GEOMETRY

GEOMETRY

The standard Competences:

Determine the position, distance, and angle that include point, line and plane in the three dimension space.

The Basic Competence:

> Determine the position of point, line and plane in the 3 dimension space

> Determine the distance from point to line and from point to plane

> Determine the size of angle, between line and plane and between two plane

1. POSITION OF POINT, LINE AND PLANE IN SPACE

Point, line and space are parts of the three dimension space. These part are named as space elements. In this chapter, we will study about position of point, line and plane in space. The beginning of this study is discussed about the meaning of point, line and plane, along with the axioms and theorems that relevant with them.

1.1 The Meaning of Point, Line and Plane

Space elements (point, line ,plane) are base term in geometry. As we had known that base term is terminology that only can be described in the following explain.

1. Point

Point only can be identified by the place, but it doesn’t have size(has no dimension). Point is shown by dot sign, then follow with the name of that point. The name of that point usually use capital letter, like A,B,C,P,Q and R.

2. Line

We can lengthen the line(means straight line ) as long as we want. But remember limit of the place that we use to draw the line, a line is drawn by just the part of line. This part of the line is called representative line. Line just has the size of long, and doesn’t have the size of wide. Name of a line can be identified by the name of representative line with use small letter, like g,h,k or say the name of segment line from starting point to the end of point.

3. Plane

Plane can be expanded. Generally a plane just drawn by the representative plane. The representative plane has two size, long and wide. Figure from plane has shape like square or rhombus, rectangle or parallelogram. The name of representative plane is written in the corner of the plane and use character a,b,c or H,U,W or with mention angle points from that plane.

4. Axioms of line and plane

Beside of the base terms, study geometry need axioms (usually called postulate). Axiom is statement that consider truth in a system and that truth receive without proof. In space of geometry, there are three important axioms. Those axioms are introduced by Euclides (300BC), he is mathematician from Alexandria.

These are the Euclides’s axioms :

Axiom 1 : Across two any kind of points, it can be made a straight line.

Axiom 2 : If a line and a plane has two alliance points, then that line is on plane.

Axiom 3 : across three any kind of points we can make a plane.

Notes:

1. In axiom 1, two kind of points are that two points are not same

2. In axiom 3, three kind of points are that points are not in one line.

According to the three axioms, then it can be 4 theorems to determine a plane.

Theorem 1: a plane is determined by three any kind of points

Theorem 2: a plane is determined by a line and a point (the point outside the line).

Theorem 3: a plane is determined by two intersection lines

Theorem 4: a plane is determined by two parallel lines.

Base term about point, line and plane, then axioms and theorems about line and plane are already known. Then we will discuss about position of point, line and plane in three dimension space.

1.2 Position of Point to Line and Point to Plane

1. Position of point to line

> The point is in the line

If line g across point A, then point A is in line g

> The point is in outside the line

If line h does not across point B, then point B is not on line h (outside the line g)

Example, look at cube ABCD-EFGH. Segment AB as a represent of line g.

a) The vertect of cube that on line g are point A and B

b) The vertect of cube that in outside line g are point C,D,E,F,G, and H.

2. Position of Point to Plane

> The point is on plane

If point A can be acrossed by plane a, then point A is on plane a.

> the point is in outside plane

If point B cannot be acrossed by plane b, then point B is in outside plane b

Example, look at cube ABCD-EFGH. Plane DCGH as the representative of plane U.

a) The vertect of cube that on plane U are point C,D,G and H.

b) The vertect of cube that in outside of plane U are point A,B,F and E.

Exercise 1.

1. Use the top of index finger as point A and pencil as line g.

a) Display the position of point A , if point A is on line g!

b) Display the position of point A , if point A in outside line g!

c) Is there any position of point A to another line g?

2. There is a cube ABCD-EFGH, BC represent line k, DE represent line l and AG represent line m. mention the angles of that cube that:

a) on line k

b) in the outside of line k

c) on line l

d) in the outside of line l

e) online m

f) in the outside of line m

3. Use the top of index finger as point A and notebook as plane U.

a) Display the position of point A , if point A is on plane U!

b) Display the position of point A , if point A in outside of plane U!

c) Is there any position of point A to another plane U?

