REPRESENTING THE VIDEO OF LEARNING MATHEMATICS

1. Solving Problem Graph

The problem is question number 13 on page 411. The question is:

The figure shows the graph of y=g(x), in the function h is defined by h(x)=g(2x)+2, what is the value of h(1)?

Answer: We are looking for h(1). The first information is the graph. The next piece of the information is that h(x)=g(2x)+2. What is h equal when x is equal to 1? So let’s substitute 1 into h(x)=g(2x)+2 à h(1)=g(2)+2. Now, how we figure out g(2) or what is g when x is equal to 2? We have the graph above. When x is equal to 2, y is equal to 1. Then g is equal to 1 when x is equal to 2. So, h(1)=g(2)+2 become h(1)=1+2 and we got h(1)=3.

The next problem is another question 13 on page 534:

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

Answer: We are looking for f(3p) or in another word what is f when x is equal to 3p? The first piece of information is equation of f(x)=x+1 and the next information is 2f(p)=20. To figure out what is f when x=3p, we need to figure out what p is. Let’s start with the equation of 2f(p)=20. Divide both side by 2, we get f(p)=10. Then, f(p) is just what is function f(x) when x is equal to p. Write that equation. Plug in p in the equation f(x)=x+1à f(p)=p+1=10àp=9. That is not the answer because we are looking for f when x=3p. So if we take p=9 to x=3p, we get x=27. We have the equation for function f which is x+1. We know that x=27, so f(27)=27+1=28.

Let’ move to the next question. Question 17 on page 412:

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?

Answer: We are looking for the greatest slope (m). In the xy-coordinate plane, the graph of x=y^2-4 is:

x=y^2-4 intersect line l at two points (0,p) and (5,t). It does not define exactly where those points are. We have the general idea of the line l on this graph.

Rewrite those two points and put it in a little table to help us.

What is the greatest slope? How do we know about the slope. We know that the slope at the line is going to be m=(y2-y1)/(x2-x1). We can put the values of x and y from the coordinate table that apply to line l. So the slope is going to be m=(t-p)/(5-0)=(t-p)/5. If we want to maximize the slope, we need to maximize the numerator (t-p). How we figure out (t-p)? We have the equation x=y^2-4 and the coordinates (0,p) and (5,t) applied to the equation. The points are the intersection of the graph and the line. So we can plug in the coordinates into the equation of x=y^2-4.

2. Factoring Polynomials

One way to define factor of the polynomials is the rule of algebraic long division that looks like the long division you learn, only harder. For an example, let’s try to see:

Is x-3 a factor of x^3-7x-6?

Answer: When dividing x-3 into x^3-7x-6, first setup the problem like a long division problem from elementary school. That is, you dividing x-3 into x^3-7x-6. Now the zero is in there because there is no second degree term. Now you must ask yourself, what times x gives you x^3? Of course it is x^2. So you write x^2 as the part of the quotient and then multiply x-3 by x^2 which gives you x^3-7x, which you subtract from x^3-0x^2 to get 3x^2. Bringing down the next term -7x, you have 3x^2-7x.

Now we begin again dividing x-3 into 3x^2-7x. Just looking at the first term x goes into 3x^2, 3x. The next part of the answer is 3x. Multiply x-3 by 3x for a product of 3x^2-9x. Subtracting with 3x^2-7x you are left with 2x-6. Now we see that x-3 divides evenly into 2x-6 which equal 2, with no remainder. So the solution to the long division problem x^3-7x-6 divided by x-3 is x^2+3x+2.

Since x-3 divided into x^3-7x-6 evenly with no remainder then x-3 is a factor of x^3-7x-6. The quotient which is x^2+3x+2 is also a factor of x^3-7x-6. We know that x^3-7x-6=(x-3)(x^2+3x+2). The quadratic expression (x^2+3x+2) can be factored into (x+1)(x+2). So x^3-7x-6=(x-3)(x+1)(x+2). Setting the factored form of the equation x^3-7x-6 to zero, we get 0=(x-3)(x+1)(x+2). Those either x-3=0, or x+1=0, or x+2=0. Solving all of this equation for x, we get x=3, x= -1, and x= -2. The roots to x^3-7x-6 are 3, -1, and -2.

Now there are three roots for the third degree equation of x^3-7x-6. On the quadratic or second degree equation we look that always have at most two roots. A fourth degree equation would have four or fewer roots and so on. The degree of a polynomial equation always limits the number of roots.

