A Few Hints How to Introduce Mathematical Concepts

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Teacher should postpone introduction of strange, exotic mathematical names and labels to, otherwise, most likely, easy to explain and easy to understand concepts in mathematics. Teacher should  first explain mathematical concepts in terms of required sequence of mathematical operations on numbers and on sets involved and then introduce labels or historically accepted names for them. Most of us will agree that many of those names are there for historical reasons, and often they are confusing, misleading, and even intimidating (if Borel, Hausdorff needed so much work to prove that theorem or formulate it, what chances do I have?). To me, it is sometimes better to refer to a theorem by a number first, like Theorem 1, Theorem 2, … and only later assign historical names or other labels to reference them.

Math lectures sometimes look to students like putting misleading, cluttered, over-detailed traffic signs on an easy to use roads and highways.

But, what are we really doing or what we want to do when we say "introducing math concepts through A or B or C real world examples"? Do math concepts need to be introduced through real world examples at all? No, they don't. They can be derived or formulated directly from axioms.  So, what would be the goal of "introducing math" through real world examples? To show that math concepts can be motivated by real life examples, but, at the same time the same math concepts can be derived inside math only, without referencing any real world domain. To me, the mandatory step of introducing some math concepts from real world example is to mandatory show that the same concept can be defined or derived from ZFC axioms. This would clearly show the border and connections, if you wish, at the same time, between applied math and pure math developed from ZFC axioms.

[introducing math, math ], math concepts, math education, mathematics, theorems ]

Mathematical Proof for Enthusiasts - What It Is And What It is Not

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With the term mathematical proof we want to indicate a logical proof, i.e. proof using logical inferences, in the field of mathematics. So, it should really be “a proof in the field of mathematics”. Also, we have to assume, and be fully aware, that proof must be “logical” anyway. There are really no illogical proofs. Proof that appears to be obtained (whatever that means) by any other way, other than using rules of logic, is not a proof at all. Assumptions and axioms need no proof. They are starting points and their truth values are assumed right at the start. You have to start from somewhere. If they are wrong assumptions, axioms, the results will show to be wrong. So, you will have to go back and fix your fundamental axioms.
When you have first encountered a need or a task for a mathematical proof, you may have asked yourself "Why do I need to prove that, it's so obvious!?".

We used to think that we need to prove something if it is not clear enough or when there are opposite views on the subject we are debating. Sometimes, things are not so obvious, and again, we need to prove it to some party.

In order to prove something we have to have an agreement which things we consider to be true at the first place, i.e. what are our initial, starting assumptions. That’s where the “debate” most likely will kick in. In most cases, debate is related to an effort to establish some axioms, i.e. initial truths, and only after that some new logical conclusions, or proofs will and can be done.

The major component of a mathematical proof is the domain of mathematical analysis. This domain has to be well established field of mathematics, and mathematics only. The proof is still a demonstration that something is true, but it has to be true within the system of assumptions established in mathematics. The true statement, the proof, has to (logically) follow from already established truths. In other words, when using the phrase "Prove something in math..." it means "Show that it follows from the set of axioms and other theorems (already proved!) in the domain of math..". Which axioms and theorems you will start the proof with is a matter of art, intuition, trial and error, or even true genius. You can not use apples, meters, pears, feelings, emotions, experimental setup, physical measurements, to say that something is true in math, to prove a mathematical theorem, no matter how important or central role those real world objects pr processes had in motivating the development of that part of mathematics. In other words, you can not use real world examples, concepts, things, objects, real world scenarios that, possibly, motivated theorems’ development, in mathematical proofs. Of course, you can use them as some sort of intuitive guidelines which axioms, or theorems, you will use to start the construction of a proof. You can use your intuition, feeling, experience, even emotions, to select starting points of a proof, to chose initial axioms or theorems in the proof steps, which, when combined later, will make a proof. But, you can not say that, intuitively, you know the theorem is true, and use that statement about your intuition, as an argument in a proof. You have to use mathematical axioms, already proved mathematical theorems (and of course logic) to prove the new theorems.

The initial, starting assumptions in mathematics are called fundamental axioms. Then, theorems are proved using these axioms. More theorems are proved by using the axioms and already proven theorems. Usually, it is emphasized that you use logical thinking, logic, to prove theorems. But, that's not sufficient. You have to use logic to prove anything, but what is important in math is that you use logic on mathematical axioms, and not on some assumptions and facts outside mathematics. The focus of your logical steps and logic constructs in mathematical proofs is constrained (but not in any negative way) to mathematical (and not to the other fields’) axioms and theorems.

