Mathematics Thinking and scientific work

The definition of Mathematics from some lecture:
•mathematics is problem solving
•mathematics is investigation
•mathematics is creation
•mathematics mean of communication
pure mathematics is formal mathematics. The characteristic of formal mathematics is axiomatic, axiomatic mathematics different mathematic.
There are some definition of deduction mathematics is concept, definition, theorem, proof, axiom, procedure to proof theorem.
To starting point mathematics is system we should have assumes. The assumes can be a concept or definition of formal mathematics. The formal mathematics have base axiom, theorem and definition. Before definition you have clear picture or concept of object mathematics. Object mathematics can be your main or idea from the possibility object meta mathematics.
How to get mathematics object from the object concert:
1.idealism
idealism is assume that something is perfect, but perfect is God’s. We assume that something is sharp, but truly there is nothing which is sharp perfectly. We assume that line is straight. But truly there isn’t a perfect straight line.
2.Abstraction
A concert object have a lot of characteristic. In mathematics we just learn a few of the characteristic. Let : to learn about plane, we just learn about the shape and the size, to learn number, we just learn about the value. Mathematics is easy because just learn a few of the characteristic.


- Characteristic of mathematic thinking
1.Consistent
same with the first formula
2.Logic
We can know about the difference of big and small, relationship, mathematic separation or arithmetic, if than statement, the table of proof, how to get conclusion (premise), thesis, antithesis, hypothesis, let, etc.
- Scientific work
Example: report, proposal, text book, etc.
The characteristic of scientific work:
1.objective
2.have criteria or standard.

The characteristic paper:
1.abstract
2.recommendation
3.free from plagiarism
4.reference

exercise

exercise I

1.we will proof that square root of two is irrational number

• square root of two is irrational number because two is not perfect square number.
• So two is irrational number.

2. we will show that the sum of angels of triangle is equal one hundred and eighty.36

• Draw a triangle between two parallel line which one of triangle side coincide in one of the parallel line. Label your triangle using ABC. The vertex of triangle is A.
• Let: angle B is interior angle which the arm is one of the parallel line and the other arm is triangle side. Angle A exterior angle of the triangle top. Angle B and angle A are called alternate interior angle, so angle B and angle A have the same degree measurement.
• Angle C is the other interior angle. So angle C and the other exterior angle of the triangle top are called alternate interior angle.
• So the sum of angle A and two exterior angle of the triangle top is one hundred and eighty degree measurement because angle A and two exterior angle are supplementary angles.

3. we will explain how to get phi

• draw a circle with radius one centimeters. Put string and place in circle.
• Find the length of string
• The length of string is circumference of a circle
• We know that circumference of a circle is two times phi times radius,
• We have number of circumference of a circle with radius one centimeter, so we can find phi.
• Phi equal perimeter over two times radius
• Substitution the perimeter and the radius, so we get phi.

4. we will find the area of region bounded by the graph of y equal x square and y equal x plus two.

• We find the intersect of two graph by find x from equation x square equal x plus two. And than add two side with negative x minus two. So we get x square minus x minus two equal zero.
• Than we can find the factor. The factor is x minus two in bracket times x plus one in bracket equal zero. So we get x equal two and x equal negative one.
• The area of region bounded by the graph is define integral over boundary negative one and upper boundary two of x plus two minus x square dx.
• The area equal open square bracket a half times x square plus two times x minus one third times x cubeb close square bracket from x equal negative one to x equal two.
• The area equal open bracket a half times two square plus two times two minus one third times two cubeb close bracket minus open bracket a half times negative one square plus two times negative one plus one third times negative one cubeb close bracket.
• The area equal two plus four minus eight third in bracket minus a half minus one minus one third in bracket.
• So the area equal four and one sixth.

5. we will find the intersection point between the circle x square plus y square equal twenty and y equal x plus one.

• Substitution y equal x plus one to the circle x square plus y square equal twenty
• We get x square plus open bracket x plus one close bracket square equal twenty
• Than find the equation, x square plus x square plus two times x plus one equal twenty.
• Add two side with negative twenty, so two times x square plus two times x minus nineteen equal zero.
• Than find the value of x with ABC formula, first x or second x equal negative b plus or minus square root of open bracket b square minus four times a times c close bracket all over two times a
• First x or second x equal negative two plus or minus square root of two square minus four times two times negative nineteen in bracket all over two times two
• First x or second x equal negative two plus or minus square root of one hundred fifty six all over four
• Square root of one hundred fifty six equal two times square root of thirty nine, so first x or second x equal negative two plus or minus two times square root of thirty nine all over four.
• So first x equal negative one plus square root of thirty nine all over two and second x equal negative one minus square root of thirty nine all over two.

Exercise II

A.Triangle
- Definition : triangle is plane figure with three straight sides.
- Pythagorean theorem,” the square on the hypotenuse of a right triangle is equal to the sum of the square on the other two sides”.
- On the basis of size of triangles, there are three types of triangles:
1. Equilateral triangle
An equilateral triangle is one with all sides equal in length
2. Isosceles triangle
An isosceles triangles is a triangle with two sides equal in length.
3. Scalene triangle
A scalene triangle is a triangle with no sides equal in length.

