slide rule and very simple electronic calculators were starting to become affordable. I remember my brother saving up several weeks of his paper round money to purchase a calculator with a square root button. The arrival of this calculator at our high school caused a sensation and my brother was asked to demonstrate this technological marvel to the headmaster. With the arrival of even more powerful devices throughout that decade, my brother and myself, and everybody else studying mathematics in the Western world, continued to be trained in the use of log tables for carrying out any calculation beyond 687 x 6578. I think the last time I used a log table Ronald Regan hadn't yet become president and computer programs were typed on cards and processed overnight.<br /><br />During this time, serious letters to the papers and educational experts lamented the fall in educational standards, my year 10 geography teacher warned that global warming would see Sydney under a foot of water by 2000 and there was a general feeling with anyone over the age of 40 that using calculators was "cheating".<br /><br /></div><div align="justify"></div><div align="justify">By the end of the 1970s and into the early 80s, calculators had advanced quickly and a range of programmable calculators were on offer. In this enlightened era, engineering students tended to be either "HP" or "Casio" adherents, though a few perverse souls identified with the reverse polish notation of the "TI" calculators. I remember quite distinctly slaving away on my Casio programmable calculator with its gigantic 2k of memory, writing quite intricate programs with the line numbering system of level 2 basic, a cute plug in ticker tape printer and an audio tape memory system. Armed with this calculating power, you felt that you could conquer the world or at least complete a pressure drop calculation for a piping system in under 10 minutes. Part of me (a very small part) still hankers for the happy chatter of my ticker tape Casio printer and the amazingly clunky graphics produced from this device. By this time, the scientific calculators familiar with modern students became standard and knowledge of the workings of a slide rule suggested either a perverted soul or a person lost in the past.<br /><br /></div><div align="justify"></div><div align="justify">The calculator was here to stay ! My arrival in the Engineering profession coincided with the great personal computer revolution and in my own small way I lead the charge, using computer programs (now written in "high" level languages like GW Basic !!) to perform complex engineering calculations that had formerly been the province of "look up" tables and approximate solutions. Even with this shift towards computing, my scientific calculator (still a Casio man) was used on a daily basis. However, by this time my career had taken a sharp turn towards research and the graphics calculator revolution bypassed me, as I was knee deep in numerics, computational thermodynamics and writing unruly "programs" in Excel. It was only when I took my current position that I was handed my first graphics calculator. It was love at first sight ! I love the fact that I can "see" the solution of an equation, that I can calculate derivatives and integrals and even form the ABC TV symbol using parametric graphics. What is there not to love ! I even accepted the transition from being a Casio man to a TI man without suffering a nervous breakdown (OK I had a little therapy).<br /><br /></div><div align="justify"></div><div align="justify">Interestingly, serious people are still lamenting the falling of educational standards, predicting that Sydney will be under a metre of water by ......, and most people over 40 think that using a CAS calculator is cheating.</div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com2tag:blogger.com,1999:blog-2635112609407238828.post-877108183743714772009-04-03T00:21:00.000-07:002012-08-11T21:28:36.551-07:00The Continuum Assumption<div align="justify">The engineering mathematics course at Swinburne is typical of most engineering mathematics courses around the world, in that, there is a heavy emphasis on the use of functions in analysing the physical world. In particular, there is underlying assumption (often unstated) that we can deal with physical data as a continuum (e.g. analysing radioactivity measurements using exponential functions). It is this assumption that underpins the "classical" paradign of engineering mathematics, which I would describe as:</div><div align="justify"><br /></div><div align="justify">A. analyse the physical relationships of the system being studied (e.g. force balance of a particle),</div><div align="justify"><br /></div><div align="justify">B. form equations that reflect these relationships, making appropriate simplifciations and assumptions (e.g. particle is spherical),</div><div align="justify"><br /></div><div align="justify">C. solve these equations for a given set of boundary conditions or limitations using either analytical or numerical techniques, and</div><div align="justify"><br /></div><div align="justify">D. analyse the solutions obtained against physical data, returning to first two steps if the solutions obtained are inaccurate or not credible.