Representing The Video of Learning Mathematics

Video 1

Solve The Problem Graph Math

Given that the next problem is question 13 on page 411.

13. The figure above 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)?

Solution:

We looking for h(1)…

The first information is we can show the graph of y=g(x) in the co-ordinate system.

The next information is h(x)=g(2x)+2. We looking for h(1), so we can substitute x=1 on the equation, so h(1)=g(2)+2. Now we have g(2), we can look on the graph, if x=2 so y=1, so we can get g(2)=1.

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

h(1)=g(2)+2

h(1)=1+2=3

The another function with no function problem, this is question 13, the another question 13 on page 534.

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

Solution :

We looking for f(3p)… What is f when x = 3p?

Now first information is f(x)=x+1, and then the next piece information is 2f(p)=20, because the question is what is f when x = 3p, so we can start from this equation down here: 2f(p)=20, we can divide by 2, so f(p)=10, and then f(p) is just what is function f(x) of f(p),so we can speak f(p)=p+1=10, then p=9. But this is not the right answer, we looking for f when x = 3p, so if we take this and answer from x = 3p,with p=9, so x=27. Then we can substitute x in the equation f(x)=x+1, so f(27)=27+8=28. And the last answer of looking for is 28.

The next question is function with no function problem, we just take it step by step, this is question 17 on page 412.

17. In the xy–coordinate plane, the graph of x equals y square minus 4 intersect line l at (0,p) and (5,t). What is the greatest possible value of the slope of l?

Solution:

We will be looking for greatest m…

The graph x equals y square minus 4 intersect line l at (0,p) and (5,t), when x=0, y=p and when x = 5, y=t. And the question is what is the greatest slope. So what do we know about the slope? We know that m equals (y-y1)/(x2-x1). The slope is going to be m=(t-p)/5. Numerator is t-p. We can play again the value into this equation intersect at (0,p) and (5,t).

Video 2

Factoring Polynomials

One way to find factors of polynomials is to formed the algebraic long division. For example lets try and see (x-3) is a factor of (x cube minus seven x minus six)? When dividing (x-3) into (x cube minus seven x minus six). First step of the problem make a long division problem for elementary school. There is you dividing (x – 3) into (x cube plus zero x square minus seven x minus six). Zero in there because is no second degree term. Now you must ask yourself what times x give you x cube? Of, course it is x square, and then you multiply (x-3) by x square, which give you (x cube minus 3 x square),you subtract (x cube plus zero x square) to get 3 x square. Bring it down to next term negative seven x, you have (3 x square minus 7 x). Now we begin again, dividing (x-3) into (3 x square minus 7 x). Just looking at the first term 3 x square dividing x is 3x. Multiply (x-3) by 3x. We can get (3 x square minus 9 x). Subtracting you have (2x minus 6). Dividing (x-3) into (2 x minus 6) which equals 2 and without a remainder. So the solution for a long division problem (x cube plus zero x square minus seven x minus six) dividing by (x-3) is (x square plus 3 x plus 2). Since (x-3) divide into (x cube minus 7 x minus 6) with no remainder. Then (x-3) is a factor of (x cube minus 7 x minus 6). The conclusion is (x square plus 3 x plus 2) is also a factor of (x cube minus 7 x minus 6). We know now that (x cube minus 7 x minus 6) equals (x-3) times (x square plus 3 x plus 2). The quadratic expression (x square plus 3 plus 2) can be factored into (x+1) times (x+2). So, (x cube minus 7 x minus 6) equals (x-3) times (x+1) times (x+2). Substitution (x cube minus 7x minus 6) to zero,so we get 0=(x-3)(x+1)(x+2) Thus either (x-3)=0 or (x+1)=0 or (x +2)=0. Solving all this equation of x we get x=3, x=-1, x=-2. The roots of (x cube minus 7 x minus 6) are (3, -1, -2).

Remember!!

¨ 3rd degree equation have 3 roots

¨ Quadratic (2nd degree) equations always have at most 2 roots

¨ A 4th degree equation would have 4 or fewer roots and so on.

¨ The degree of polynomials equation always limits the number of roots.

