Sunday, April 17, 2011

How math can be applied to so many different fields and how we can use math in real life

How to differentiate between pure math and applied math? How we can use math in real life? How math can be applied to so many different fields? Where all those volumes of math come from? Can I revisit my math from primary and secondary school and finally understand what is it about? Can I learn math now? How I can teach my kids or students effectively primary and secondary school math?

I will be trying to answer these questions in the next several articles.

I encourage you to read the whole article, especially the action with index cards, where I explain, essentially, how you can use math in real world situations, and two steps in real world math solving method.

In order to “apply” math, we have to decide we want to quantify something. Quantification is the essence of math application. Note that nonmathematical relationships between objects exist long before we decided  to quantify. If the quantification is applicable to the field of our interest then we can say we can “apply” math to it, i.e. pure mathematical results may be applicable to that field. When we say mathematics can be applied to many fields, it really means, counting, quantification can be applied and done to those fields. However, quantitative relations are not the only one of importance in the field we chose to study. Knowledge of all mathematics will not help you a bit to discover a single physics law unless you first think in terms of non-math concepts.

As you can see, from the picture, 2 + 3 counts can come from many different fields.

You can then abstract counting and numbers 2 and 3 as common to all those fields. Moreover, once you’ve done counting,  you can deal with counts alone, not even thinking where they come from. You conclude that 2 + 3 = 5 no matter what you have been counting. That's why math can take off as a separate discipline, called pure math (not that applied math is dirty!). Pure means that suddenly we are not interested where the quantities, counts, numbers come from, we are specifically and only interested in the numbers, counts properties.

When you see a person writing down 2 + 3 =...   you don't know what objects she has on her mind, what objects she is adding! But you know the result will be 5! That's, actually, pure math!  You see, a person is writing down 2 + 3 =… Now,  let's say it again, you don’t know what that person has in mind, which objects she was counting. But, you know that the result will be 5! You abstracted the counts and counting from objects being counted. That’s, so called, pure math. Even, probably, without knowing it, you did a great job abstracting numbers, and operations on them, from real world situations. It is important to note how you dealt here with counts only, and not with particular objects. But, if you keep, on a separate sheet, what objects you are counting, you can transfer the result back to the real world.

Once you are aware that you can work with numbers, counts only, you are in position to investigate properties of these counts, without considering at all where they come from. You may now want to continue to develop math as a separate discipline! Make more examples, think of more numbers, operations on them! 5 + 3 = 8. 3 x 4 = 12. 8 - 5 = 3. You can say now, that if you have 7 and you add 8 you will get 15, just by dealing with numbers, no matter what objects and rules about them were involved. You realize that you can work with numbers only! And operations you have at your disposal, addition, subtraction, division, multiplication. Now, be sure, there are no other operations in math. These four are sufficient to define all "kinds" of numbers and all mathematics. Other, so called, more complex operations (although they are not so complex, in many cases) are just different sequences of these basic four, +, -, x, /, it’s just that mathematicians like give them exotic names, that's all.

This is probably the most important step you have made in learning math so far.

You can play with other numbers and math operations (addition, subtraction, division, multiplication) on them and be sure they will be common and be in intersection of other fields (outside math). Mathematics could have been also called Countology, a science about counts. Or even Setology, a science about sets, because sets are fundamental objects in mathematics. Calling math Setology or Countology is emphasizing the fact that no matter which "fancy" names mathematicians use inside math, those names always represent sets, or counts, or sets of counts, or sets of pairs of counts, sums, or some other relationships between them!

At one point you may want to play with numbers 3 and 7, like 3 + 7 = 10, and perhaps think what are the fields where these counts can come from. Or, in other words, what can be counted to get those numbers. Once you find what can be counted, you then investigate the relationships between objects counted. These relationships need not to be mathematical, but can influence counts.  Note though that you used those numbers, 3 and 7, without knowing what you have counted! That's pure math! Once you are aware that you can deal with numbers by themselves, as separate entities, as we just did, you enter the area of pure math research :-) Congratulations!

One important note about math and applied math. Looking at 2 + 3 it is impossible to say to which objects these numbers are referring to. Of course, you can try to find examples from real life for the counts 2 and 3 and their addition, but, again, going the other way, from math to real world,  just looking at 2 and 3 you can not say what you have counted. That' pure math. You can add 2 and 3 and then you can return the result to the real world, to the objects you have been counted. So, if you counted apples, it will be 5 apples, if you counted cars, it will be 5 cars. Note how math result can be reused for many different objects you have counted. In this "re-usability" lies the value of math. But, on the other hand, since you can deal with numbers only, you can investigate properties of pure numbers, their relationships, their magnitudes, you can investigate operations on them, different sets of counts, i.e. numbers. Result may be applicable to some real world scenario, and frequently is.
Let’s start with clarifying even more and explaining in more detail the difference between pure and applied math. Let’s say you have a task at hand and some mathematical calculations to complete. Say, you want to calculate how much money you will have to pay for 5 CDs if each one costs $3.00. Now, I know it is easy to calculate, it’s 5 x 3 = 15 dollars. But, I will show you how you can tell apart what is pure and what is applied math, and from these “rules”, which are more than rules, how you can apply math to many different fields.
Take a set of blank, white index cards, size 5" x 8" (inches), say, 50 of then. You might not use them all for our investigation, but I want you to have enough for the examples that will come.
Take two cards now. On one card we will write down the task what we have talked about and a required calculation, on another we will write down only actual mathematical calculation. For instance, you can write, on the first card: “How many dollars I will pay for 5CDs if one CD costs $3?”. Then, on this first card, which I will call Field Card, write down the calculation requirements together with “units” you are dealing with, i.e. objects you are counting. Like this,
                           5 CDs x 3 $ = ? on Field Card
Now, take another card. I will call this card Math Card. On this card you only write down the actual mathematics you have used, without any units, or objects counted, or explanation what the task was, like this
                           5 x 3 = 15 on Math Card

