Monday, June 6, 2011

Motivation to Use Graphs in Math, Physics and to Know Arbitrary Surface Area Calculations

Now, you may ask, are there any other reasons to learn calculus and integration, other than calculating the surface area of arbitrary surfaces of physical objects (see my previous post below). Yes, there are. The point is that mathematics uses graph as a tool a lot. Graphical representation of functions is the most common use. Also, mathematicians, physicists, economists like to represent a number, a count, as a length of a line in a coordinate system, where length matches the count in question. Then, since we use line to represent counts or numbers obtained, the product of two numbers will automatically be shown as surface, bounded by these lines. Then, volumes can come into picture too. It is just pure convenience, call it visual convenience, if you wish, that we have found lines and surfaces to represent our quantities. We are still interested in numbers which these graphical elements represent.

As you know you can count many, many things. Moreover, what you have also seen, you can play with numbers without knowing where they come from. For you know that, for instance, 2 + 3 is equal 5 no matter whether you counted apples, CDs, meters, kilograms, pears, cars, pencils, rockets, measure length etc. You can work with numbers, 2, 3, and 5 now independently of any objects from real world. And it is helpful to know in advance that 2 + 3 = 5, so you can use in any possible counting situation.

Now, let’s say you want to count how many cars are driving by each minute on the street. These can be the result of your measurements: 1st min 15 cars, 2nd minute 23 cars, 3 rd minute 10 cars, 4th minute 12 cars, 5th minute 22 cars. Of course you can draw a table with the columns showing minutes and associated count of cars.

Time (mins)

That’s an excellent way to do it. Now, let’s say you want to represent count of cars not only by numbers written on paper, or by table,  but with marbles. This means that for 10 cars you will have 10 marbles. For 22 cars you will have 22 marbles. Maybe you did not have a paper at hand, so you needed to use marbles. Then you can use the marbles to count minutes. One marble for one minute, two marbles for two minutes etc. You realize that you can use marbles to represent the count of anything, apples, pears, oranges, rockets, cars, cars, etc. But, marbles doesn’t seem to be practical too much. You have to carry a lot of them, and, although they carry exact information about the counts of things they represent, they are too heavy to carry around. What else can be used to represent conveniently numbers of anything? It was discovered that length can be used to represent counts of other things. You draw the line of certain length and say that it represents a count of some objects. In our example we will draw a line of 10 mm to represent 10 cars, then line of 23 mm to represent 23 cars. The similar things can be done for time measurements. We will draw a line of say 1 cm for one minute then 2 cm for 2 minutes etc. We are using length of line to show the numbers, counts we have obtained. Please note, selecting line as a tool is our choice, because of convenience since its length represents different counts, numbers, quantities. Why it is convenient? Look at the graph we can make with lines. You can immediately see when a quantity is bigger than the other, you can see the trend between quantities and their other relationship, like minimum, maximum, etc. Of course all this things can be seen from a table too, but you have to go row by row, and do a lot of comparison. Table is one of the ways to represent pairs of numbers as we did here. Moreover, we can use either table or graph! But note how graph is more convenient, because it visually shows relationship that can be harder to spot in the table. Let’s repeat again, we used length of a line  (as we first did with marbles, or as you used fingers to count objects) as a universal way to represent counts of all other objects we are interested in any particular measurements or calculation. You also can line up numbers along the line, matching the length of line with a corresponding number. Note that you use line, length of line for pure convenience! There is no law that tells you need to use line, length of line to represent numbers and counts. It just happen to be a convenient visual way to use line to represent numbers.
There is nothing wrong using marbles or fingers, it's just seem to be impractical to communicate some ideas, however, these two methods of representing numbers are completely correct too. For instance while you think you can represent only integers with you fingers, you can actually represent any number! How? Say, you want to show number 3/5 with your fingers. You will show first 5 fingers and say this is how many times I divide number one. Then you show 3 fingers and say this is how many times I will multiply the quantity I've just obtained by dividing number one into 5 parts. The result of these two simple operations gives exactly 3/5 and it's communicated with fingers only!

Graph of a function serves the purpose to represent and show all pairs of numbers. Line on a graph should be seen as a continuous set of PAIRED NUMBERS.

Here are some examples of lines in action! Of course these are called graphs in mathematics, physics, engineering, economics, or in any other discipline where we have to show some quantitative relationship.

Fig 1. Examples of graphs

We are, perhaps, used to look at the graphs and focus on their shape, or on, even, their aesthetics values or characteristics. But, aesthetic value is of far less importance, as graphs are concerned. No matter how attractive graph looks, the line of it represents the PAIRED NUMBERS which are the lengths from each coordinate. That's the main purpose of graph. The shape of graph tells us how these pairs of numbers differ from one to another by their magnitudes.

So, using graph is about calculating surface area that can represent not only physical objects, like Pirate Island, airplane vertical stabilizer, or ship’s sail, but any surface that can come from graphical representation of various amounts, measured, counted, obtained by any way, and their quantitative relationships. We came up to the point that surface will represent something because we agreed, at the first place, that LENGTH represents something, that length of line we draw in graph corresponds to a number, count, obtained by measurement (or by any other mean).

Perhaps, it is a good place to describe steps that we use in these scenarios and how we can use thus acquired knowledge.

  1. First we measure quantities of interest and note their relationships.
  2. We represent the counts of these different quantities by lengths of line. We draw a graph.
  3. Since we started dealing with lines, automatically then the surface bounded by these lines may represent a quantity of interest. Example is speed, v, and time, t. If we plot graph v(t), then automatically surface area under the graph will represent path taken.
  4. We use our “method of rectangles” to calculate surface area, if required!

Note how we first focus, without considering the usage of graph, on relationships and measurements, mathematical treatment, of different objects. Then, we can use numbers by themselves, or use  table, or graph, for our further analysis. In graph we represent all those counts and numbers by lines of certain length, to show us, visually, quantitative relationships between measured objects.

Not also that lines, curves generated can be analyzed in their separate worlds of geometry or analytical geometry for example.

Note that coded color scheme can be introduced as the third dimension on the two dimensional graph. Volume integration is possible even here, but first you have to have a map, an association between a color and the number it represents.

[ graphs, math, math concepts, math education, math graphs, math, mathematics, physics concepts, tutoring calculus, schoo,l education, mathematics, physics, real world math, real world rational numbers ]

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