Feeling that something is "obvious" in mathematics can still be a useful feeling. It can guide you towards new theorems. But, those new theorems still have to be proved using mathematical concepts only, and that has to be done by avoiding the words "obvious" and "intuition"! Stating that something is obvious in a theorem is not a proof.However, using own intuition to construct a proof or to formulate theorem is definitely useful.

Again, proving means to show that the statement is true by demonstrating it follows, by logical rules, from established truths in mathematics, as oppose to established truths and facts in other domains to which mathematics may be applied to.

As another example, we may say, in mathematical analysis, that something is "visually" obvious. Here "visual" is not part of mathematics, and can not be used as a part of the proof, but it can play important role in guiding us what may be true, and how to construct the proof.

Each and every proof in math is a new, uncharted territory. If you like to be artistic, original, to explore unknown, to be creative, then try to construct math proofs.

No one can teach you, i.e. there is no ready to use formula to follow, how to do proofs in mathematics. Math proof is the place where you can show your true, original thoughts.

[ set, set theory, concept of a set, sets in mathematics, real world, applied math, applied mathematics, axioms, math education, math proof, mathematical axioms, mathematical proof, mathematical theorems, mathematics, theorems, tutoring ]

How to Teach Your Kids and Yourself to Think More Freely About Math and Real World Math Applications

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When I was a kid, I remember, I was often asked, during a math chat, “How many apples are there?” or “If you subtract 3 apples from 10 apples how many apples are left?”. Or, "Peter has 5 apples. How many apples he will have if he gives Susan 3 apples"? Then, there were many more examples of counting objects.

I am sure that many of us have had this kind of first encounters with mathematics.

When I was 4th grade, an interesting idea came to my mind. Why do we need to know concept of an apple in order to subtract 3 from 10? How and when we become aware that an object is an apple, and not, say pear? And what if Peter doesn't own an apple? Isn't the ownership a legal concept? Does Peter really own an apple? Then, if Susan is really hungry and Stephen is not, then giving Susan 3 apples is way more important than actually counting apples. Also, turning left and drive 2km is not the same as turning right and drive 2km. Yet, left and right were not part of math. They are not quantities, yet they are so important. Those were my thoughts when I was in 4th grade..

After a while, and after many examples involving apples, cherries, cars, marbles I have been shown, it became clear to me that 3 and 10 are separate, abstracted concepts of numbers, and hence they can be dealt with independently of any apples, oranges, pears, and other objects used as counting example. Teachers are constantly talking about universal nature of math and even about pure math. Yet, somehow, teachers were always sticking to real world examples when explaining math concepts. At the same time they were talking where the math can be applied, but miss to inform us what is it we are actually applying, i.e. pure math. Fortunately, I have realized that math can be independent discipline and that the math tasks and problems can be tied to the numbers only, without any real world examples. But, that was a trouble, when I looked how the math is taught. I  did not talk to teachers about that, they were very strict, and I was afraid to raise any issue. Yet, I was the best student in primary school, in my class, and one of the best in school, in math. And the issue was why would you expect from kids to know non mathematical concepts in order to add two numbers? Why kids should know social structures and social relationships, legal concepts and then learn math from those examples? Math can be taught without them. The best approach is to say that math can be motivated by real world affairs and also it can be developed from math axioms.

Then, there was another puzzle. The opposite (of knowing non mathematical concepts) was also true. As a kid, I realized that I didn't know how the automobile engine works, yet I can count engines. Then, in 7th grade when physics kicked in, we have been counting objects that we only vaguely knew what they are, like, electrons, atoms, waves, particles. To me, if math says that definition what actually belongs to a set, in order to be counted, must be clear and non ambiguous, it should be followed. Yet, we do not know what is electron, but it was apparently sufficient to stick to "it looks like it's a particle, but, it can be a wave too, and it may appear and disappear time after time". So, apparently we can define set "of objects we don't know exactly what they are" and have that count too.

Throughout “schooling” teachers (not all), most likely, failed to educate us what math really is. What is left from so called math education, for many of us, is a bitter, frustrating, scary feeling, tight test schedules, unresolved problems, and many confusion about math concepts. Moreover, teachers were running through material without giving any deeper insights.