4. There is a regular pyramid T.ABCD

a) Mention the vertect of pyramid that are located in the flanks of side.

b) Mention the vertect of pyramid that are located in the flanks of base.

c) Mention the vertect of pyramid that are located in the planes of side.

d) Mention the vertect of pyramid that are located in the base of plane.

e) Is there any vertect of the pyramid that in the outside of flanks

f) Mention the agle of pyramid that in the outside of the base flank

g) Mention the agle of pyramid that in the outside of base plane

5. There is a cube KLMN-PQRS, plane KLMN represent plane a, plane KLQP represent plane b and plane KMRP represent plane c. Mention the vertect of that cube that:

a) are located on plane a

b) are located in the outside of plane a

c) are located on plane b

d) are located in the outside of plane b

e) are located in plane c

f) are located in the outside of plane c

1.3 Position of Line to Line and Line to Plane

A. Position of Line to Another Line

The position possibility of a line to another line in three dimension space are intersect, parallel or cross.

> Two intersect lines

Line g intersect to line h if line g and line h are located in a plane (remember theorem 3) and have alliance point. In geometry of plane, the alliance point is called the intersection point between two lines.

Notes:

If two lines are intersect in more than one point, so that two line are same.

> Two parallel lines

Line g and line h are parallel lines, if line g and line h are located in a plane (remember theorem 4) and have not any alliance point

> Two across lines

Line g across (not intersect and not parallel) to line h, if line g and line h are not located in a same plane.

Example :

Look at the cube ABCD-EFGH

Side AB represent line g

a) The sides of that cube that intersect to line g are AD,AE,BC and BF.

b) The sides of that cube that parallel to line g are BC, EF and HG.

c) The sides of that cube that across to line g are CG ,DH, EH and FG.

d) Is there any side of cube that same with line g?

> Axiom of two parallel lines :

Axiom 4 :

Across a point that is located in the outside a line , it only can be made a line that parallel with that line.

> Theorems of two parallel lines

According to the position of two parallel lines, it can be pulled same conclusion. The conclusions are described in these theorems:

- Theorem 5 :

If line k parallel to line l, and line l parallel to line m, so line k parallel to line m.

- Theorem 6 :

If line k parallel to line h and intersect line g, line l parallel to line h and intersect line g too, so line k, l and g are located in a plane.

- Theorem 7 :

If line k parallel to line l, and line l pierce plane a, so line k pierce plane a.

THE VIDEOS

Exercise 4:

THE VIDEOS

A. Grammar

> Tipe dasar dari kalimat sederhana (simple sentence).

Kalimat sederhana adalah semua elemen yang paling dasar pada kalimat yang terdiri dari subjek atau predikat.

> Subjek adalah yang melakukan tindakan dari kata kerja pokok.

Subjek sederhana adalah kata benda khusus yang menunjukkan sebuah kegiatan.

Contoh :

The happy little child kicked the gnome over fence.

Subjek : child

Simple subjek : the happy little child

Predikat : kicked

> Predikat dari sebuah kalimat terdiri dari : kata kerja utama dan yang mengikutinya.

Contoh:

The happy little child kicked.

The gnome over fence.

> kalimat sederhana dapat diperoleh tanpa subjek dan predikat.

Kalimat perintah -> memerintah seseorang untuk melakukan sesuatu.

Contoh:

Kick that gnome over the fence.

Kalimat di atas tidak ada subjeknya.

B. Verb ( Kata Kerja)

Kata kerja adalah untuk menjelaskan suatu kejadian.

Kata kerja adalah kata terpenting dalam suatu kalimat.

Contoh :

Dave runs.

Runs adalah kata kerja, karena menjelaskan kejadianyangdilakukan Dave.

Dalam tata bahasa inggris, perubahan bentuk kata kerja menunjukkan ssiapa yang sedamg melakukan kegiatan tersebut.

They, we, I,you -> verb (kk)

She, he and it -> verb (kata kerja +s)

Kata kerja – to be :

- I am , you are, he is, she is, it is -> kata ganti tunggal

- We are, they are -> kata ganti jamak.

D. Basic Trigonometry

Trigonometry is from Greek, that is from trigon and metron.

To remember the function of trigonometry we can use

Soh cah toa

Soh = sine is opposite over hypotenuse

Cah = cosine is adjacent over hypotenuse

Toa = tangent is opposite over adjacent

Sine a = opp/hyp

Cos a = adj/hyp

Tan a = opp/adj.

E. Compound Sentence (Kalimat Majemuk)

Contoh :

It is the end of the world as we know it and I feel fine.

> kalimat di atas terdiri dari 2 klausa dan dihubungkan dengan kata penghubung(and).

> Independent clause adlah klausa yang dapat berdiri dalam kalimat.

> jika dalam kalimat terdiri dari 2 klausa, maka kalimat tersebut disebut kalimat majemuk.