<br /><br /></div><div align="justify"></div><div align="justify">This approach, and many subtle variations, has proved to be very powerful in analysing engineering problems, though complex systems where subtle changes in geometry and boundary conditions can produce large variations in behaviour (e.g. turbulence in fluids, movement of fine particles and "chaotic systems" in general) have proved difficult to model using this approach. Stephen Wolfram, in his book "A New Kind of Science" (2002) (see <a href="http://www.wolframscience.com/">http://www.wolframscience.com/</a>) argued that the classical approach was fundamentally flawed and need replacing with a new approach called "cellular automata". At the heart of Wolfram's claims was this central observation:<br /><br /></div><div align="justify">All of our measurements of the world are made discretely, that is, we obtain discrete numbers from our instruments (e.g. the temperature measurement from a thermometer) including our senses, and artificially impose continuous relationships upon the world by forming equations around fundamentally discrete phenomena. We could more easily, and naturally, use discrete mathematical models to describe the physical world and dispense with the classical approach.<br /><br />Quite a claim !! As you imagine this book caused much debate, some of it polite and in some cases, quite inpolite ! You might find the overheads of a lecture I gave on the book interesting - see <a href="http://www.swin.edu.au/feis/mathematics/staff/gbrooks_pres.html">http://www.swin.edu.au/feis/mathematics/staff/gbrooks_pres.html</a> - and there are literally hundreds of sites on the web discussing this book. You may also interested to read a much earlier (and more modest) version of the same idea by Konrad Zuse (1910-1995) who published "Computing Space" in 1967. An English translation of this pioneering work on "digital physics" is available at <a href="http://www.idsia.ch/~juergen/digitalphysics.html">http://www.idsia.ch/~juergen/digitalphysics.html</a>. Zuse was also an early pioneer in the development of the computer and was, clearly, a highly imaginative and interesting thinker.<br /><br /></div><div align="justify"></div><div align="justify"></div><div align="justify">I think the claims, details and repercussions of Wolfram's claim are a bit detailed to discuss here but I do think the first part of his central claim is uncontroversial, that it, the measurements we make of the world are discrete and the equations we impose on this discrete data reflect our intellectual choices not an underlying physical connection between equations and nature (i.e. cannon balls do not have a parabolic equation written into their structure, it is "us" that chooses a parabola to model the motion of the ball). I think this is underlying assumption to appreciate as we continue along our path of differentiating/integrating/ etc. continuous functions to describe the physical world.</div><div align="justify"></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com0tag:blogger.com,1999:blog-2635112609407238828.post-46813295282926014202009-03-27T21:05:00.000-07:002009-03-28T06:49:10.034-07:00My Favourite Function<div align="justify">People have their favourite colours, football teams (go dogs !) and beaches. Why not your favourite function ?</div><br /><div align="justify">For me there are many attractive candidates for "my favourite function". For example, I enjoy the simplicity of mx + c, the up and down of x<sup>2</sup>, the surprising plateauing of x<sup>3</sup>, the lovely endless symmetry of the sinx and cosx and even the quirkiness of complex polynominals (x<sup>4</sup> + x<sup>3</sup> + x<sup>2</sup> + x). One of my associates is very fond of the hyperbolic functions but personally find their curviness rather artificial (they are just a compound of two other functions). I must admitt that I find the limited domain of most inverse functions a little off putting. Why choose a function with a limited range when you can have the whole number line !