Long division process for 3rd order polynomial:
1. Find a partial quotient of x square, by dividing x into x cube to get x square.
2. Multiply x square by the divisor and subtract the product from the dividend.
3. Repeat the process until you either “clear it out“ or reach a remainder.

Video 3

Graph of A Rational Function

Graph of a rational function which can have discontinuities because has polynomial in the denominator.
Is possible value x divide by 0
Example :

f(x) equals (x+2) over (x-1)

when x=1,so f(1)= (1+2) over (1-1) equals 3 over zero. That is bad idea.
f(1)=(1+2) over zero is break in function graph.
f(x) = ( x+ 2) over (x-1)
for x=0, f(0)=(0+2) over (0-1) equals -2

insert x=1,so f(1)=1+2 over (1-1) equals 3 over 0, it is impossible.
Rational functions don’t always work in this way! Take graph f(x) = 1 over ( x square plus 1 ). Not all rational functions will give zero in denominator because of the (+1) is never zero.
Rational functions denominator can be zero!
Polynomial have smooth and unbroken curve and for rational function x : zero in the denominator that impossible situation. There is no value for the function.

A break can show up in 2 ways. A simply type break is missing point on the graph.

Example :

y= (x square minus x minus 6) over (x minus 3)

The graph loose like this if x = 3, so (3 square minus 3 minus 6) over (3 minus 3) equals 0 over 0. That is not possible, not feasible, and not allowed.

So that is no way if x = 3. This is a typical example to the missing point syndrome.

y = (3 square minus 3 minus 6) over (3 minus 3) equals 0 over 0

When you see result of 0 over 0 and also tell you direction be possible factor top and bottom of rational function and simplify.

For example:

y = (x square minus x minus 6) over (x minus 3) equals (x minus 3) times (x plus 2) all over (x minus 3) equals (x plus 2).

Video 4

Invers Function

F(x,y) = 0

Function y = f(x) : VLT

1.1 function x=g(y) : HLT : invertible

Look at the graph of function y equals x square

y=2x-1

Look at the graph function!

y=2x-1

1+x=2x

x=1

2x-1=y

2x=y+1

x=1/2(y+1)

x=(1/2)y+1/2

This is a equation, so:

y=(1/2)x+1/2

We have function

f(x)=2x-1

g(x)=(1/2)x+1/2

What is the value of f(g(x))?

So, f(g(x))=2g(x)-1

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

=x+1-1 = x

What is the value of g(f(x))?

So, g(f(x))=(1/2)f(x)+1/2

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

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

g=f^-1

f(g(x))=f(f^-1(x))=x

g(f(x))=f^-1(f(x))=x

Example:

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

Solution:

y(x+2)=x-1

yx+2y=x-1

yx-x=-1-2y

(y-1)x=-1-2y

x=(-1-2y)/(y-1)

y=(-1-2x)/(x-1)

@ x=0, y=-1

@ y=0,

-1-2x=0

-2x=1

x=-(1/2)

V Asym x=1

H Asym @ y=-2

Look at the graph function!

Translate The Mathematics Articles

1. English into Indonesian

JOHANNES KEPLER (1571-1630)

Johannes Kepler was born in Weil der Stadt in Swabia, in southwest Germany. His paternal grandfather, Sebald Kepler, was a respected craftsman who served as mayor of the city; his maternal grandfather, Melchior Guldenmann, was an innkeeper and mayor of the nearby village of Eltingen. His father, Heinrich Kepler, was "an immoral, rough and quarrelsome soldier," according to Kepler, and he described his mother in similar unflattering terms. From 1574 to 1576 Johannes lived with his grandparents; in 1576 his parents moved to nearby Leonberg, where Johannes entered the Latin school. In 1584 he entered the Protestant seminary at Adelberg, and in 1589 he began his university education at the Protestant university of Tübingen. Here he studied theology and read widely. He passed the M.A. examination in 1591.

Kepler's teacher in the mathematical subjects was Michael Maestlin (1550-1635). Maestlin was one of the earliest astronomers to subscribe to Copernicus's heliocentric theory, although in his university lectures he taught only the Ptolemaic system. Only in what we might call graduate seminars did he acquaint his students, among whom was Kepler, with the technical details of the Copernican system.