> untuk menggabungkan dua klausa digunakan :

1.titik dua (:)

Contoh : I love my two sisters : they bake me pie.

2.titik koma (;)

Contoh : It’s the end of the world as we know it; I feel fine.

3. garis hubung (-)

Terdiri dari beberapa elemen yang mengejutkan.

> Klausa dependent adalah klausa yang tidak bida berdiri sendiri dan bukan kalimat lengkap.

> Kalimat kompleks adalah kalimat yang terdiri dari klausa independent dan klausa dependent.

F. Limit by Inspection

Limit by inspection can be two condition :

1. x goes to positive or negative infinity.
2. limit involves a polynomial divided by a polynomial

We must see the power of x in the numerator and in the denominator.

1. If the highest power of x is greater in numerator, limit is positive or negative infinity.

Example: limit x -> inf [(x^3+4)/(x^3+x+1)] = inf

2. If the highest power of x in numerator less than in the denominator, the limit is zero.

Example : limit x-> inf [(x^3+3)/(x^3+1)] = 0

3. If the highest power of x in numerator is same as highest power of x in denominator, the quotient of the coefficient of the two highest powers.

Example : limit x -> inf [(4x^3+x^2+1)/(3x^3+4)] = 4/3

G. Pre Calculus Graph

> Graph of a rational function which can have dis continuities because has polynomial in the denominator.

Example :

f(x) = (x+2)/(x-1)

f(1) = the value became (1+2)/(1-1)=3/0 -> bad idea, it will break in the graph

f(0) = (0+2)/(0-1)=-2

> Rational function do not always work this way.

Take graph f(x) = 1/(x^2) not all rational function will give zero in denominator.

> Rational function denominator can be zero.

> example:

y = (x^2-x-6)/(x-3) , if x = 3

y = 0/0

when the result 0/0, we can simplify that equation,

y = (x^2-x-6)/(x-3)

= [(x+2)(x-3)]/(x-3)

y = x+2

H. Trigonometry Function

Trigonometry is study about right triangle. To remember the function of trigonometry we can use soh cah toa.

Trigonometry function is ratio of different sides of triangles.

Trigonometry function :

1. sine theta = opp/hyp

2. cosine theta = adj/hyp

3. tangent theta = opp/adj

4. cosecant theta = hyp/opp

5. secant theta = hyp/adj

6. cotangent theta = adj/opp

notes:

opp = side opposite theta

adj = side adjacent to theta

hyp = hypotenuse

Do You Believe in Me

Exercise 2:

Do You Believe in Me

There is a student, a boy that has no fear to talk in front of a lot of people. He is from Dallas. He stand on stage, talk to all audience to affect them to believe what did he says. He studies in Dallas ISD and joins Charles Rice Learning Centre.

He says that he can do anything, become anything. He says that every single of us can realize our dreams. He says that we must not give up to learn in Dallas. He always says ‘do you believe in me?’.

He tells that each of Dallas people can be graduate, ready for college and get a work place. He ask to the audience to believe that we can reach a highest potential.

He says that people in Dallas must believe that they can be anything what they want. He believes that all of people in Dallas can be a success people with their ability.

Exercise 6:

The definition, explanation and the example of the term questioning by :

1. Tiada yang lain selain dirimu yang akan selalu aku tunggu

-There is no body beside of you that I will always waiting for.

2. Garis singgung lingkaran dalam segitiga.

-tangent of a circle in triangel.

3. Juring lingkaran : sector of circle

Ex : The sector of that circle is not known.

4. berboncengan disaat gerimis dating.

-Give aride whwn drizzle is coming.

5. Garis yang saling bersilangan

-Line that across each other

6. Tali busur : khord

Ex : AB is the khord of circle O

7. Proyeksi orthogonal : orthogonal projection.

Ex : Find the orthogonal projection of x^2+y^2=0

8. Sudut yang mengapit diameter lingkaran.

-Angle that imposed tha diameter of the circle.

9. Menghitung peluang dengan pendekatan frekuensi relative.

- Find the vacant with approximation of relative frequency.

10. Jangkauan interquartil : interquartil range.

Ex : Find the interquartil range from these data.

Representing The Video of Learning Mathematics

1. Video 1

SOLVING PROBLEM IN GRAPH

Problem 1:

The figure shows the graph of y = g(x). if the function h is defined by h(x)= g(2x)+2. What is the value of h(1)?

We are looking for h(1).

We can see in the graph. The next pieces information is h(x) = g(2x)+2. we can say this equation with equation 2.