Let’s summarize the long division process for third order polynomial:

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

2. 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. Pre-Calculus Graph

Let’s begin by discussing the graph of a rational function which can have discontinuities. A rational function has a polynomial in the denominator which means you are dividing by something that is valuable quantity.

It’s possible that some value of x will meet to division be zero. Example:

If f(x)=(x+2)/(x-1), when x=1, the function value become f(x)=(1+2)/(1-1)=3/0 with 0 in the denominator. For this function, choosing x=1 is a bad idea.

When there is a bad choice of x when it makes the bottom of the rational function, it shows up a break in a function graph. For example, suppose you to finding the graph f(x)=(x+2)/(x-1). Start with inserting 0 for x. So now we have f(0)=(0+2)/(0-1)=2/(-1)=-1. So, you put off point down on the graph at (0,-2).

Next you try x=1. This time you get f(1)=(1+2)/(1-1)=3/0. That is you know is impossible and it means that you can not compute y-value when x=1.

It also mean that the graph of this function will not have any point for x=1. The graph is separated into two disconnected pieces at x=1.

Take the graph of f(x)=1/(x^2+1). No matter what numbers you choose for x, the denominator will never zero. The graph is smooth and unbroken.

Don’t forget that the general rule on the rational function, you must expect the possibility that the denominator may turn out to be zero.

Review: For polynomials, the graph is a smooth unbroken curve. For a rational function, sometimes the value of x may be zero in the denominator. That is an impossible situation because there is no y for that x. At that point, there is no value for the function and there is a break in the graph.

A break can show up in two ways. The simpler type of a break is just a missing point. This function is an example of this type of break: y=(x^2-x-6)/(x-3).

The gate that appear in the graph is at points where x=3. If you try to substitute 3 for x in the equation, the result is y=(3^2-3-6)/(3-3)=0/0. That is not possible, not feasible, and not allowed. So, there is no y for x=3. That is the typical example of the missing point syndrome and conveniently it always goes with the result like 0/0.

When you see the result of 0/0, it also tells you that it should be possible to factor the top and the bottom of the rational function and simplify. For an example: y=(x^2-x-6)/(x-3)=(x-3)(x-2)/(x-3)=(x+2).

For this kind of the break, the missing point is a loop hole. For the original function without simplify, x=3 is a bad point because it leads to the division be zero, y=0/0. But if you simplify first, then there’s no problem with x=3, y=x+2. It can be one idea or key in calculus.

Removable singularity appears simplest missing points on the graph when x leads to 0/0. For this kind of a break, if you factor and simplify the rational function, the division by zero can be avoided.

4. Inverse Function

We will talk about the inverse function. To talk about it, we need to review the definition of the function. If we have the relation of f(x,y)=o, then function y=f(x). That is 1.1 function if we can represent it in x=g(y). The function y=f(x) is “VLT” (Vertical Line Test). Every vertical line intersect the graph in at most one point. The 1.1 function satisfy “HLT” (Horizontal Line Test).

If we look at function y=x^2, this is not 1.1 function because if we graph any horizontal line on the graph, we get two intersection points.

To make it as a 1.1 function, we need a domain of 0<=x, so the graph will be:

We can figure out the other squared root of the left side so if we have function of 1.1 then the function is “invertible”.

Let’s start out with the function y=2x-1. And let’s look at the graph of that function.

This is a straight line with y intersect (-1) and x intersect (1/2). Look at the line y=x. That line intersect the graph of y=2x-1. So, we get x=2x-1à 1+x=2x à 1=x. So, the intersection of the line y=x and y=2x-1 with x=1 is (1,1).

See this relation for y=x. Write 2x-1=y à 2x=y+1 à x=(y+1)/2 à x=(y/2)+1/2. Changing x to y in the last equation, we get y=(x/2)+1/2. Looking back on the graph, then we get another line. That line contain the point (1,1). The x-intercept is (0,-1) and the y-intercept is (0,1/2). So, the invers line to the given line y=2x-1 passes through the same point.

We have f(x)=2x-1 and on the other hand, we have g(x)=(x/2)+1/2. We want to compute f(g(x)). f(g(x))=2(…)-1=2((x/2)+1/2)-1à x+1-1=x. g(f(x))=1/2(…)+1/2à g(f(x))=x-1/2+1/2=x.

So, the important problem is g=f-1 à f(g(x))=f(f^(-1)(x))=x and g(f(x))=f^(-1)(f(x))=x.

