the 5th posting

I
1. To prove that square root of two is irrational we can use the contradiction method (proof by Contradiction), which assumes that the opponent's statement is correct. Then we shows that the assumption is that any means of proof is correct.

First, assume that square root of two is rational number that can be formed into fraction (a) (b).
square root of two equal fraction (a) (b)
Move segment and quadrate this equation,it’s be:
two times b square equal a square
Because the left segment is even, the right segment must be even too. So the example a equal 2 times k.
two times b square equal two times k in bracket square
b square equal two times k square
Then lead to b square is also even. This means that b must be even.
This means, on the assumption that this resulted in both a and b must be even. In fact, the a and b is to be relatively prime. If both numbers must be even, it’s meaning that the number should not be simplified. So, there would not be a and b satisfy the conditions square root of two equal fraction (a) (b). So, square root of two is irrational number.

2. To solve that the some angle of triangle is equal to one hundred eighty degree.
triangle is cornered A,B,C.
Then pull the line from the corner to a point in the triangle (it's up to the direction whereit’s line, make sure that point lies in the triangle)
Then, eg in the corner:
Adivided into A1,A2
Bdivided into B1,B2
Cdivided into C1,C2
(written in chronological order)

Then we called that point Oand is certainly O equal three hundred sixty degree
O equal three hundred sixty degree equal open bracket cne hundred eighty degree minus A2 plus B1 in bracket close bracket plus open bracket one hundred eighty degree minus A2 plus C1 in bracket close bracket plus open bracket cne hundred eighty degree minus C2 plus A1 in bracket close bracket.
Three hundred sixty degree equal five hundred fourty degree minus open bracket A1 plus A2 in bracket plus B1 plus B2 in bracket plus C1 plus C2 in bracket close bracket equal five hundred fourty minus open bracket A plus B plus C close bracket
so evident A plus B plus C equal one hundred eighty degree.

3. to get phi we can use a circle with circumference of two times phi times r(radial). If this circle we looking for of lt’s length, we can divide this circumference with this diameter, and the result is phi.

4. find the area of region boundered by the graph y equal x square and y equal x plus two.
First we must looking for the intersection point of y equal x square and y equal x plus two. One of this way is make a equation of y equal x square and y equal x plus two it’s y equal y, we can write it x squre equal x plus two. Move all the variable in one side, x square minus x minus two equal zero. It’s a general equation. From this equation we get tge intersection point. Looking for number which make zero function.
Open bracket x minus two close bracket times open bracket x plus one close bracket equal zero
The result are x equal two and x equal negative one.
This result be limit to looking for area.
After that, we can write it on formula
L equal integral from x equal negative one until x equal two of open bracket x plus two in bracket minus x square close bracket dx, and the result is fraction (thirty seven) (six)

5. To determine the intersection point between the circle x square plus y square equal twenty and y equal x plus one
Substitute y equal x plus one in x square plus y square equal twenty
It’s write
X square plus open bracket x plus one close bracket square equal twenty
X square plus x squareplus two times x plus one equal twenty
Two times x square plus two times x plus one equal twenty
Two times x square plus two times x minus nineteen equal zero
To get the root of this equation we use abc formula.
X one-two equal negative b plus minus square root of open bracket b square minus four times a times c close bracket divided all by two times a
X one-two equal negative two plus minus square root of open bracket four minus four times two times negative nineteen divided all by two times two
X one-two equal negative two plus minus square root of one hundred fifty six divided all by four
X one equal negative two plus square root of one hundred fifty six divided all by four
X two equal negative two minus square root of one hundred fifty six divided all by four
So to get the intersection point we must substitute the result of x to y equal x plus one
Y one equal negative two plus square root of one hundred fifty six divided all by four plus one. And the result is two plus square root of one hundred fifty six divided by four
Y two equal negative two minus square root of one hundred fifty six divided all by four plus one. And the result is two minus square root of one hundred fifty six divided by four.


II
1. modus
definition : the most frequently observed value of the measurments in the relevant set of data.
collect data
modus equal L plus fraction (d1) (d1 plus d2) times c
L is down limit of class modus
D1 is difference frequently of class modus and before class
D2 is difference frequently of class modus and after class
C is length of class
Application
Found the modus from data that given
data freq
41-45 10
46-50 20
51-55 28

56-60 42
61-65 24

Modus equal L plus fraction (d1) (d1 plus d2) times c
Modus equal fifty five point five plus fraction (fourteen) (fourteen plus eighteen) times five
Modus equal fifty five point five plus two point one nine
Modus equal fifty seven point sixty nine

2. trapezium
Definition
Trapesium is a square that has exactly two parallel sides that.
Theorems
In isoceles trapesium , has same base angle.
Theorems
If in a trapesium, has same base abgle it’s isoceles trapesium.
Theorems
If in a trapesium, it’s diagonal has the same length it’s called isoceles trapesium




Formula of the area
L equal half times open bracket length on the top plus length on the bottom close bracket times trapesium high.
Application
long side parallel trapesium are eight and eighteen. the area of trapesium is one hundred and fifty six centimiters square. Count high trapesium.

The answer:
high trapesium equal L times two divide all by open bracket length on the top plus length on the bottom close bracket.
High trapesium equal one hundred fifty six times two divide all by open bracket eight plus eighteen close bracket.
High trapesium equal three hundred twelve divide all by twenty six
High trapesium equal twelve.


.
3. rational number
definition: Real numbers that can be reproduced in the form of a fraction (a) (b) where a and b must be integer.

Application
1.Number of 4. This number can be re-arranged in fraction (4) (1).a = 4 and b = 1. Thus, 4 is the rational number.
2. 0,98787768638 is the rational number because it’s can be re-arranged in fraction (98787768638) (100000000000)