We are looking for h(1), so we can substitute to the equation 2, and we get

h(x) = g(2x)+2

h(1) = g(2)+2

Then, we will looking for g(2) in the graph. There are 2 in axis x, so we can see the value in axis y, that is g(2) is 1.

Then we change g(2) with 1, so h(1) = 1+2 = 3

So we get h(1) = 3

Problem 2:

Let the function be defined by f(x) = x+1.If 2f(p) = 20. What is the value of f(3p)?

We are looking for f(3p). it means what is f when x = 3p.

We have already known that f(x) = x+1 and 2f(p) = 20.

To find f(3p), first we must find the value of p.

2f(p) = 20, so f(p) = 10

f(p) is just same with f(x) since p equal to x. So let plug-in p as x. Then we can write f(x) = x+1 equal with f(p) = p+1.

In above shows that f(p) = 10 , then f(p) = p+1 = 10.And we get p = 9

We are looking for f(3p), so take p = 9 to x = 3p.

And we get x = 3.9 = 27.

Then we substitute x = 27 to the equation

f(x) = x+1

f(27) = 27+1 = 28

And the answer of f(3p) is 28.

Problem 3:

In the xy-coordinate plane, the graph of x = y^2-4 intersect line l at (0,p) and (5,t). What is the greatest possible value of the slope of l?

We are looking for the greatest slope (m) of l. In the equation shows that x = y^2-4. So we can draw the graph. There are two points that intersect the axis of y, then the graph intersect line l in coordinate (0,p) and(5,t). we rewrite the value of x and y.

If x = 0 y = p

If x = 5 y = t

We are looking for the slope, we know that the slope of a line is the different of y coordinate divided by the different of x coordinate.

m = (y2-y1)/(x2-x1)

We can substitute to the equation of slope

m = (t-p)/5

and what is the maximize, so we must maximize the nominator (t-p). We figure t-p with equation x = y^2-4, so we plug-in the pair of coordinate to the x = y^2-4.

2. Video 2.

FACTORING POLYNOMIAL

One way to find factoring polynomial is with Algebraic long division. For example: Is x-3 a factor of x^3-7x-6.

First we can use method in elementary school by dividing x^2-7x-6 by x-3. Because there is no second degree term, then x^3-0x^2-7x-6 divide by x-3.

Now, what times x to be x^3, of course is x^2 and then multiply with x-3. There is x^3-3x^2. Then subtract the x^3-0x^2-7x-6 with x^3-3x^2, we get 3x^2-7x-6. And we must divide that by x-3, that must 3x. Multiply 3x with x-3, that is 3x^2-9x. Subtracting 3x^2-7x-6 with 3x^2-9x. So it must be 2x-6, then 2x-6 divided by x-3, that is 2. And multiply 2 with x-3, it is 2x-6. then subtract 2x-6 with 2x-6 the answer is 0. So there is no remainder. So the solution of long division problem x^3-7x-6 divide by x-3 is x^2+3x+2.

Since x-3 division to x^3-7x-6, evenly with no remainder the x-3, is a factor of x^3-7x-6. The solution of that problem is x^2+3x+2 is also a factor of x^3-7x-6.

We now know that x^3-7x-6 = (x-3)( x^2+3x+2). The quadratic can be factor into (x+1)(x+2), so x^3-7x-6 = (x-3) (x+1)(x+2).

Since x^3-7x-6 = 0 , we get 0 = (x-3) (x+1)(x+2). Thus x-3 = 0 or x+1 = 0 or x+2 = 0 solve the equation we get x = 3, x = -1 and x = -2.

There are 3 roots for 3^rd degree equation and quadratic (2^nd degree) equation always have at most 2 roots. A 4^th degree equation would have 4 or fewer roots and so on. The degree of a polynomial equation always limits the number of roots.

Let the summary the long division for a 3^rd order polynomial:

1. Find a partial quotient of x^3 by dividing x into x^3 to get x^2.

2. Then multiply x^2 by the divisor and subtract the product from the dividend

3. Repeat the process until you either “clear it out” or reach a remainder.

3. Video 3

GRAPH OF RATIONAL FUNCTION

Graph of a rational function can have discontinuities because a rational function has a polynomial in the denominator.

Example if f(x) = (x+2)/(x-1), when x = 1 the equation become (1+2)/(1-1) which is 3/0, with 0 in the denominator , so that is bad idea. It is bad choice to chose x = 1, it will break in function graph. Example f(x) = (x+2)/(x-1), try to inserting 0 for x, we get -2. So we put point down on the graph at (0,-2). Now try x = 1, equal 3/0. It is impossible, it means that the graph not have any point at x = 1. Rational functions don’t always work this way, not all rational function will give zero in denominator.