For another example, take y=(x-1)/(x+2). This is the line with a vertical asymptot at the line x=2 and x=1. The x-intercept is equal to 1, so the point is (1,0). The y-intercept is equal to (0,-1/2). And otherwise the function graph is a hyperbola in a second and fourth quadrant and respect to its asymptoth.

Take a look at the equation of y=(x-1)/(x+2). Multiply both side with (x+2), we get y(x+2)=(x-1) à yx+2y=x-1 à yx-x=-1-2y à (y-1)x=-1-2y à x=(-1-2y)/(y-1). Now, interchange the x to y and y to x à y=(-1-2x)/(x-1). When x=0, we get y=-1 and when y=0 we get -1-2x=0 à -2x=-1 à x= -1/2.

With the vertical asymptot at x=1 and horizontal asymptot at y=-2, again we get the hyperbola. If we look carefully, we will see two functions that reflect each other.

The invers function of y=2^x is: .

TRANSLATE A MATHEMATICS ARTICEL

1. Translate an English mathematics article into Indonesian

ARITHMETIC

Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. In common usage, the word refers to a branch of (or the forerunner of) mathematics which records elementary properties of certain operations on numbers. Professional mathematicians sometimes use the term (higher) arithmetic when referring to number theory, but this should not be confused with elementary arithmetic.

a. History

The prehistory of arithmetic is limited to a very small number of small artifacts indicating a clear conception of addition and subtraction, the best-known being the Ishango bone from central Africa, dating from somewhere between 18,000 and 20,000 BC.

It is clear that the Babylonians had solid knowledge of almost all aspects of elementary arithmetic by 1800 BC, although historians can only guess at the methods utilized to generate the arithmetical results - as shown, for instance, in the clay tablet Plimpton 322, which appears to be a list of Pythagorean triples, but with no workings to show how the list was originally produced. Likewise, the Egyptian Rhind Mathematical Papyrus (dating from c. 1650 BC, though evidently a copy of an older text from c. 1850 BC) shows evidence of addition, subtraction, multiplication, and division being used within a unit fraction system.

Nicomachus (c. AD60 - c. AD120) summarised the philosophical Pythagorean approach to numbers, and their relationships to each other, in his Introduction to Arithmetic. At this time, basic arithmetical operations were highly complicated affairs; it was the method known as the "Method of the Indians" (Latin "Modus Indorum") that became the arithmetic that we know today. Indian arithmetic was much simpler than Greek arithmetic due to the simplicity of the Indian number system, which had a zero and place-value notation. The 7th century Syriac bishop Severus Sebhokt mentioned this method with admiration, stating however that the Method of the Indians was beyond description. The Arabs learned this new method and called it "Hesab" or "Hindu Science". Fibonacci (also known as Leonardo of Pisa) introduced the "Method of the Indians" to Europe in 1202. In his book "Liber Abaci", Fibonacci says that, compared with this new method, all other methods had been mistakes. In the Middle Ages, arithmetic was one of the seven liberal arts taught in universities.

Modern algorithms for arithmetic (both for hand and electronic computation) were made possible by the introduction of Hindu-Arabic numerals and decimal place notation for numbers. Hindu-Arabic numeral based arithmetic was developed by the great Indian mathematicians Aryabhatta, Brahmagupta and Bhāskara I. Aryabhatta tried different place value notations and Brahmagupta added zero to the Indian number system. Brahmagupta developed modern multiplication, division, addition and subtraction based on Hindu-Arabic numerals. Although it is now considered elementary, its simplicity is the culmination of thousands of years of mathematical development. By contrast, the ancient mathematician Archimedes devoted an entire work, The Sand Reckoner, to devising a notation for a certain large integer. The flourishing of algebra in the medieval Islamic world and in Renaissance Europe was an outgrowth of the enormous simplification of computation through decimal notation.

b. Decimal arithmetic

Decimal notation constructs all real numbers from the basic digits, the first ten non-negative integers 0,1,2,...,9. A decimal numeral consists of a sequence of these basic digits, with the "denomination" of each digit depending on its position with respect to the decimal point: for example, 507.36 denotes 5 hundreds (10²), plus 0 tens (10¹), plus 7 units (10⁰), plus 3 tenths (10^-1) plus 6 hundredths (10^-2). An essential part of this notation (and a major stumbling block in achieving it) was conceiving of zero as a number comparable to the other basic digits.