And that view I questioned through primary school. Later, it became clear that math can be developed as a discipline without waiting for real world examples. However, it is true that certain directions in math were motivated by physics problems, like calculus, but still, calculus can be developed from scratch without physics input. You can not expect from kids to fully know concept of force, energy, electrical field, in order to introduce calculus. Calculus can be introduced way earlier, right after irrational numbers.

Well, I did have my personal “fight” with math, and I think I won! And this happened relatively early, around 4^th grade, when I realized that you can work with numbers independently from any real world examples, but I did question how we define sets, and can we really always define clearly what belongs to a set. Later, that proved to be related to Russell's paradox. That was, at least for me, key point when I started to like math, to appreciate its elegance, its independence as a discipline. After primary school I enrolled in Mathematical Gymnasium, then later, finished my Bachelor's Degree in Electrical Engineering (later redefined as Masters Degree, Bologna, around 2000), studied Astronomy along the way, and have exciting career in these fields.

Here, I would try to keep examples simple, intuitive, and I would always emphasize that particular mathematical concept can be defined and derived without real world examples, but can be motivated by them.

In this article I want to explain a method how to teach kids, or primary school students, or anyone who is interested to understand math better, to think more freely about math, and to see that math is a game, puzzle game sometimes, whose variations can be created by you as well. You do not need to wait for someone else to give you mathematical problems and tasks. I hope that, after you read this post, you will have an unstoppable desire to revisit other parts of math, and to go on your own journey which will also include YOU creating new mathematical problems to solve.

Let’s start. Counting objects is the core of mathematics. Let’s say we have two kids, Peter and Stephen, who wants to learn math. Usually the learning starts with showing some objects, say apples, and asking kids to count them. Then, it comes addition of apples, subtraction, and perhaps later multiplication and division. Now, after, say one hour or so, or after even a few days, Peter and Stephen will always expect to see objects then to count them. That’s where the word “given” in math comes from, or even word “suppose”. Peter or Stephen will always expect to have a problem in front of him, to be “given” that he needs to solve. Needless to say, kids may be, at this point, scared to asked where these problems come from? Who “gives” them? Here is the answer.

It would be nice to show to kids that 2 + 3 = 5 is independent of any objects counted. This can be done by going through a number of examples with apples, oranges, cars, CDs, pencils, even time, like hours and seconds, and then show separately that 2 + 3 equals 5 no matter what are you counting. You can reuse this numerical answer for any objects you may be counting in future. At this point kids should be aware of numbers as separate concepts. They have to be aware that numbers can exist without any real world examples. They have to be aware that they can do two things with numbers. Deal with them as pure numbers, like 2 + 3 = 5, and, obtain these numbers by counting objects. It is very important to explain this fact to kids.

And, now, here is the main step!

Get a bunch of objects together, say 30 apples, in a basket. Ask Stephen to go outside of room. Then, tell Peter to specify how many apples he wants to be taken of the basket and put on table. Suggest to Peter that the number he specified is completely arbitrary, it’s up to him to tell the number. Or, he may have his own logic to do that. Let’s say Peter chose number 5, for whatever reason. Then 5 apples will be taken from the basket and put on table. Ask Peter again to chose another number. Peter says 3 and he takes 3 apples from the basket to table, in another group.

Now, explain to Peter how his arbitrary, starting selection, a starting number of apples dictated how many pencils will be counted. It also dictated the two numbers Stephen will be dealing with. Also, explain to Peter that it is him who specified how many apples were to be counted. There was no pressure from anyone else, no “given” number of apples.  It was Peter who “gave” the numbers.

So, in mathematics, not only that you can count some objects and tell how many of those objects are there, but also you can use a pure number to start with, and tell how many of things is to be counted. This count, specified by you, can be a “given” starting point for someone else’s calculation! Please feel free to select any “starting” number again and play with mathematical operations, addition, subtraction etc, using other numbers by your choice. The numerical results you get is important knowledge, and can be reused in real world applications.

Let’s say, after a while, you have a few pages with numbers chosen by you, and a number of results obtained by multiplication, addition, etc. You may want to sort them, as you wish, maybe by magnitude, smaller numbers first, bigger numbers later. You may ask yourself, how this can be applied in real life? But you can not go from math back to real objects because, as we have seen, when you write 2 + 3 you do not know what is counted. Can be anything. However, there is a way for real world application! If you start with “IF I counted CARS, and get 2, and then IF 3 more cars came by” what is the result? Hence, once you start keeping track WHAT you have counted, and WHY, you can use your mathematical results to match the numbers. Note here that math does not see where the numbers 3 and two came from or why. You can count cars passing by, you can count cars on parking lot, or these can be numbers you tell car dealer to move cars in the garage. Math can’t see that. Math only see you provided to it number 2 and 3 to add. It is you that you have to carefully keep track what did you count and why and where the result will go.