Another example, f(x) = 1/(x^2+1). No matter what we chose for x, the denominator will never to be zero. But don’t forget that rational function, denominator can be zero. For polynomial, the graph is smooth and unbroken curve. But for rational function, somehow for x it can be zero in denominator. That is impossible situation, because there is no value for the function and break in the graph.

The break graph can show up in two ways. It is missing point on the graph. Example y = (x^2-x-6)/(x-3), at point x = 3 the graph is break, because if we substitute the x-3 for the equation, the result is 0/0, it also tell we that it should be possible to factor the top and the bottom on rational function and simplify. Other example, y = (x^2-x-6)/(x-3) we can factor, so y = (x-3)(x+2)/(x-3), and simplify y = x+2 .

If y = x +2 it is no problem if substitute x=3. It is the one of the key idea on calculus. Removable singularity is missing point on the graph, then we factor and simplify rational function, division by zero can be avoided.

4. Video 4

INVERS FUNCTION

Let, f(x, y) = 0

y = f(x)

function x = g(y) invertible

Example:

y=2x+1

the invers of y=2x-1 is
2x-1=y
2x =y+1
x =1/2(y+1)
x =(1/2)y+(1/2)

so y=(1/2)x+(1/2)
Then f(x)=2x-1
g(x)=1/2x+1/2
substitute that x equal to g(x)
f(g(x)) =2(1/2x+1/2)-1
=x+1-1
=x
substitute that x equal to f(x)
g(f(x)) =1/2(2x-1)+1/2
=x-1/2+1/2
=x
Then g=f ^-1
f(g(x))=f( f ^-1 (x))
g(f(x))= f ^-1 (f(x))=x

Next Example:

y=(x-1)/(x+2)
We are looking for the invers of y=(x-1)/(x+2)
y(x+2) =x-1
yx+2y =x-1
yx-x =-1-2y
(y-1)x =-1-2y
x=(-1-2y)/(y-1)

so y=(-1-2x)/(x-1)
when x=0 then y=-1
y=0 then -1-2x =0
-2x =1
so we get x =-(1/2)

Translate Mathematics Articles

English to Indonesia

History of Mathematic by Shelley Walsh (2000)

http://homepage.mac.com/

The name given by mathematicians to the numbers we count with is the natural numbers. For a long long time in history these were simply the numbers.

The details of how arithmetic was invented are not known, but it is clear that people have not always done it. The rudiments of addition and subtraction probably go back even to before people had the kind of idea that we have nowadays of number. In early times the idea of number was much more concrete than it is nowadays. Instead of giving a name to how many there was of something, they used something smaller and cheaper to model it like a pile of pebbles or sticks or tally marks.

Archaeologist have found a number of clay tokens that they think were used by the ancient Babylonians for keeping track of how many or various things that they had. Some of these are nearly 10,000 years old. At around 8000 BCE these people started domesticating plants and animals and started being able to produce more that they could eat at once, so they felt the need to keep track of how much they had of various things. It is not known if they even had words for numbers at this time, but that needn't have stopped them. Clay was plentiful and easy to use, so they formed it into various shapes to represent the things that they had. One small ball of clay might represent a certain measure of grain and different shaped piece of clay might represent a lamb.

Using these sorts of methods of accounting one might easily imagine that the idea of adding and subtracting in the physical sense of adding a certain pile of tokens to another piles of pebbles or tokens or taking away some tokens as a way of modeling the similar action with the bigger more valuable items that they represented. Multiplication and Division are more complicated, so they were probably invented much later. Even in early civilizations they didn't view them quite the way we view them nowadays. In ancient Egypt, for example, the basic operations were addition, subtraction, and doubling, and the various problems that we would solve by multiplying were probably viewed as separate individual problems to be solved by the use of these operations.

Throughout history various notations for numbers have been used. The notation system we finally settled on came originally from India. Roman Numerals were used in Europe as late as the early Renaissance, but the dark ages were really only the dark ages for Europe. Even before the fall of Rome the Indians were making good use of this superior system of number notation that we use today and call the Hindu-Arabic system. The reason Arabic is part of the name is that it was introduced to Europe by the Arabs, the people who really were having a good time in the 'dark' ages.

One of the most important of these Arabs for elementary mathematics is Al-Khwarizmi. He wrote an arithmetic book where he explained the Indian number notation system and the methods for doing arithmetic with it. This book became so popular that systematic methods for doing things got named after him and became called algorithms.