Now when you have the result, you go back to first, Field Card, and write down 15 with “units” i.e. objects counted to it, in this case dollars: 
                           5 CDs x 3 $ = 15$ on Field Card
Take another blank card. It will be our new Field Card. Let’s do another calculation. You want to buy 3 model airplanes and each model costs $10. How much you will have to pay for these airplanes? Again, it is easy to calculate the correct number (3x10=30 airplanes), but we still want to separate the task description and actual calculation. Hence, Field Card, the new one will have
                          3 models x 10$ = ?$ on Field Card
On our Math Card we will write the calculation we are performing, without mentioning objects we have been dealing with:
                          3 x 10 = 30 on Math Card
Our Math Card, at this point, will have two lines on it:
                          5 x 3 = 15
                          3 x 10 = 30
How about a new task? There are five cars and each car can Have 3 people inside. Whatis the total number of peoplethat can use the cars? 5 cars x 3 people = ? people
Let’s take our Math Card to write down only the calculation required. But wait! We already have 5 x 3 = 15 written on it. We can use that, we don’t have to write it again! It, apparently, doesn’t matter if we multiplied 5 CDs and 3 dollars or 5 cars and 3 people. In both cases the result was count of 15 or number 15. We can conclude that no matter what objects we have counted, if we have 5 things and 3 other things, the result of multiplication will always be 15! As we have just done on Math Card, we have written only numbers without objects counted. Whether we count CDs, dollars, cars, or people, 5 x 3 will always be 15. We can reuse that calculation for next and any other example requiring to multiply counts of 5 objects with counts of 3 objects! At this point you can guess that if we have another example where we have to calculate say, 3 things x 10 things, and if we note what we have written on our Math Card, we can see that we have already written there 3 x 10 = 30. And we can reuse this result as well, for any new examples involving 3 objects and 10 objects.
We have now three Field cards, with three tasks outlined on them. Let’s put a title on each card, in the top centre area. First card will be called “Music CDs”, second card will have title “Airplane Models”, and the third card will have the title “Transport”. You may sense that there can be similar calculations say in biology, chemistry, sales reports, your grocery list, your monthly spending, airport schedules etc. And, all these new tasks may reuse the same calculations we have written on the Math Card. In other words, we have one Math Card and many Field cards.
We have two reusable mathematical results which we can use again and again. Let’s try! Let’s talk Environment! If I have 3 trees and each one needs 10 minutes to plant, how much time I will need to plant these 3 trees? Take a new card, write the title “Environment” at the top and write down the task I have just described. Now, take a look at the Math Card? Is there a line 3 x 10? Yes there is! We can reuse the calculation there, 3 x 10 = 30, we can go back to “Environment” card and write down calculation in units,
3 trees x 10 min = 30 min
We will need 30 minutes to plant 3 trees.

Now, stack away the Field Cards, and keep in front of you only Math Card. There is one interesting thing we can say about it. We can predict, that perhaps, in future, in real life, we can have not only 3 or 5 or 10 objects. We can sense there can be calculations required for other numbers as well. Do we really need to wait for these “real world” examples to pop up so we can fill in another row in Math card!? No! We can play with numbers and fill in some new rows on Math card by ourselves! Let’s do it! You can chose any combination of numbers you like. Let’s say, we chose 7x5. It will be 35. New row on Math Card will be 7 x 5 = 35. How about 3 x 4? Sure, let’s add another row, 3 x 4 = 12. We are certain these calculations can be reused for many different real time tasks. Even we can modify slightly our own examples with CDs. Say, now, we want to calculate how much we will pay if we want to buy 7CDs and each one costs $5. We can look in Math Card and find the row with actual calculation we just entered, 7 x 5 = 35. Hence, the answer is we will need to pay $35 for 7 CDs.