Back to Peter and Stephen. Now, it is time to ask Stephen to specify how many apples will be taken out of the basket while Peter is out of the room. Now Peter will have “given” numbers to add, and Stephen will become aware that it is him who can specify these initial conditions for counting.

These exercises should be continued with measuring and specifying length. First Peter will be the one who will specify the distance between two objects in the room, by moving those objects apart. Then, Stephen will come in the room and measure the distance. Second exercise will be for Stephen and Peter, to specify the distance and then measure and move two objects to match that distance. It is of fundamental importance here to make Peter and Stephen aware that it’s them who specify the initial numbers, counts, and then the change is done in real world by their own specification! During these exercises kids will start thinking more freely about mathematics, and starting creating their own mathematical problems, instead of only dealing with what is “given” to them.

In next post I will write how we separate and connect two logical worlds. First one is how we create rules and keep track what and why is counted and the second world is the world of mathematics that will deal with these numbers. With the second world we are already familiar at this post.

Real World Math Applications

Peter may want to ask “Ok, I have done all this math calculations, and what I get is a number as the result. Where is the real world application?”

Here is the answer. Let’s say Peter obtained number 5 as counting pencils. What now? What possible application can be created using number 5 and pencils?

Here is how the real world examples can be created. Make this sentence:

“If I have 5 pencils AND IF 'something else A' THEN 'something else B .”

Note first “AND IF”! Following that it can be a true statement about anything you can imagine that can depend on the number of pencils counted. You have a true statement that you counted and obtained 5 pencils. Then, you may say, IF I have 5 pencils AND IF it’s sunny outside THEN I will decide to go out and draw landscapes. Note how the number of pencils influenced your decision to go outside and draw landscapes. Please note how open ended logical connective “AND IF ” can be related to any specified fact YOU have in mind! The very true and accurate calculation of 5 pencils can lead to an action that you can attach to this number! It is really up to you or up to your analysis what will depend on number five. It can be your creative new statement, new initiative or new decision that will start once you have 5 (and not 3 or 6) pencils in. That connective on the other side of "AND IF" can be another count, obtained from counting other objects too.

Now, teachers will rarely ask you to do this! Teachers will just stop after you calculated 5 (pencils or other objects) and leave you hanging there asking “what for?” and “what now?” I want to change that. I want to show you how the math calculations are actually applied, used, and utilized in the real world.

[ applied math, applied mathematics, learn math, math, math and real life, math concepts, math education, math tutoring, mathematics ]

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

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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, 2^nd minute 23 cars, 3 rd minute 10 cars, 4^th minute 12 cars, 5^th minute 22 cars. Of course you can draw a table with the columns showing minutes and associated count of cars.

Time (mins)	Cars
1	15
2	23
3	10
4	12
5	22

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 ]

Calculus for Kids - How to Calculate Surface Area of an Irregular, Arbitrary Shape - Story of the Pirate Island

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My goal here is to introduce the ideas of calculus, and methods of summation and integration in the way that can be explained and thought to primary school students. In addition, anyone interested in knowing more about calculus and ideas behind integration can read and, most likely, greatly benefit from this chapter.

There are only a few prerequisites for this integration introduction.

In order to achieve this goal, students, or readers, should have to be familiar with integers, and at least with rational numbers, fractions. Concept of a function would be good to know, but it is not required. It will be introduced here. Readers should know how to calculate surface area of an rectangle (I am sure everyone knows that!), some triangles, and a circle. That's it! It's my goal to make new concepts’ explanations easy and clear, with minimum initial knowledge required! It has to be fun all along too! Rigorous proofs are left out. They can be found in any book on calculus.

Fig. 1. The Pirate Island at night.




Fig. 2. The Pirate Island during the day.


This chapter will introduce readers with a method to calculate the surface area of an object of an irregular, arbitrary shape.