Another important person for arithmetic is a man living in Italy in the 13th century named Leonardo of Pisa, known also as Fibonacci . He is probably the European most responsible for the fact that we use Hindu-Arabic numerals and compute the way the Indians and Arabs did instead of on a counting board. His father was a diplomat and he traveled widely with his father and was educated in North Africa and learned about the Hindu-Arabic number notation and computation methods and became convinced that they were greatly superior to the systems used in Europe. When he returned to Italy he wrote a book called Liber Abacci explaining these numbers and how to do arithmetic with them and the book was very influential.

The advantages of this system for addition and subtraction are not a lot, but it is much better for multiplication. From the 13th c. onward the commercial world was getting increasingly complicated, so such things were increasingly important. Because of this Leonardo's book influenced very much the merchants of the time as well as the mathematicians. So a number of people were able to make their living teaching this new arithmetic and writing books about it and new innovations developed so that eventually the methods that you were taught in school evolved.

Sejarah Matematika oleh Shelley Walsh (2000)

Nama yang diberikan oleh matematikawan-matematikawan untuk angka-angka yang kita hitung adalah bilangan-bilangan alami. Sejak jaman dahulu dikenal dengan bilangan sederhana. ( seperti yang kita kenal).

Penemuan secara terperinci tentang bagaimana aritmatika( ilmu hitung) ditemukan tidak diketahui, namun itu telah jelas bahwa orang-orang tidak harus selalu melakukannya. Dasar-dasar penjumlahan dan pengurangan mungkin mundur sebelum orang-orang memiliki bermacam-macam pemikiran tentang bilangan yang telah kita miliki sekarang ini. Di masa dahulu pemikiran ide tentang bilangan lebih konkret dari pada sekarang ini, malahan pemberian nama untuk berapa banyaknya sesuatu, mereka menggunakan sesuatu yang lebih kecil dan murah untuk memperagakannya seperti tumpukan kerikil atau tongkat.

Arkeolog telah menemukan sejumlah mata uang dari tanah liat yang mereka perkirakan digunakan oleh Babilonia kuno untuk menyimpan beberapa banyak atau bermacam benda yang mereka miliki. Beberapa dari itu mendekati 10.000 tahun. Sekitar 8000 BCE masyarakat mulai membudidayakan tanaman dan binatang, dan mulai memproduksi lebih apa yang dapat mereka makan, sehingga mereka merasa perlu untuk mengawasi berapa banyak benda yang mereka miliki. Tidak diketahui jika mereka memiliki kata-kata untuk bilangan pada waktu itu, tetapi itu tidak menghentikan mereka. Tanah liat sangat berlimpah dan mudah untuk digunakan, lalu mereka membentuknya dengan berbagai macam bentuk untuk merepresentasikan apa yang mereka miliki. Sebuah bola kecil dari tanah liat mungkin mewakili sebuah ukuran untuk padi dan potongan bentuk berbeda dari tanah liat mewakili seekor domba.

Meode perhitungan ini mungkin mudah dibayangkan, yaitu pemikiran dari penjumlahan dan pengurangan. Pada pengertian fisika tentang penjumlahan tumpukan mata uang untuk tumpuksn kerikil lainnya atau mengambil beberapa mata uang sebagai jalan penggambaran perbuatan yang sama dengan nilai yang lebih besar dari yang mereka wakili. Perkalian dan pembagian lebih kompleks lagi, maka mungkin mereka menemukannya labih setelah itu. Pada Mesir kuno contohnya, dasar operasi adalah penjumlahan, pengurangan dan doubling dan bermacam-macam masalah yang kita akan selesaikan dengan mengalikan yang mungkin diperlihatkan sebagai permasalahan individu untuk diselesaikan dengan menggunakan operasi ini.

Seluruh sejarah bermacam notasi untuk bilangan telah digunakan. Sistem notasi akhirnya telah kita tetapkan asli datang dari India. Bilangan Roman telah digunakan di Eropa hingga awal Renaissance, tetapi jaman kegelapan hanya untuk zaman kegelapan untuk Eropa. Bahkan sebelum jatuhnya Roma, orang India telah menggunakan system notasi bilangan seperti yang telah kita gunakan sekarang dan dikenal sebagai system Hindu-Arab. Alasan Arab diambil sebagai nama adalah untuk memperkenalkan ke Eropa oleh orang-orang Arab, yaitu oaring-orang yang memiliki waktu yang baik pada “masa kegelapan”.