• On the basis of the measures of the angles, there are three types of triangle:

1.Obtuse triangle
An obtuse triangle is a triangle which has one obtuse angle and two acute angles.
2.Acute triangle
An acute triangle is a triangle which all of angle is acute angles.
3.Right triangle
A right triangle is a triangle which has one right angle and two acute angle.
- The sum of angels of triangle is equal one hundred and eighty.
- the area of triangle is a half times the length of base times the height.
Problem solves:
1.We have right triangle with hypotenuse seventeen centimeters and height fifteen centimeters.Find the area of right triangle!
Solution:
•According to Pythagorean theorem, the square on the hypotenuse of a right triangle is equal to the sum of the square on the other two sides.
•So we can find the length of base. The length of base is square root of open bracket hypotenuse square minus height square close bracket.
•The length of base equal square root of seventeen square minus fifteen square in bracket. Seventeen square equal two hundred and eighty nine. And fifteen square equal two hundred and twenty five.
•So the length of base is square root of two hundred and eighty nine minus two hundred and twenty five in bracket.
•The length of base equal square root of sixty four. Square root of sixty four equal eight.
•So the length of base is eight.
•the area of triangle is a half times the length of base times the height.
•The area equal a half times eight times fifteen
•a half times eight times fifteen equal sixty
•so the area of square triangle is sixty.
2.Find the perimeter of triangle in number one
Solution:
•The perimeter of triangle is the sum of all sides.
•The perimeter of triangle equal seventeen plus fifteen plus eight
•seventeen plus fifteen plus eight equal forty
•so the perimeter of triangle is forty.

B.Parabola
•Definition : a parabola is the set of point in the plane that are equidistant from a point (the focus) and a line (the direction)
•We can draw parabola graph from square equation
•How to draw parabola graph??
-First find the x intercept of curve, we can find the x intercept of curve when the value of y is zero so the equation equal zero. Than we calculate the factor of the equation, if we can not find the factor, we can use ABC formula. ABC formula,” first x or second x equal negative b plus or minus square root of open bracket b square minus four times a times c close bracket all over two times a. so we can find the x intercept of curve.
-Than find y intercept of curve, we can find the y intercept of curve when the value of x is zero. Than substitution the value of x in the equation so w can find the y intercept of curve.
-We also find point of the top. Let point of the top is (x,y). Value of x is negative b over two times a and value of y equal b square minus four a times c all over negative four times a in bracket.
-Than draw the x intercept of curve, y intercept of curve and point of the top, and draw curve which contain the points.
•Characteristic of parabola:
Let: the equation is a times x square plus b times x plus c
-Above open parabola, when the value of a ( coefficient of x square) is negative.
-Bottom open parabola, when the value of a (coefficient of x square) is positive.

C.Exponent equation
•exponent equation is equation which the base or the power contain x variable.
•let: a to the m, a is the base and m is the power
•the formula of exponent equation:
-if the value of the base more than zero and m,n is rational number.
a.a to the m times a to the n equal a to the m plus n in bracket.
Example: calculate four to the six times four to the three !
four to the six times four to the equal four to the six plus three in bracket, so the solution is four to the nine
b.a to the m over a to the n equal a to the m minus b in bracket
example: calculate seven to the eight over seven to the two!
seven to the eight over seven to the two equal seven to the eight minus two in bracket, so the solution is seven to the six
c.a to the m in bracket to the n equal a to the m times n in bracket.
Example: calculate three to the four in bracket to the three!
three to the four in bracket to the three equal three to the four times three. So the solution is three to the twelve.
d. a times b in bracket to the m equal a to the m times b to the m
e. a over b in bracket to the m equal a to the m over b to the m
f. if a not equal zero than a to the zero equal one
g. if a more than zero and m is rational number than a to the negative m equal one over a to the m
h. if m,n is integer, n more than one and m over n is rational number, than a to the over n in bracket equal n-th root of a to the m.
•thorem:
1. a to the f(x) equal one
if a more than zero, a not equal one and a to the f(x) equal one, than f(x) equal zero
example: find the value of x in equation three to the open bracket x square plus three times x minus ten close bracket equal one!
Answer: if three to the open bracket x square plus three times x minus ten close bracket equal one, so x square plus three times x minus ten equal zero. The factor of x square plus three times x minus ten equal zero is x plus fife in bracket times x minus two in bracket. we get x plus fife equal zero, so the value of x is negative fife. And we get x minus two equal zero, so the value of x is two. The solution x equal fife or x equal two.
2.A to the f(x) equal a to the p
If a more than zero, a not equal one and a to the f(x) equal a to the p, than f(x) equal p
Example: find the value of x in equation: two to the x square minus fife times x in bracket equal two to the six.
Answer: two to the x square minus fife times x in bracket equal two to the six mean x square minus fife times x equal six. Than add two side by negative six. Than we get equation x square minus fife times x minus six equal zero. Factor of the equation is x minus six in bracket times x plus one in bracket. So the value of x is six or negative one.
3.A to the f(x) equal a to the g(x)
If a more than zero, a not equal one and a to the f(x) equal a to the g(x), than f(x) equal g(x)