Tuesday, April 19, 2011

Math, Real World Examples, and How You Can Understand Math Even Better

You can read more "How math can be applied to so many different fields and how we can use math in real life".

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, 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!). 

Many students are not puzzled so much with real world math applications. Students know it's there! Students appear to be puzzled how math can exist as a separate discipline. When and how did it take off from all those apples, pears, and other counting objects? How math can be a part of physics, economics, engineering, chemistry, finance, commerce, trading, accounting, and yet, mathematics can exists as a separate and independent discipline, from all of the fields it is applied to? I am answering this question in my blog here, and in this article as well.

When you see a person writing down 2 + 3 =...   you don't know what is she adding! But you know the result will be 5. That's 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! That’s, so called, pure math. You did a great job abstracting math from its real world applications. You may now want to continue to develop math! Make more examples! 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. be sure, there are no other operations in math. All other, so called, operations are just different sequences of these basic four, +, -, x, /.  That’s pure mathematics, as the mathematicians want to say. This is probably the most important step you have made in learning math.

As I mentioned, we can separate counts 2 and 3 into different area and play with them exclusively.

These "pure" math means playing only with numbers. It's just about making new counts, i.e. numbers,  (like 2 + 9 = 11 or 7 x 5 = 35) and doing operations on them, without worrying where they come from. Motivation to do operations is up to you since real world at this point is out of our interest! This play with numbers only, is very similar to playing with marbles and counting how many marbles you have after some operations, like, addition, subtractions etc. However, if you are interested WHAT you are counting, then you can associate objects names (the ones you are counting) to the actual numbers, like 5apples, 12cars, 15$, 7CDs, songs, steps, kilometers. You keep track what you are counting on one paper, but on another paper you do actual addition, subtraction etc only with numbers! Don't forget, the object names do not influence actual numerical operations. Object names are for you only, to keep track what you have been counting. It can be, again, compared with marbles. If you say, these 5 marbles represent 5 CDs you have counted, and then you add 3 more you can say that you have 8 marbles, or, since they represent CDs that you have 8 CDs. The trick here is that, when counting marbles, YOU can decide that they represent various things you are interested to count, like apples, cars, CDs, balls, etc. Marbles are in some sense very similar to numbers, they are universal method of representing the counts of different objects.

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) too.

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. Note 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!

Now, let's take a look at that intersecting area where the numbers are. You can enlarge that area and add more numbers. Moreover, you can add any counts you want and investigate any operations on them as you wish. You can add, 1, 5, 7, 10, 25, 26, 27, ...102, 1237. They are all in that central area. But you can, now, look at that area independently from what intersects it. You can add more numbers, like 2/3, 4/5, 6/7, 1/3, 1/129. You see how you can create new numbers (which are called rational numbers, because they are ratios of integers). That is all math, pure math. Other fields that use math and counts can use your results instead of rediscovering them each time when they need to calculate something. These fields can be any of academic fields and the math they use is called "applied math". But, you see that mathematical results can be motivated by real world examples, but also they can be devices by you, when you just play with numbers. Essentially, you do not need real world examples to develop mathematics. One example is geometry of Lobachevsky that has been developed first and only then discovered and used by Einstein in his Theory of Relativity. On the other hand, Newton's problems in Physics, like calculating speed, led to development of calculus. But be aware, calculus could be developed even without looking at physical problems.

At this point I would like to show how mathematics can be an independent discipline and also can be, and is, used in real life. I will make a comparison between lines and some shapes and numbers. When you look at the building, and you want to draw it, you  will draw some lines, squares, rectangles, trying to mimic the shape of the building as truthfully as possible. Look at the lines you draw. You draw vertical, horizontal, and lines at any arbitrary angle. It appears that which line and where you will draw it will depend on the shape of the building. And that's true. Now, look away from the  building and get a new, blank sheet of paper. Draw a line on it. Draw horizontal lines, vertical, and at arbitrary angles. You are not required to draw a building. Just draw lines. You see how you abstracted lines, and, moreover you can play with them without taking care whether they represent anything in real world. Moreover, you can draw completely new building with your lines and call a company to actually build a new object from your drawings!

Similar things is with numbers. You can count real objects and get their numbers and deal with them. You can add, subtract, multiply numbers following the real world examples, like, "how many liters of gas I will use if I travel 125 km with the car that consumes 8.9L/km..." etc. When you calculate this or any other example, you deal only with numbers. You keep track of units, like what you have counted, aside. But, then, you can notice that 7 x 5 = 35 regardless what is counted! Similar thing happened with lines! You could draw lines without worrying if they represent any object in real world. Now, you can play with numbers, any number, and use ANY operations on them without worrying do they currently represent anything in real world. That's the essence of math. When you do applied math you still do pure calculations while keeping aside the units, what you have counted. But, in pure math you start with some numbers, it's up to you with which ones, without providing reasons why they are there, and do calculations on them. Try both scenarios and you will see the point! 

Here are more links you might like as well:
[ applied math, applied mathematics, education, math concepts, math education ]

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