In 1594 Kepler accepted an appointment as professor of mathematics at the Protestant seminary in Graz (in the Austrian province of Styria). He was also appointed district mathematician and calendar maker. Kepler remained in Graz until 1600, when all Protestants were forced to convert to Catholicism or leave the province, as part of Counter Reformation measures. For six years, Kepler taught arithmetic, geometry (when there were interested students), Virgil, and rhetoric. In his spare time he pursued his private studies in astronomy and astrology. In 1597 Kepler married Barbara Müller. In that same year he published his first important work, The Cosmographic Mystery, in which he argued that the distances of the planets from the Sun in the Copernican system were determined by the five regular solids, if one supposed that a planet's orbit was circumscribed about one solid and inscribed in another.

Except for Mercury, Kepler's construction produced remarkably accurate results. Because of his talent as a mathematician, displayed in this volume, Kepler was invited by Tycho Brahe to Prague to become his assistant and calculate new orbits for the planets from Tycho's observations. Kepler moved to Prague in 1600.

Kepler served as Tycho Brahe's assistant until the latter's death in 1601 and was then appointed Tycho's successor as Imperial Mathematician, the most prestigious appointment in mathematics in Europe. He occupied this post until, in 1612, Emperor Rudolph II was deposed. In Prague Kepler published a number of important books. In 1604 Astronomia pars Optica ("The Optical Part of Astronomy") appeared, in which he treated atmospheric refraction but also treated lenses and gave the modern explanation of the workings of the eye; in 1606 he published De Stella Nova ("Concerning the New Star") on the new star that had appeared in 1604; and in 1609 his Astronomia Nova ("New Astronomy") appeared, which contained his first two laws (planets move in elliptical orbits with the sun as one of the foci, and a planet sweeps out equal areas in equal times). Whereas other astronomers still followed the ancient precept that the study of the planets is a problem only in kinematics, Kepler took an openly dynamic approach.

In 1610 Kepler heard and read about Galileo's discoveries with the spyglass. He quickly composed a long letter of support which he published as Dissertatio cum Nuncio Sidereo ("Conversation with the Sidereal Messenger"), and when, later that year, he obtained the use of a suitable telescope, he published his observations of Jupiter's satellites under the title Narratio de Observatis Quatuor Jovis Satellitibus ("Narration about Four Satellites of Jupiter observed"). These tracts were an enormous support to Galileo, whose discoveries were doubted or denied by many. Both of Kepler's tracts were quickly reprinted in Florence. Kepler went on to provide the beginning of a theory of the telescope in his Dioptrice, published in 1611.

During this period the Keplers had three children (two had been born in Graz but died within months), Susanna (1602), who married Kepler's assistant Jakob Bartsch in 1630, Friedrich (1604-1611), and Ludwig (1607-1663). Kepler's wife, Barbara, died in 1612. In that year Kepler accepted the position of district mathematician in the city of Linz, a position he occupied until 1626. In Linz Kepler married Susanna Reuttinger. The couple had six children, of whom three died very early.

In Linz Kepler published first a work on chronology and the year of Jesus's birth, In German in 1613 and more amply in Latin in 1614: De Vero Anno quo Aeternus Dei Filius Humanam Naturam in Utero Benedictae Virginis Mariae Assumpsit (Concerning the True Year in which the Son of God assumed a Human Nature in the Uterus of the Blessed Virgin Mary"). In this work Kepler demonstrated that the Christian calendar was in error by five years, and that Jesus had been born in 4 BC, a conclusion that is now universally accepted. Between 1617 and 1621 Kepler published Epitome Astronomiae Copernicanae ("Epitome of Copernican Astronomy"), which became the most influential introduction to heliocentric astronomy; in 1619 he published Harmonice Mundi ("Harmony of the World"), in which he derived the heliocentric distances of the planets and their periods from considerations of musical harmony. In this work we find his third law, relating the periods of the planets to their mean orbital radii.

In 1615-16 there was a witch hunt in Kepler's native region, and his own mother was accused of being a witch. It was not until late in 1620 that the proceedings against her ended with her being set free. At her trial, her defense was conducted by her son Johannes.