Algorism comprises all of the rules of performing arithmetic computations using a decimal system for representing numbers in which numbers written using ten symbols having the values 0 through 9 are combined using a place-value system (positional notation), where each symbol has ten times the weight of the one to its right. This notation allows the addition of arbitrary numbers by adding the digits in each place, which is accomplished with a 10 x 10 addition table. (A sum of digits which exceeds 9 must have its 10-digit carried to the next place leftward.) One can make a similar algorithm for multiplying arbitrary numbers because the set of denominations {...,10²,10,1,10^-1,...} is closed under multiplication. Subtraction and division are achieved by similar, though more complicated algorithms.

c. Arithmetic operations

The traditional arithmetic operations are addition, subtraction, multiplication and division, although more advanced operations (such as manipulations of percentages, square root, exponentiation, and logarithmic functions) are also sometimes included in this subject. Arithmetic is performed according to an order of operations. Any set of objects upon which all four operations of arithmetic can be performed (except division by zero), and wherein these four operations obey the usual laws, is called a field.

(Taken from http://en.wikipedia.org/wiki/Arithmetic)

In Indonesian:

ARITMETIKA

Aritmetika berasal dari bahasa Yunani αριθμός yang berarti “bilangan”. Aritmetika merupakan cabang matematika paling dasar yang tertua. Cabang ini digunakan oleh hampir setiap orang untuk melakukan perhitungan sederhana hingga bisnis dalam kemajuan ilmu pengetahuan. Aritmetika merupakan cikal bakal matematika yang menghasilkan operasi dasar pada bilangan. Ahli matematika profesional kadang menggunakan istilah aritmetika untuk mengartikan kata teori bilangan yang seharusnya tidak rancu dengan aritmetika dasar.

1. Sejarah

Prasejarah aritmetika ditemukan pada sedikit artifak yang mengindikasikan konsep yang jelas tentang penjumlahan. Hasil terbaik yang diketahui ditemukan pada tulang Ishago di Afrika Tengah sekitar tahun 18.000-20.000 SM.

Orang-orang Babilonia mempunyai pegetahuan yang kuat pada hampir setiap aspek dasar aritmetika sekitar tahun 1800 SM. Akan tetapi, ahli sejarah hanya dapat menduga kegunaan metode yang digunakan untuk membangkitkan hasil dari aritmetika, seperti pada tablet tanah Plimpton 322 yang kemudian menjadi daftar tripel Pythagoras, tanpa cara kerja untuk menunjukkan bagaimana daftar tersebut dapat dihasilkan pada awalnya. Papirus Rhind Mesir (tahun 1650 SM, merupakan salinan dari teks yang lebih tua pada 1850 SM) menunjukkan bukti dari penjumlahan, pengurangan, perkalian, dan pembagian yang digunakan pada sistem pecahan satuan.

Nicomachus meringkas filosofi Pythagoras dengan bilangan dan hubungan antara keduanya pada bukunya yang berjudul “Introduction to Arithmetic”. Pada masa ini, operasi dasar aritmetika memiliki tingkat kesulitan yang tinggi. Metode orang India “Modus Indorum”-lah yang kemudian menjadi aritmetika yang kita kenal saat ini. Aritmetika India jauh lebih sederhana dari aritmetika Yunani, seperti yang terlihat pada sistem bilangannya (India) yang mempunyai angka nol dan notasi nilai tempat. Pada abad ke-7, uskup Siria bernama Severus Sebhokt memuji metode sistem bilangan ini dengan menyatakan bahwa metode India adalah deskripsi yang luar biasa. Orang-orang Arab mempelajari metode baru ini dan menamakannya “Hesab” atau “Ilmu Pengetahuan Hindu”. Fibonacci (dikenal sebagai Leonardo dari Pisa) mengenalkan metode India ini ke Eropa pada tahun 1202. Pada bukunya “Liber Abaci”, Fibonacci menyatakan bahwa semua metode memiliki kesalahan jika dibandingkan dengan metode India. Pada zaman pertengahan, aritmetika merupakan satu dari tujuh seni bebas di universitas-universitas.