So, it’s clear that Math Card is reusable. The more numerical entries we put on it, the more chance is that some real life example will be a match for them.
Now, let’s look again to Math Card. It looks nice. It looks neat, clean. We deal only with numbers there while our example cards are stacked away. Then, we look closer to the actual multiplications. I can see now, that I can deal and play only with numbers! I can generate another row on Math Card without knowing which real life example will be a match. Let’s add one more row, say, 2 x 9 = 18. Now, let’s say there are several people outside our room waiting for calculations to be done. One is in “Transport”, but another one is in “Movie Business” (want to calculate how many tickets she sells each month), and yet another person is in another field. Field!? Yes! And all of them have their index cards, Field Cards with field title written on them. We can lend, to each of these people our ready-to-use Math Card so, they can find the actual calculation that is relevant to their requirements.
Our Math Card is right here. We look at the rows we have added. Again, since we have concluded we can add more rows, let’s examine what is the next step. For sure we have to get more blank cards from the stack. Let’s take two more. Now we have three Math Cards. I know that I can play only with numbers now. I can add more numbers. I can add more operations, like subtraction, addition, knowing for sure there are plenty of examples in real life that these calculations can relate. Moreover, I can start to begin to notice some common properties of these numbers. I can group them which one is even which one is odd. I can add new Math Card to which I will write only multiplications, and another card which will contain only subtractions. Hmm, interesting. My index cards start too look like a math book! Moreover, since they deal with pure numbers only, it can be called pure math book. And you are right. This is a point when we can see how pure math books can exists without any “real” life examples in it. It’s because we can investigate numbers properties just looking at numbers. If we want a “real world” example we can ask people with Field Cards to come forward.
Our Math Cards can be a sole focus of our studies. Since we know we can add any number we wish, and do any sequence of math operations we wish, we might as well add to Math Card all what we have been discovering as properties of these numbers and operations. Our results will be written on new math cards, and we will detail which numbers we have used, what operations have been performed and what results we have obtained. If you open any pure math book, you will see that it is exactly the case. They are like index cards notes collected together and given Table of Contents!
Now back to Field Cards. The important point I want to make here is that we separate and collect index cards for each field. No mix up. If we want to do “Movie Business” calculations, we will fill in index cards with calculations requests related only to movie business. For instance, we will write on index cards tasks like how many tickets is sold during summer in three theatres downtown. What is an average number of visits in the Theatre number 3? Which movie has the biggest number of tickets sold? As we have seen, for all these requirements we can chose related Math Card with ready to use pure calculations, and get the calculated results back to “Movies Business” card. This is the reason mathematics can be “reused” in many fields. As the time passes, people dealing with Field Cards may want to record their logic related to their particular field in a book and publish it! Those books usually will have titles “Mathematics for Transport Studies”, “Mathematical Biology”, “Quantitative Physiology”, “Applied Mathematics in Food Industry”, “Financial Mathematics”, “Math in Economics”. But, they are, in essence, nothing more than summarized Field Index Cards, enriched by many examples in the particular field, and with hidden “pure” Math Cards. They will rarely show that common math is reused here. But, if you have these books at hand, and compare mathematics used in them, you will see that math is one and the same, it’s just objects counted that were different.

Real world reasoning will tell you what to count and with which mathematical operations you will start with. Mathematics will accept these starting points, but will see only numbers and operations, and not what is counted or measured. An example can be any physical law. In each formula, which is within physics, is important what you measure or count, and relationships between the objects or concepts, like mass, speed, electrical charge, distance. But when you start calculations, you will deal with pure math, counts, numbers, keeping aside what is counted (kilograms, meters, seconds, temperature, etc). For instance, in finance, buying or selling is not part of the math. That's part of financial domain. It's separate logic. When you count how much you have sold or bought, those numbers, or counts are actual mathematics. It is YOU who will have to keep track who "buys" and who "sells", while math will keep track of counts and give you results in them.

There are two parts of an applied math problem. 1) Correctly establish relations between non-mathematical objects 2) Solve newly defined numerical, i.e. mathematical problem. The first action will require the knowledge of the real world domain, which is, in most cases, extraneous to mathematics, and, in many cases you will deal with relations and definitions that has little to do with math. For instance, when you "buy" or "sell" something and in certain quantity, buying and selling are not mathematical objects. In Physics, object moving through space, or water going through turbine, are not mathematical relations nor objects. They are non mathematical things whose actions can be quantified, that's all. But these things themselves or their actions are not mathematical objects, they are physical objects. In economics, you will have demand for 500 cars, and supply of 470 cars. But, demand and supply are not mathematical objects, and their relations are not mathematical. Their relations are defined by human will, decision making, sociological laws, and specific logic associated with it. Only when we start quantifying them, keeping track which numbers are "demand" and which numbers are "supply" we enter mathematical world. Note that without keeping track what we have counted, we would not know to which objects the numbers 500 and 470 belong too.

Once we established what needs to be calculated, we can deal with numbers only, with mathematics only. And that's the second part of problem solving. Given these initial conditions from the world of economics or physics or trading, we now deal with calculations, perhaps solving some equations, which is pure math, and knowledge of pure math can be very beneficial at this point. Once the calculations are done (of course, we should keep, all along, track of units) we go back to the real world domain and use the newly obtained solution and results.

One answer to the question how math can be applied to so many different fields is because so  many things can be quantified. But even before that, we have to make decision to quantify something. The very fact that we can distinguish different objects through our psychological, cognitive, abstraction, categorization capacities and abilities gives a rise to an important choice of action - to count what we can distinguish. The very fact that we can distinguish so many objects means that we can put them in sets, and count them is the core reason why mathematics can be applicable to so many real world situations.

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