1618 marked the beginning of the Thirty Years War, a war that devastated the German and Austrian region. Kepler's position in Linz now became progressively worse, as Counter Reformation measures put pressure on Protestants in the Upper Austria province of which Linz was the capital. Because he was a court official, Kepler was exempted from a decree that banished all Protestants from the province, but he nevertheless suffered persecution. During this time Kepler was having his Tabulae Rudolphinae ("Rudolphine Tables") printed, the new tables, based on Tycho Brahe's accurate observations, calculated according to Kepler's elliptical astronomy. When a peasant rebellion broke out and Linz was besieged, a fire destroyed the printer's house and shop, and with it much of the printed edition. Soldiers were garrisoned in Kepler's house. He and his family left Linz in 1626. The Tabulae Rudolphinae were published in Ulm in 1627.

Kepler now had no position and no salary. He tried to obtain appointments from various courts and returned to Prague in an effort to pry salary that was owed him from his years as Imperial Mathematician from the imperial treasury. He died in Regensburg in 1630. Besides the works mentioned here, Kepler published numerous smaller works on a variety of subjects.

Taken from http://galileo.rice.edu/sci/kepler.html

In Indonesian:

JOHANNES KEPLER (1571-1630)

Johannes Kepler lahir di Weil der Stadt di Swebia, di bagian barat daya Jerman. Kakek dari pihak ayahnya, Sebald Kepler, seorang tukang yang dihormati yang menjabat sebagai walikota di kotanya; kakek dari pihak ibunya, Melchior guldenmann, seorang pengurus rumah penginapan dan walikota di dekat desa Eltingen. Ayahnya, Heinrich Kepler, “adalah seorang yang tuna susila, kasar, prajurit yang suka bertengkar,” berdasarkan Kepler, dan dia mendiskripsikan ibunya juga serupa tak menyenangkannya. Dari tahun 1574-1576 Johannes tinggal bersama eyangnya; tahun 1576 orang tuanya pindah ke dekat Leonberg, dimana johannes masuk ke Sekolah Latin. Tahun 1584 dia masuk Sekolah Menengah Protestan di Adelberg, dan tahun 1589 dia memulai pendidikannya di Universitas Protestan Tubingen. Disana dia belajar teologi. Dia lulus ujian M.A. tahun 1591.

Michael Maestlin (1550-1635) adalah guru matematika Kepler. Maestlin adalah salah satu dari pendahulu ahli astronomi yang menggunakan teori Heliosentrik Copernicus, meskipun di kuliahnya dia hanya mengajar sistem Ptolemaic. Hanya murid lulusan sekolah menengah saja yang dia perkenalkan, yaitu diantaranya Kepler, dengan rincian teknis dari Sistem Copernican.

Tahun 1594 Kepler diangkat sebagai profesor matematika di Sekolah Menengah Protestan di Graz (di Austria, Syria). Dia juga menunjuk daerah matematika dan membuat kalender. Kepler tinggal di Graz sampai tahun 1600, ketika seluruh Protestan dipaksa untuk berpindah agama ke Katolik atau meninggalkan daerah itu, bagian dari Counter Reformation. Selama 6 tahun Kepler mengajari aritmatik, geometri(ketika disana banyak murid yang tertarik) dan berpidato. Di waktu luangnya dia mengajarkan astronomi dan astrologi pada murid-murid privatnya. Tahun 1597 Kepler menikah dengan Barbara Muller. Di tahun yang sama dia juga menerbitkan karya besarnya yang pertama yaitu “The Cosmographic Mystery”, dimana dia membantah bahwa jarak planet dari matahari di sistem Copernican yang dideterminkan oleh 5 zat padat biasa, jika satu diandaikan orbit sebuah planet yang dibatasi oleh satu zat padat dan menggoreskan pada yang lain.

Kecuali air raksa, hasil gagasan Kepler sungguh luar biasa akurat. Karena bakatnya sebagai ahli matematika yang menunjukkan prestasi maka Kepler diundang oleh Tycho Brahe ke Prague untuk menjadi asistennya dan menghitung orbit baru untuk planet-planet dari pengamatan Tycho. Kepler pindah ke Prague tahun 1600.