<br /><br /></div><div align="justify"></div><div align="justify">I think looking for a favourite in any area involves the formation of various vanities and snobberies, which is what makes competitions like "the top ten albums of all time" alot of fun. It is an opportunity to laugh at your own prejudices whilst studying the quirky choices of others.<br /></div><div align="justify"></div><div align="justify">So what is my favourite ?<br /><br />e<sup>x</sup> is definitely my favourite function. Why ?<br /><br /></div><div align="justify"></div><div align="justify">Certainly, the exponential function forms a pleasing curve but it is more its amazing characteristics that draws me to e<sup>x</sup>. I love the fact the function is based on an irrational number but calculates something commonly observed in nature (e.g. radioactive decay, rates of chemical reactions, etc.). I find the idea that the slope of any point of the line is the value at that point (dy/dx = e<sup>x</sup>) amazing and totally fascinating. For me, e<sup>x</sup> is number one ! (which is only true when x = 0)<br /><br /></div><div align="justify"></div><div align="justify">What is your favourite function ?</div><div align="justify"></div><div align="justify"></div><div align="justify"></div><div align="justify"></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com1tag:blogger.com,1999:blog-2635112609407238828.post-67999847596578482042009-03-19T02:56:00.000-07:002009-03-20T02:42:16.867-07:00The End of Elegance<div align="justify"><br />I think there are three breakthroughs in mathematics that have really shook the foundations of the field, the first is the discovery of irrational numbers (formerly accredited to Hippaus, a member of Pythagoras's school around 500 BC but Indian mathematicans are now thought to have been earlier), the second, relates to the work of Cantor in the 19th Century in showing that infinity comes in different sizes, and, thirdly, Godel's incompleteness theorem in the first part of the 20th century, which demonstrated that attempts to form systems of axioms that are entirely logically consistent are doomed. These amazing feats of insight, intellectual rigour and imagination, initially triggered rejection and a strong counter reaction from their peers. In the case of Hippaus, legend has it, that this discovery cost him his life, as Pythagoras's followers incensed with his proof that the square root of 2 is irrational threw him into the sea ! After time, these ideas were accepted, incoporated into the mathematical mainstream and built on by thinkers who followed in the wake of these tidal waves.<br /><br />Lets address the first intellectual tsunami; the discovery of irrational numbers. Why was this so important ? This discovery was important because it challenged a central notion of the type of mathematics that Pythagoras and his followers were seeking to establish. Pythagoras viewed mathematics as sacred and capable of explaining the deepest ideas and describing the natural world around them. For the school of Pythagoras, shapes and numbers were elegant expressions of profound ideas. In this intellectual climate, they assumed that numbers could always be expressed in terms of simple ratios of integers (e.g. 1/7), which they understood in geometric terms - a feature of Greek mathematics that makes it hard for modern reader to appreciate their arguments directly.</div><div align="justify"></div><div align="justify"><br />What did Happaus show ? We don't have access to the original proof but we can assume that his proof followed this type of argument:<br /><br /></div><div align="justify"></div><div align="justify">If the SQRT (2) is rational, than is follows:<br /><br />SQRT (2) = a/b where a and b are integers.</div><div align="justify"></div><div align="justify"><br />It also follows:<br /><br /></div><div align="justify"></div><div align="justify">2 = a<sup>2</sup>/b<sup>2</sup>, which can easily be turned around to 2b<sup>2 </sup>= a<sup>2</sup></div><div align="justify"></div><div align="justify"><br />We know that 2 times any number will result in an even number and that square root of any even number results in an even number, therefore, "a" must be an even number. If "a" is an even number than we can express it as 2r, where r is another integer, and we can re-arrange the equation above to:<br /><br />2b<sup>2</sup> = 4r<sup>2</sup>, which can be simplified to b<sup>2</sup> = 2r<sup>2</sup>.