Mathematical Thinking and Scientic Works

What is Mathematical thinking?

Mathematical thinking is the way to think mathematically. Honestly, mathematics objects are in our mind and its characteristics are absolute.

How to think mathematically?

According to Mr. Marsigit, we must do two steps. A first step is idealization. We must think and assume that everything is the ‘same’ and ‘perfect’. For example, there is a basket ball (a ball which is usually use in playing basketball). For the real, the ball is not full circle in its shape. If we see it clearly and sharply (especially using microscope), we’ll find the roughness of this ball. So, the ball becomes not really ball and soft.

Secondly, we must do an abstraction. It is means that we must take a part of all the characteristics of the object we learn about. For example, the ball has many characteristics, it’s heavy, it’s hard, its shape is circle, etc. but in thinking mathematically, we don’t need all of that characteristics. We just need to take a part of them (for example the volume). So, we just concern and think to the volume of the ball. The objects in mathematical thinking are always consistence, logic and systematic.

While, the characteristics of scientific works are impersonal (objective, based on the truth) and also consider on the ethical code (the most important thing is free from plagiarism –said Mr. Marsigit-).

A. _Desil

What is Desil?
Do U know what the meaning of desil?
Refers into Ronald E. Walpole's book, "Pengantar Statistika" (3rd edition), desil is value that divide a compound of observations results into ten parts to be the same. Desil symbolized with D1,D2, etc. Now, let's see the example! --> to be continued (in the next posting), I'm sorry all, I'm sorry Mr. Marsigit for posting lately.

Exercising

On Thursday, April 2^nd 2009, Mr. Marsigit commanded us to do some Mathematics exercises. There were 5 questions that we must do.

1. Explain how to prove that the square root of 2 is irrational number!
2. Explain how to show or to indicate that the sum angles of triangle are equals to 180^o!
3. Explain how you are able to get π (phi) !
4. Explain how you’re able to find out the are of region bounded by the graph of y = x² and y = x + 2 !
5. Explain how you’re able to determine to the intersection point between the circle x² + y² = 20 and y = x + 1 !

The answers:

1. We’ll approve that the square root of 2 is an irrational number. First, we must know that an irrational number is a number that the value cannot be expressed exactly as a fraction a/b, where a and b are integers and relatively prime.

Let’s assume that the square root of 2 is a rational number, so we can state the square root of 2 equals a per b. If x is a positive integer, x² is even if and only if x is even.

Then, we can say that the square root of 2 is equals a per b. Multiply each side with b and we get new equation, a equals to b times the square root of 2. Multiply each sides by itself and we get that a square is equals to b square.

From this equation we ensure that a square is even because a square is twice of integer. Because of a square is even, so it’s sure that a is also an even. And then, we may write a equals to 2 times c (where c is an integer) and substitute this (a equals to 2 times c) in to previous equation. We get that 4 times c square equals to 2 times b square. We can divide each side with 2 and get equation 2 times c square equals to b square.

From this last equation, we get another clue that b square is also even. Why do I say that? Because as we know that b square is equals to twice square of integer, so, b is an even, of course.

Finally, it’s clear for us that a is even and b is even. As we know that the square root of 2 is equals to a per b (where a and b is even, both of them are not relatively prime) so the square root of 2 is irrational number.

2. We can draw a triangle to prove that the sum of angles of triangle is equals 180 degree.

First, let’s assume that the point angles is A, B and C (where AC is the base). From point angle C we can make a parallel line with line AB, so there is a new angle (we call it angle D) which it’s side as big as angle A, because angle A and angle D is same side ( I mean “sehadap” –sorry, I don’t know sehadap in English-). There is an angle between angle D and angle C (we call it angle E), whish has same size with angle B because angle E and angle B is angled in the opposed. Then, it shows that angle C, angle D and angle E form a straight line which is the size is 180 degree. It’s mean that the sum of angle C plus angle D (it substituted by angle A) plus angle E (it substitude by angle B) equals to 180 degree. In the other words, the sum of angle A plus angle B plus angle C equals to 180 degree. It’s enough to prove that the sum angles of triangle is equals to 180 degree.