Students are, for a long time, trained to calculate areas of rectangles, trapezoids, circles, maybe ellipse, and other 2D and 3D objects. They are pushed to memorize formulae in order to pass the tests or math exams. Yet, when they look around, in the real world, there are examples that has nothing to with all that mathematics material the students have been through for many years. Look at the stocks graph, temperature changes, wind direction, car driving, etc. Even kids in 4^th grade will have good understanding what is mechanically say, going on in these cases, but, quantification and mathematics are not shown to be, somehow, applicable for those instances yet. One example is that students will be most likely puzzled what would be the formula to calculate a surface area of an object of a completely irregular shape, like the shape of an fried egg, or the shape of an island. This is excellent question, and, moreover, there is no need to keep students away from the answers for years, until they come to secondary school, to get the calculus and pre-calculus classes. Basic ideas are very easy to grasp, and I am sure, this chapter will be an eye opener and new ideas generator for the readers.

Let’s start. You do not need to hunt for a formula in mathematics all the time! Method, approach is way more important than finding, or even worse, memorizing formulae in many cases. And, it's the actual method to calculate a surface area of an arbitrary shape that is the essence of integration, the idea that represents one of the fundamental concepts in calculus.

Let's take our focus from a boring math text, and look around. Where we can apply math? Let's consider the following example. A pirate island. We want to calculate a surface area of a pirate island, let's call it The Pirate Island.

There may be many reason why we need to do that. Maybe we want to see how much of it is covered by forest. Or, we want to make an accurate map for future treasure hunters. Here I want to make one important point. Math can't distinguish between real world and fictional one, like the fairy tale! Check this. Two dragons ate 53kg of coal each. How much coal they ate together? It's up to you to develop reasoning what and why to count, and provide these requirements to math. Math will do the calculations, but can’t distinguish whether the numbers are coming from real or fictional world.

The Island has a completely arbitrary and irregular shape. If we try to search for a formula ready solution, we will not find it, most likely. So, how we calculate the surface area of an arbitrary shape?
To start with, let’s draw the island from the bird’s perspective. Let’s focus only on shape and leave out vegetation, animals, sand, rocks etc. Just the shape of the island in one instance of time. Of course, we have to do that, because the shape of island is constantly changing because of the waves or tides. But, we will assume one shape for our purpose.

Fig. 3. The Pirate Island shape from the bird’s perspective.


Since we know how to calculate the surface of an rectangle, let’s use rectangle to calculate the surface of the island. Let’s cover it with rectangles. We can cover it with triangles too, or with circles, for which we know how to calculate surface area. That will change a little in our approach, and, in addition, rectangles are quite simple too. So, let’s do it. Here is how the Island looks like covered with rectangles.

Fig. 4. The Pirate Island covered with rectangles.

We want to try to cover, as closely as possible, the shape of the island with rectangles, and then compare their surface areas.

If we label each rectangle and note its dimensions we can add them up and obtain the surface area covered by all the rectangles we have used so far. We will immediately notice several things. First is that our covered area is not the actual area of the island, but only close to it. There are errors, there are missed surfaces where the curvatures are. On the good side we can have some idea, from our calculation, after adding rectangles surfaces, what the total area approximately is. It’s a bit bigger than what we got, but we also note that it has a definite limit value, which is actual surface area of the Island. Don’t worry that we are talking here about estimated value. We are doing a good estimate, we have a good approach to estimate. Moreover, the estimation will lead us to determine the exact value of the island surface area.

At this point, let’s see what we can do to get closer to the actual surface area. The first thing that comes to mind is to add more (smaller) rectangles to fill in gaps between present rectangles and the curvature line of the island. That’s correct. Let’s magnify one part of the island and rectangles in it, to see clearly the gaps on surface between rectangle and the curve.

Fig. 5. A magnified portion of the Island.

With the addition of more and more, smaller and smaller rectangles we fill in the gaps denser and denser, and our covering surface area is getting more and more accurate, closer to the real value. However, because of the smoothness of the shore curve, the smoothness of the island shape, there appear to be always gaps between rectangles and the island curve! That is actually very true. Let’s magnify again the portion of the island we have just filled with more rectangles and see how it looks like.

Fig. 6. Again magnified portion of the island with more, smaller, finer rectangles in.