</div><div align="justify"><br />Using exactly the same argument as the one above, we can say that "b" must also be even. Now, we have a contradiction, because any ratio of two even integers can be reduced to a ratio involving an even and a odd number (e.g. 2/8 = 1/4). Therefore, it is not possible for the square root of 2 to be expressed as a ratio of two integers. In fact, this intriguing qauntity can not be directly calculated but only approximated.<br /><br /></div><div align="justify"></div><div align="justify">This is still a somewhat shocking result. A physical representation of the square root of 2 can be easily visualised by constructing a right angle triangle with two sides the length of 1 m. The hypotenuse of the triangle must be the square root of 2 (using Pythagoras's famous theorem) .... we can see it, we can easily esimate the length using a ruler, how can it be that we can not calculate it ? This is exactly the intellectual dilemna that haunted Pythagoras, disturbed many mathematicians since the Greeks (notably Newton) and still causes one to shake your head and muse that God must be playing some elaborate joke on us. The later discovery of the irrational nature of pi and <em>e</em>, and Cantor's discovery that that there are many more irrational numbers than rational on the number line, just serves to deepen the shock. The type of elegance visualised by the early Greeks was over. No wonder they metaphorically shot the messenger by throwing him into the sea.<br /></div><div align="justify"></div><div align="justify"></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com0tag:blogger.com,1999:blog-2635112609407238828.post-83174177813376514252009-03-19T02:29:00.000-07:002009-03-19T02:53:22.326-07:00Sum of Geometric Series<div align="justify"><br />In our proof that repeating numbers are rational, we used the following relationship:<br /><br />S = Sum of the geometric series ar<sup>n-1</sup><br /> = a + ar<sup>1</sup> + ar<sup>2</sup> + ar<sup>3</sup> + ar<sup>4</sup>...... <br /> = a/(1-r)<br /><br />Where does this rather elegant and surprising relationship come from ? Certainly, this simple realtionship is rather unexpected .... why would an infinite series converge on this simple ratio ?<br /><br />Like many relationships in mathematics, the proof is beautifully simple. Firstly, form the equation S - Sr = a + ar<sup>1</sup> + ar<sup>2</sup> + ar<sup>3</sup> ..... - ar<sup>1</sup> + ar<sup>2</sup> + ar<sup>3</sup> .... = a<br /><br />Therefore, rearranging we arrive at S = a/(1-r).</div><div align="justify"></div><div align="justify"><br /><br />QED<br /><br /></div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com1tag:blogger.com,1999:blog-2635112609407238828.post-4648559749619444422009-03-16T20:18:00.001-07:002009-03-19T02:55:31.761-07:00Are repeating numbers irrational ?<div align="justify"><br />The question of the nature of repeating numbers comes up when we convert fractions into binary, as even apparently simple fractions in decimal becomes an infinitely long string in binary, for example:<br /></div><div align="justify"></div><br /><div align="justify">0.1<sub>10</sub> = 0.0001100110011 ....<sub>2</sub> = 0.00011<sub>2</sub>.<br /></div><div align="justify"></div><br /><div align="justify">On first appearances, we seem to have "changed" the type of number we are representing, just through the change of base. Have we in effect converted a rational number into an irrational number ?<br /><br />No, we haven't ! This new representation of the number is still rational. The proof is as follows:</div><div align="justify"></div><div align="justify"></div><div align="justify"><br />A rational number is defined as a number that can expressed as the quotient of two integers (e.g. 0.1 = 1/10).</div><div align="justify"></div><div align="justify"><br />We can express an repeating number as a geometric series:<br /></div><div align="justify">e.g. 0.997997997997 ..... = 0.997 + 0.997 (1/1000)1 + 0.110(1/1000)2 + ...... etc.</div><div align="justify"></div><div align="justify"><br />where a = 0.997 and r = (1/1000)</div><div align="justify"></div><div align="justify"><br />It is well know that the sum of geometric series of this type = a/(1 -r), which will result in a ratio of integers (in this example 997/999).</div><div align="justify"></div><div align="justify"></div><br /><div align="justify">This, because an infinitely repeating numbers sequence can be represented as a geometric series and the sum of a geometric series can expressed as a ratio of integers, such numbers must be rational.<br /><br /></div><div align="justify">QED (Quite easily done for non Latin speakers)</div><div align="justify"></div><br /><div align="justify">Note: Thank you to associate Sergey Suslov for his thoughts on this topic.</div>Geoffrey Brookshttp://www.blogger.com/profile/04656568550653375222noreply@blogger.com0