Algoritma modern pada aritmetika (baik perhitungan manual atau elektronik) dapat disusun berkat pengenalan notasi tempat desimal dan angka Hindu-Arab pada bilangan. Aritmetika angka dasar Hindu-Arab dikembangkan oleh para matematikawan besar India, yaitu Aryabhatta, Brahmagupta, dan Bhaskara. Aryabhatta berusaha untuk membedakan notasi nilai tempat dan Brahmagupta menambahkan nol pada sistem bilangan India. Brahmagupta mengembangkan perkalian, pembagian, penjumlahan, dan pengurangan modern yang berdasar pada angka Hindu-Arab. Walau kini dianggap sebagai dasar, penyederhanaan ini merupakan puncak dari ribuan tahun perkembangan matematika. Archimedes, seorang matematikawan, telah mencurahkan seluruh hasil kerjanya, “The Sand Reckoner”, untuk menemukan notasi pada angka yang lebih besar. Perkembangan aljabar di dunia Islam dan Renaissance Eropa merupakan perkembangan penyederhanaan besar dari perhitungan notasi desimal.

b. Aritmetika desimal

Notasi desimal menggagasi semua bilangan real dari digit yang paling dasar, yaitu 10 bilangan positif pertama: 0, 1, 2,…,9. Bilangan desimal berisi rangkaian dari digit dasar ini dengan “denominasi” (penyebutan) dari tiap digitnya yang tergantung pada posisinya. Misalnya bilangan 507,36 merupakan 5 ratusan (10²), ditambah 0 puluhan (10¹), ditambah 7 satuan (10⁰), ditambah 3 seperpuluhan (10^-1), ditambah 6 seperratusan (10^-2). Bagian penting dari notasi ini (dan hambatan utama untuk mencapainya) adalah penyusunan nol sebagai bilangan yang dapat diperbandingkan dengan digit dasar lainnya.

Algoritma terdiri dari semua aturan dalam menyajikan perhitungan aritmetika dengan menggunakan sistem desimal untuk menampilkan bilangan, dimana bilangan yang ditulis dengan 10 simbol bernilai 0-9 dikombinasikan memakai sistem nilai tempat (notasi posisi). Setiap simbol tersebut mempunyai bobot 10 kali lipat dari satu simbol ke simbol lainnya ke arah samping kanannya. Notasi ini memperbolehkan penjumlahan bilangan sebarang dengan menambah digit pada tiap tempatnya yang disempurnakan dengan tabel penjumlahan 10x10 (jumlah digit yang melebihi 9, 10 digitnya dibawa ke posisi kirinya). Angka 1 dapat membentuk algoritma yang mirip untuk mengalikan bilangan sebarang karena himpunan dari denotasi {…,10².10,1,10^-1,…} mengakhiri sebuah perkalian. Pengurangan dan pembagian didapat dengan cara serupa dengan algoritma yang lebih sukar.

c. Operasi Aritmetika

Yang termasuk operasi aritmetika tradisional adalah penjumlahan, pengurangan, perkalian, dan pembagian. Operasi yang lebih modern (seperti manipulasi persentase, akar kuadrat, eksponensial, dan fungsi logaritma) terkadang termasuk dalam operasi aritmetika tradisional. Aritmetika ditampilkan tergantung pada perintah operasi. Himpunan dimana keempat operasi aritmetika ini dapat ditampilkan (kecuali pembagian dengan nol) dan mengikuti aturan yang berlaku dinamakan himpunan semesta.

2. Translate an Indonesian mathematics article into English

BILANGAN KOMPLEKS

Dalam matematika, bilangan kompleks adalah bilangan yang berbentuk:

a+bi

dimana a dan b adalah bilangan riil, dan i adalah bilangan imajiner tertentu yang mempunyai sifat i ² = −1. Bilangan riil a disebut juga bagian riil dari bilangan kompleks, dan bilangan real b disebut bagian imajiner. Jika pada suatu bilangan kompleks, nilai b adalah 0, maka bilangan kompleks tersebut menjadi sama dengan bilangan real a.

Sebagai contoh, 3 + 2i adalah bilangan kompleks dengan bagian riil 3 dan bagian imajiner 2.

Bilangan kompleks dapat ditambah, dikurang, dikali, dan dibagi seperti bilangan riil; namun bilangan kompleks juga mempunyai sifat-sifat tambahan yang menarik. Misalnya, setiap persamaan aljabar polinomial mempunyai solusi bilangan kompleks, tidak seperti bilangan riil yang hanya memiliki sebagian.