Salah satu orang Arab yang paling penting untuk matematika dasar adalah Al-Khwarizmi. Dia menulis sebuah buku aritmatika, dimana dia menjelaskan system notasi bilangan Indian dan metode untuk mengerjakan aritmatika dengan itu. Buku ini menjadi sangat popular karena metode yang sistematik untuk menyelesaikan telah memiliki nama dan menjadi algoritma.

Orang penting lainnya untuk aritmatika adalah orang Italia pada abad ke 13 bernama Leonardo, dikenal juga sebagai Fibonacci. Dia adalah orang Eropa yang paling bertanggung jawab untuk fakta bilangan Hindu-Arab. Ayahnya adalah seorang diplomat dan dia berkelana dengan ayahnya dan menimba ilmu di Afrika utara dan mendalami tentang notasi bilangan Hindi-Arab dan metode perhitungan, dan dipastikan bahwa mereka menjadi superior untuk system yang digunakan di Eropa. Ketika dia kembali ke Italia, dia menulis sebuah buku berjudul Liber Abacci yang menjelaskan bilangan ini dan bagaimana mengerjakan aritmatika dengan itu dan buku ini sangat berpengaruh.

Keuntungan dari system ini untuk penjumlahan dan pengurangan tidak banyak.

Dari abad ke 13 kemajuan yang luar biasa dan penting, karena buku Leonardo sangat berpengaruh pada pedagang-pedagang dan para ahli matematika. Sehingga banyak masyarakat mulai mempelajari aritmatika baru dan menulis buku-buku tentang itu dan membuat metode-metode yang sudah kita ketahui.

Indonesia to English

Sumber : Hadiwidjojo,moeharti.1973.Ilmu Ukur Analitik Bidang.Yayasan Pembina FKIE IKIP Yogyakarta .

SUSUNAN KOORDINAT

1. Letak suatu titik pada suatu garis lurus

Untuk menunjukkan letak suatu titik pada suatu garis lurus diambil pada garis itu suatu titik tertentu O yang diberi nama titik asal. Maka letak suatu titik T pada garis itu akan tertentu pula, apabila diketahui jarak titik T dari titik asal O ini. Untuk membedakan apakah titik T terletak di sebelah kiri atau kanan dari O, maka jarak dari T ke O disebelah yang satu dari O diberi tanda positif dan di sebelah lainnya diberi tanda negative.

Pada umumnya diambil jarak-jarak disebelah kanan O positif dan di sebelah kiri O negative. Jika titik asal O ini dinamakan titik nol dan digunakan satuan-satuan panjang , misalnya cm maka setiap titik T pada garis g dapat ditunjukkan letaknya oleh suatu bilangan yang menyatakan jarak OT. Sebaliknya suatu bilangan nyata menunjukkan letak suatu titik T pada garis g tersebut. Bilangan ini disebut koordinat titik itu atau dalam hal ini disebut absis titik itu.

Jika garis g disebut sumbu x, maka untuk menunjukkan letak suatu titik T dapat ditulis T(x) dan x ini adalah absis titik T.

Contoh :

O(0) berarti O berjarak 0 dari titik asal.

T1 (+4)berarti T1 berjarak 4 satuan panjang dari O dan terletak disebelah kanan O.

T2 (-1)berarti T2 berjarak 1 satuan panjang dari O dan terletak disebelah kiri O.

Jadi suatu titik pada suatu garis lurus mempunyai satu koordinat yang disebut absis titik tersebut.

Tampak bahwa setiap titik pada garis menentukan satu bilangan nyata yaitu absisnya, dan sebaliknya setiap bilangan nyata menentukan letak suatu titik. Jadi ada koraspondensi satu-satu antara titik-titik pada garis dan himpunan bilangan-bilangan nyata.

2. Jarak dua buah titik

Apabila kordinat-koordinat 2 buah titik pada suatu garis diketahui, maka jaraj kedua titik itu dapat dihitung. Jarak kedua titik itu tidak diberi tanda positif atau negative, tetapi diambil harga mutlaknya, artinya:

| x | = x , bila x>=0

| x | = -x , bila x<>

Misalnya :

4ada dua buah titik T1 (+5) dan T2(+2). Jarak T1 T2 = 3 ini didapat dari (+2) – (+5) = -3 , yang harga mutlaknya 3.

4 P1 (-1) and P2 (4), jarak P1 P2 = 5. ini didapat dari (4) – (-1)= 5.

4O1 (-3) and O2 (-7), jarak O1 O2= 4. ini didapat dari (-7)

-(-3)= -4, yang harga mutlaknya 4.