3. As we know that π’s (phi) value is 3.14. After I read some references, now I’ll tell the history of phi. The value of phi (π = 3.14 or 22/7) found by Egyptian people in 1650 BC. We can get the value of phi by make a circle; calculate the perimeter and the diameter. If we make any circle, we’ll get that the value of the comparison between the perimeter and the diameter is always constant: 3.14. So, if we divide the circle perimeter with its diameter it is equal to 3.14 (phi). That is why the circle perimeter formula is equal to diameter time phi.
4. First step is we must draw the graph of y equals x square. We try to substitute x with a number, for example x equals 1, so we’ll get the value of y equals one or we can write it down in a table of x and y (the substitution and the result then). We can substitute x or y with any numbers. So do the equation y equals to x plus 2. So, we are able to draw the graph and get the intercept point between those two graphs. Mathematically, the interception point can be find by assume that y (in the first equation) equals to y (in the second equation. We get a new equation, x square equals to x plus 2. By reducing each side with x plus 2, we’ll get that x square minus x minus 2 equals to zero. Then we get the values of x, x is equals negative1 or x equals 2.

The area of the region bounded by the graph of y equals to x square and y equals to x plus 2 is equals to integral of x plus 2 minus x square dx from x equals negative 1 to x equals 2 (we call the region with A). After we calculate, we get the result that A equals to four half area unit.

5. To determine the intersection point between y equals x plus 1 and x square plus y square equals twenty is, first, we change the equation form x square plus y square equals twenty into form y equals square root of twenty minus x square in bracket. Assume that y equal to y, so x plus one is equal to square root of twenty minus x square in bracket. Then, square each side and we’ll get that two x square plus two x minus nineteen equals zero. We can use abc pormula to find the values of first x and second x, and we get the first x equals minus one plus square root of thirty nine per two and second x is equal to minus one minus square root of thirty nine per two: both of them are the intercept points.

Mozaic Videos

On last Thursday, March 26^th 2009 Mr. Marsigit ask me and my friends watched some English videos. Honestly, these videos are very interesting. But I want to say before that frankly, I don’t know well about the content of these videos, because the pronunciation was very good, so I just understood a little (I’m sorry Mr. Marsigit). There were 6 videos and in this blog, I’ll write what the videos about.

The first video is about motivation and the title is Death Poets Society. It told about a teacher who gave some motivations to his students. He told to his student that we must try to find our own “voice”. It’s mean that we must try to become the real ourselves and not just follow what people do. We also have to look at thing in different point, different way. Don’t overcome a problem just with a solution, but we must open our mind to find the others solution. Try to get the better ones.

The second video told about a boy (+ 8-10 years old) who stand up on the stage, in front of thousands of adult people and he said wonderfully in front of them. The boy motivated the audience about a trust. He asked the audience to believe themselves. We must believe in ourselves, to dream anything, do anything. I think it was the most interesting video because it was very amazing. Imagine! What a wonderful presentation! It done by a small children, in front of thousands people, and the audiences claps their hands when the boy said something powerful and “magic” for them. I’m very proud to that child. He could show up his ability to influence the people with his said. I doubt that this powerful presentation could be presented by every child. Excellent!

Next video was a song about Mathematics. Maybe the title was “What You Know about Math?” because the face lyrics was “…what you know about Math, what you know about Math,,,”. It’s a hip hop music, I think and it’s contain about everything in Mathematics, about trigonometry, logarithm, numbers, exponents, dividing, etc.

The fourth video is about how to solve differential equation. The example is dy per dx equals 4 x square dx. The way is integrate each of side and calculate, so we get the result. Next video is about solving linear inequality. We can get the value of a variable (for example x) by add or reduce each sides with a same numbers. So, we get the result for final. The last video was about the explanation of logarithm equation and it’s solving.