Kepler membantu sebagai asisten Tycho Brahe sampai kemudian Brahe meninggal pada tahun 1601 dan kemudian menunjuk pengganti Tycho sebagai kaisar ahli matematika, pangkat paling tinggi di matematika Eropa. Dia sibuk dengan kedudukannya sampai tahun 1612, Kaisar Rudolph II dipecat. Di Prague Kepler menerbitkan buku-buku penting. Tahun 1604 Astronomia pars Optica (“The Optical Part of Astronomy”) muncul, yang menyuguhkan pembiasan atmosfer tetapi juga menyuguhkan lensa-lensa dan memberikan penjelasan modern dari cara kerja mata; tahun 1606 dia menerbitkan De Stella Nova (“Concerning The New Star”); dan pada tahun 1609 Astronomia Nova (“New Astronomy”) muncul, yang berisi dua hukum pertama(planet-planet bergerak dengan lintasan elips dengan matahari berada di salah satu fokusnya dan Luas daerah yang disapu pada selang waktu yang sama akan selalu sama). Mengingat ahli astronomi lain masih mengikuti aturan kuno yang mempelajari planet-planet bermasalah hanya pada ilmu geraknya, Kepler mengambil pendekatan dinamik.

Tahun 1610 Kepler mendengar dan membaca penemuan Galileo. Dia menggubah sebuah surat panjang dukungan yang dia terbitkan dengan cepat sebagai Dissertatio cum Nuncio Sidereo (“Conversation with the Sidereal Messenger”) dan tahun kemudian, dia menemukan penggunaan dari teleskop, dia menerbitkan hasil pengamatannya yaitu satelit Jupiter dengan judul Narratio de Observatis Quatuor Jovis Satellitibus (“Narration about Four Satellites of Jupiter observed”). Sistem itu sangat mendukung Galileo, orang yang penemuannya diragukan atau ditolak oleh beberapa orang. Keduanya yang dicetak ulang dengan cepat di Florence. Kepler pergi untuk menyediakan permulaan dari teori teleskop pada Dioptrice-nya, yang diterbitkan pada tahun 1611.

Selama periode itu Kepler mempunyai tiga orang anak (dua lahir di Graz tetapi meninggal), Susanna(1602), yang menikah dengan asisten Kepler Jacob Bartsch tahun 1630, Friedrich(1604-1611), dan Ludwig(1607-1663). Istri Kepler, Barbara, meninggal pada tahun 1612. Pada tahun itu Kepler menerima jabatan daerah ahli matematika di kota Linz, sampai tahun 1626. Di Linz Kepler menikah dengan Susanna Reuttinger. Pasangan itu menghasilkan 6 anak, 3 meninggal.

Di Linz, Kepler menerbitkan karya yang pertama “Kronologi” dan “Tahun Kelahiran Jesus”, di Jerman pada tahun 1613 dan lebih banyak di Latin tahun 1614: De Vero Anno quo Aeternus Dei Filius Humanam Naturam in Utero Benedictae Virginis Mariae Assumpsit (“Concerning The True Year in which The Sound of God assumed a Human Nature in the Uterus of The Blessed Virgin Mary”). Kepler menunjukkan bahwa kalender orang Kristen terdapat kesalahan selama lima tahun, Jesus lahir di 4 BC, sebuah kesimpulan hingga sekarang dapat diterima secara universal. Diantara tahun 1617-1621 Kepler menerbitkan Epitomo Astronomiae Copernicanae (“Epitome of Copernican Astronomi”) yang menjadi pengaruh perkenalan Heliosentrik Astronomi; tahun 1619 dia menerbitkan Harmonic Mundi (“Harmony of The World”), dimana dia mendapat jarak Heliosentrik dari planet-planet dan periodenya dari mempertimbangkan pertunjukan musik harmoni. Di pekerjaan itu dia menemukan hukum ketiganya, hubungan periode planet untuk rata-rata orbitnya.

Tahun 1615 seorang tukang sihir wanita memburu daerah Kepler, dan ibunya dituduh jadi tersangkanya. Tahun 1620 terjadi perlawanan. Pada akhirnya, anaknya Johannes Kepler, yang menyelamatkan dirinya.