Tampak bahwa jika ada dua titik A(x1) and B(x2), maka jarak kedua titik itu sama dengan harga mutlak dari x1- x2 atauAB = | x1- x2 | = | x2- x1|.

Jarak 2 buah titik pada suatu garis sama dengan harga mutlak dari selisih absis-absisnya. Jika ada T1 (x1) dan T2 (x2), maka T1 T2= | x1- x2 | = | x2- x1|.

Catatan :

Jarak 2 buah titik dapat pula diberi tanda positif atau negative jika yang dimaksud dengan T1 T2 adalah jarak dari T1 ke T2 dan yang dimakdud dengan T2 T1 ialah jarak T2 ke T1. dengan jaan ini sekaligus dapat dilihat apakah T1 terletak disebelah kanan atau kiri dari T2, sesuai dengan arah positif atau arah negative dari sumbu x.

Untuk cara ini berlaku : T1T2= x2- x1 and T1T2= x1- x2.

Dapat diperiksa pada contoh-contoh di atas:

T1T2= (+2) – (+5) = -3, berarti T2 di sebelah kiri T1.

P1P2= 5, berarti P2disebelah kanan P1.

COORDINATE STRUCTURE

1. The location of point on straight line

For indicate the location of a point on straight line, we must take a point, for the example point O, that we call domain point. So location for point T on that line will appear, if the distance for point T and O is detected. For differentiate, wether point T on the left side or the right side of point O, we must give plus sign if that point in the right side of point O, and give minus sign if that point in the left side of point O.

Generally the distance in the right side of O is positive , and negative in the left side of point O. If domain point O, we call point zero and use length like cm, so each point T in the line g can be shown that location by the number that explain the distance of OT. On the contrary, each real numbers show the distance of point T on the line g. This number is called coordinate that point or we can call absis that point.

If line g we call axis x, then for indicate point T can be written T(x) and x is an absis for point T.

Example:

O(0) means point O has distance 0 from domain point.

T1 (+4) means T1 has distance 4 from 0 and in the right side of O.

T2 (-1) means T2 has distance 1 from 0 and in the left side of O.

So every point in a straight line has one coordinate that called absis that point.

Appear that each point on straight line determine a real number, it is absis . On the contrary, each real numbers determine location a point. So it has a correspondence between points in the line and real numbers compilation.

2. The distance of two points

If we found coordinates two points on line, so the distance of that two points can be calculated. The distance of two points must not has positive or negative sign, but we take the absolute value, it means

| x | = x , if x>=0

| x | = -x , if x<>

Example:

4There are two points T1 (+5) and T2 (+2). The distance of T1 T2= 3, we can get it from(+2) – (+5) = -3 , has absolute value 3.

4 P1 (-1) and P2 (4), the distance of P1 P2 = 5. It is from (4) – (-1)= 5.

4O1 (-3) and O2 (-7), the distance of O1 O2= 4. Its from (-7)

-(-3)= -4, has absolute value 4.

It shows that if there are two points A(x1) and B(x2), so the distance of both points is equal with absolute value from x1- x2 or AB = | x1- x2 | = | x2- x1|.

The distance of two points in a line equal with absolute value from the difference from that absis. If T1 (x1) and T2 (x2), so T1 T2= | x1- x2 | = | x2- x1|.

Notes:

Distance of two points can contain positive or negative sign, if T1 T2 is distance from T1 to T2. With this way we can see whether T1 on the right side or in the left side of T2, suitable with positive or negative direction from axis x.

For this method equal : T1T2= x2- x1 and T1T2= x1- x2.

We can check from the example in above.

T1T2= (+2) – (+5) = -3, means T2 in the left side of T1.

P1P2= 5, means P2in the right side of P1

This method is not use in middle school.

how to express mathematics

DWI RETNO SARI

07305141026

MATH R 07

A. Bersilangan: Cross

Definition:

In mathematics, cross means two things, like line or figure that they are not intersect and not parallel.

Example :

Line g and line h cross each other.

B. Jangkauan Interquartil: inter quarter range

Definition:

Quarter means one of four equal parts of data in sequence. Inter quarter range means the distance of two quarter.

Example :

Find the inter quarter range from these data.

C. Rudiments

Definition:

The most basic parts or principles of something

Example:

Mathematics is the rudiments of sciences.

D. Perceived

Definition:

Think of something or someone in a particular way

Example:

He perceived how to find the formula of cylinder volume.