Still more gaps. But, they are smaller too. Hence, the error is smaller. We have a sense, that even if we add more and more rectangles, there will be always gaps, though smaller and smaller, because the curve is smooth and our rectangles have right angles. However, there is one powerful thing that we can notice at this point. While there is a difference between the surface area we are getting by adding smaller and smaller rectangles and the real island surface area, also is the error between the surface area we are getting, and some correct value we are after. Moreover, with this approach, by adding more and more smaller and smaller rectangles we can approach to the correct value as close as we wish. We can get close to any number of decimal places. That sounds like a good achievement, right? If you calculate the surface area of an island that is say 10 square miles with the accuracy of 0.00000000000005 square mile, which is half of the grain of sand area, you seem to have quite a good result. But, you may ask, what is the exact value, and not just approximate, no matter how close to that value, using our rectangles, we get.

There are two answers to this question. One is that, when we add more and more, smaller and smaller rectangles to fill in the gaps, we may be able to see to which value our successive sums are aiming at! We can be 100% certain that there is a value we are after, and sometimes we can actually name that limiting value, and that will be the actual surface area of the Island. Note again, that even we do not “reach” that value with our process of getting rectangles, we still can approach it as close as we wish, and as accurate as we wish, even to millionth of decimal places. When we can say that we can do that kind of approach to some limiting value, the value in question is actual correct value for our way of summation. Moreover, that’s exactly how mathematicians define a limit value of a sum in the process of integration. Integration is nothing more than summation taking into account this limit summation method of our rectangles.

The other answer has to do with the definition of irrational numbers. Irrational means numbers that can not be represented by a ratio of two integers (rational numbers). For instance, square root of 2 or Pi, are some of irrational numbers. Their main property is that they are represented by infinite number of non repeating decimal places. You may think then that these numbers will be infinitely large. But, they are not, because each time smaller and smaller fractions are added, like we added smaller and smaller rectangles in our island and we know that these rectangles are always between island limits and our present calculated surface area. As long as you add smaller and smaller fractions each time, and you know that there is upper limit that can not be exceeded you are dealing with irrational numbers.  Try it by yourself. Get any number and start adding more and more decimal places to it! You will see that every time you, actually, can have the upper limit of your number, if you wish, or, you can add more and more decimal places, noting that each time you added obviously a smaller fraction! From the very strict mathematical point of view, what we have just done is the actual definition of an irrational number. They must be defined as limit values as you add smaller and smaller fractions. Sometimes these limit values are clearly visible, sometimes they may not been known, but in each case it has to be proved that they exist and that we can approach them to any accuracy we wish and as close as we wish.

If you still think that, at least theoretically, the coastal island line is still too jagged to be smooth (but continuous!) on any scale, you are right. Another approach is to consider fractal theory, in calculating the coastal line length. Good starting point is Koch snowflake or Mandelbrot works on fractals. That is the subject outside the scope of this post.

Now, back to Island. While we explored our accuracy of calculating surface area, don’t forget what we have started with! We have started even with a thought to try to find ready to use, plug in, formula for the surface of the island. But, that was wrong approach. Then we thought of method of covering rectangles, and by using them we got the correct and very powerful method to calculate, to any accuracy we wish, the island surface area.

We can now generalize our result. Using this method of covering rectangles we can calculate essentially surface area of any shape. Interesting example is calculating surface area of an aircraft tail and rudder section.
Here is illustration. Principle is the same.

         


Fig. 7. Numerical methods of calculation of a surface area of an aircraft vertical stabilizer and rudder section.

This method of covering rectangles, to call it like that, is the fundamental idea behind calculus and numerical methods in engineering applications. Numerical methods in engineering simply means, in many cases “since there is no ready to use formula, let’s use some approximate method with limiting process for summation, like method of covering rectangles”.

You are now familiar with one more fundamental idea of calculus. Taking the calculation to the limit, in our case we are adding more and more smaller rectangles to get to the limit of  the island surface area. Note that! Taking to the limit is what is the basis of the calculus. That’s very powerful thought we came with and powerful method. For instance, if we did not take it to the limit, we may not be able to see what the actual surface area value may be, nor we will be in position to conclude that we will approach that area with any accuracy we want. When you take process to the limit you are already mastering the most fundamental idea in calculus.

Using "covering rectangle" method, method of dividing, then summing the area or line length, or surface areas, (limiting process is involved here, as we have seen)  is a common method of solution for arbitrary, irregularly shaped lines, surfaces and volumes. Also, the method is used as well when a function is defined in a closed form, frequently by one or more formulae (where possible) to describe (as a continuous, smooth function) lines, surfaces, and volumes.

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