Tahun 1618 permulaan dari 30 tahun perang yang merusak Jerman dan daerah Austria. Jabatan Kepler di Linz menjadi lebih buruk, sebagai Counter Reformation mempengaruhi kaum Protestan di bagian Austria ibukota Linz karena dia adalah pegawai negeri pengadilan Kepler dibebaskan dari dekrit tetapi walaupun demikian dia tetap dianiaya. Saat itu Kepler mencetak Tabulae Rudolphinae (“Rudolphine Tables”), tabel baru berdasarkan pengamatan akurat Tycho Brahe, menghitung keserasian astronomi Kepler. Ketika terjadi pemberontakan, api memusnahkan tempat percetakan, toko, dan banyak edisi cetakan. Prajurit yang ada di dalam kota mengamankan rumah Kepler. Dia dan keluarganya meninggalkan Linz tahun 1626. The Tabulae Rudolphinae diterbitkan di Ulm tahun 1627.

Hingga akhirnya Kepler kehilangan jabatannya dan tidak mendapatkan gajinya. Dia mencoba untuk mendapatkan jabatannya kembali dari berbagai pengadilan dan dikembalikan di Prague untuk membongkar gaji yang dihutang selama bertahun-tahun sebagai kaisar ahli matematika dari perbendaharaan kekaisaran. Dia meninggal di Regensburg tahun 1630. Kepler juga menerbitkan karya-karya yang lebih kecil pada berbagai macam subjek lainnya.

2. Indonesian into English

ILMU KOMPUTER

Disiplin Ilmu Komputer sudah muncul sejak era tahun 1940, seiring dengan berpadunya teori algoritma dan logika matematika, serta ditemukannya komputer elektronik dengan kemampuan penyimpanan program. Adalah Alan Turing dan Kurt Godel, yang pada tahun 1930-an berhasil memadukan algoritma, logika, dan penghitungan matematika serta merealisasikannya dalam sebuah alat atau rule system. Prinsip algoritma yang digunakan adalah dari Ada Lovelace, yang dikembangkan 60 tahun sebelumnya.

Penemu algoritma sendiri yang tercatat dalam sejarah awal adalah dari seorang yang bernama Abu Abdullah Muhammad Ibnu Musa Al-Khwarizmi. Al-Khwarizmi adalah seorang ahli matematika dari Uzbekistan yang hidup di masa tahun 770-840 masehi. Di literatur barat ia lebih terkenal dengan sebutan Algorizm. Kata algoritma sendiri berasal dari sebutannya ini. Sedangkan komputer analog diciptakan oleh Vannevar Bush pada tahun 1920, dan disusul dengan komputer elektronik yang dikembangkan oleh Howard Aiken dan Konrad Zuse tahun 1930.

Kemudian John Von Neumann mendemonstrasikan salah satu karya fenomenalnya pada tahun 1945, yaitu sebuah arsitektur komputer yang disebut "von Neumann machine", dimana program disimpan di memori. Arsitektur komputer inilah yang kemudian digunakan oleh computer modern sampai sekarang.

Tahun 1960 adalah babak baru dimulainya formalisasi Ilmu Komputer. Jurusan Ilmu Komputer pada universitas-universitas mulai marak dibangun. Disiplin ilmu baru ini kemudian terkenal dengan sebutan Ilmu Komputer (Computer Science), Teknik Komputer (Computer Engineering), Komputing (Computing), atau Informatika (Informatics).

Dewasa ini banyak sekali peneliti yang mencoba membuat kajian dan melakukan pendefinisian terhadap Ilmu Komputer. Bagaimanapun juga, dasar Ilmu Komputer adalah matematika dan engineering (teknik). Matematika menyumbangkan metode analisa, dan engineering menyumbangkan metode desain pada bidang ini.

CSAB [3] (Computing Sciences Accreditation Board, http://www.csab.org) membuat definisi menarik tentang Ilmu Komputer:

Ilmu Komputer adalah ilmu pengetahuan yang berhubungan dengan computer dan komputasi. Di dalamnya terdapat teoritika, eksperimen, dan pendesainan komponen, serta termasuk didalamnya hal-hal yang berhubungan dengan: teori-teori untuk memahami komputer device, program, dan system Eksperimen untuk pengembangan dan pengetesan konsep Metodologi desain, algoritma, dan tool untuk merealisasikannya,
Metode analisa untuk melakukan pembuktian bahwa realisasi sudah sesuai
dengan requirement yang diminta.

Beberapa definisi lain yang lebih abstrak adalah:

Ilmu Komputer adalah ilmu yang mempelajari tentang representasi pengatahuan (knowledge representation) dan implementasinya. Atau definisi Ilmu Komputer adalah ilmu yang mempelajari tentang abstraksi dan bagaimana mengendalikan kekompleksan.

Kita bisa simpulkan dari persamaan pemakaian terminologi dan hakekat makna dalam definisi yang digunakan para peneliti diatas, bahwa:

Ilmu Komputer adalah ilmu pengetahuan yang berisi tentang teori, metodologi, desain dan implementasi, berhubungan dengan komputasi, komputer, dan algoritmanya dalam perspektif perangkat lunak (software) maupun perangkat keras (hardware).

Ilmu Komputer adalah ilmu yang mempelajari tentang komputer.

Ilmu Komputer adalah ilmu yang mempelajari tentang bagaimana menulis
program komputer.

Taken from:

http://groups.google.co.id/group/itmania/browse_thread/thread/db3ddc4c5b636630?hl=id&ie=UTF-8&q=aplikasi+logika+matematika#eebe75ce6257f4ce

In English:

COMPUTER SCIENCE

Discipline Computer Science have appeared since 1940, in step with associated of algorithm theory and mathematics logic and also found computer electronik with program storage capacity. He is Alan Turing and Kurt Gudel, which in 1930 successfully combine algorithm, logic and calculation of mathematics and also realize it in an implement or rule system. Algorithm principle that used is from Ada Lovelace, which is developed 60 years ago.

The finder of algorithm which noted in the first history from Abu Abdullah Muhammad Ibn Musa Al Kwarizmi. He is mathematician from Uzbekistan who life in era 770-840 M. In west literature, he more famous with predicate Algorizm. Algorithm came from this. In the act, computer analog was created by Vannevar Bush in 1920, end then computer electronik which developed by Howard Aiken ands Konrad Zuse in 1930.

Then John Von Neumann demonstrate the one of his fenomenal work in 1945, a computer architectur which then used by computer modern until now.

In 1960, new generation is started for fomalitation of Computer Science. The way of Computer Science in the university was established. Discipline this new science then famous with named Computer Science, Computer Engineering, Computing or Informatics.

Nowdays, oodles scientist who try meeting and definite for Computer Science. However, Computer Science principle is mathematics and engineering. Mathematics dedicated analysis method and engineering dedicated desain method for this sector.

Computing Sciences Accreditation Board definite about Computer Science: Computer Science is science that related with computer and computation. There are theoritics, experiment and component desain, and also included the matter that related with: theory-theory to comprehend device computer, program and experiment system for development and examination desain methodology concept, algorithm and tool to realize it, Analysis method for proof that realize is according with requirement that required.

Several other definition for more abstract is: Computer Science is science that learn knowledge representation and this implementation, or definition of Computer Science is science which learn abstraction and how to drive complexation.

We can generalize from terminology usage equation and substance significance definition, that: Computer Science is science that included theory, methodology, desain and implementation, which related with computation, computer and algorithm on the software or hardware.

Computer Science is science to learn how to write computer program.

Computer Science is science to learn using of the applications of computer.

How to Express Mathematics

A. Constraints

Definition : 1. Thing that limits or restricts something, or your freedom to do something

2. Strict control over the way you behave

Sentence : It seems that the unsuccessful of the project for promoting educational change in Indonesia due to the constraints such as the complexity of the educational environment.

B. Merely

Definition : only, simply

Sentence : The result of initiated Lesson Studies can be perceived as merely a starting point.

C. Arah jarum jam : Clock-wise

Definition : it is rotated to the right direction

Sentence : The point P(2,4) is rotated by 90 degrees a direction with the clock-wise.

D. Fungsi tangga : Staircase function

Definition : The function that was defined by
= the biggest smaller integer or be the same as x

Sentence : Draw the staircase function curve of and find the continuitas function in !