Dalam bidang-bidang tertentu (seperti teknik elektro, dimana i digunakan sebagai simbol untuk arus listrik), bilangan kompleks ditulis a + bj.

a. Notasi dan operasi

Himpunan bilangan kompleks umumnya dinotasikan dengan C, atau . Bilangan real, R, dapat dinyatakan sebagai bagian dari himpunan C dengan menyatakan setiap bilangan real sebagai bilangan kompleks: a = a + 0i.

Bilangan kompleks ditambah, dikurang, dan dikali dengan menggunakan sifat-sifat aljabar seperti asosiatif, komutatif, dan distributif, dan dengan persamaan i ² = −1:

(a + bi) + (c + di) = (a+c) + (b+d)i

(a + bi) − (c + di) = (a−c) + (b−d)i

(a + bi)(c + di) = ac + bci + adi + bd i ² = (ac−bd) + (bc+ad)i

Pembagian bilangan kompleks juga dapat didefinisikan (lihat dibawah). Jadi, himpunan bilangan kompleks membentuk bidang matematika yang, berbeda dengan bilangan real, berupa aljabar tertutup.

Dalam matematika, adjektif "kompleks" berarti bilangan kompleks digunakan sebagai dasar teori angka yang digunakan. Sebagai contoh, analisis kompleks, matriks kompleks, polinomial kompleks, dan aljabar Lie kompleks.

b. Definisi

Definisi formal bilangan kompleks adalah sepasang bilangan real (a, b) dengan operasi sebagai berikut:

(a, b) + (c, d) = (a + c, b +d)

(a, b) . (c, d) = (ac - bd, bc + ad)

Dengan definisi diatas, bilangan-bilangan kompleks yang ada membentuk suatu himpunan bilangan kompleks yang dinotasikan dengan C.

Karena bilangan kompleks a + bi merupakan spesifikasi unik yang berdasarkan sepasang bilangan riil (a, b), bilangan kompleks mempunyai hubungan korespondensi satu-satu dengan titik-titik pada satu bidang yang dinamakan bidang kompleks.

Bilangan riil a dapat disebut juga dengan bilangan kompleks (a, 0), dan dengan cara ini, himpunan bilangan riil R menjadi bagian dari himpunan bilangan kompleks C.

Dalam C, berlaku sebagai berikut:

1) Identitas penjumlahan ("nol"): (0, 0)

2) Identitas perkalian ("satu"): (1, 0)

3) Invers penjumlahan (a,b): (−a, −b)

4) Invers perkalian (reciprocal) bukan nol (a, b): [a/(a^2+b^2),-b/(a^2+b^2)]

(Taken from http://id.wikipedia.org/Bilangan_Kompleks)

In English:

COMPLEX NUMBER

In mathematics, the complex number are the number with the form:

a+bi

where a and b are real numbers, and i is the imaginary number which has a characteristic i² = -1. The real number a is called the real part of the complex number and b is the imaginary part. The complex number is equal to the real number a if the value of b is 0.

For example, 3 + 2i is a complex number with the real part 3 and the imaginary part 2.

Complex number can be added, subtracted, multiplied, and divided as the real number, but it has some interesting additional characteristics. Such as, polynomial algebraic equation has a solution of the complex number, unlike the real number which has a half only.

In some diciplines (such as electrical engineering, where i is a symbol for current), the imaginary unit are written as a + bj.

a. Notation and Operation

The set of the complex number is denoted by C, or . The real number, R, can be called as a subset of C by considering every real number as a complex number :

a = a + 0i.

Complex number are added, suntracted, multiplied, and divided with the laws of algebra such as associative, commutative, and distributive, with the equation i² = -1 :

(a + bi) + (c + di) = (a+c) + (b+d)i

(a + bi) − (c + di) = (a−c) + (b−d)i

(a + bi)(c + di) = ac + bci + adi + bd i ² = (ac−bd) + (bc+ad)i

The division of the complex number can be defined. So that, the set of the complex number can build a mathematics plane that different with the real number as an enclosed algebra.

In mathematics, the adjective of the “complex” has the meaning that the complex number is used as the based of used number theory. For example, complex analysis, complex matrix, complex polynomial, and complex Lie algebra.

b. Definition

The formal definotion of the complex number is a pair of the real number (a, b) with the operation of:

(a, b) + (c, d) = (a + c, b +d)

(a, b) . (c, d) = (ac - bd, bc + ad)

The complex numbers forms the set of complex number denoted by C from above definition.

Since complex number a + bi is an uniquely specified by the real number pair (a, b), the complex number has a relation of of on-on-one corespondence with the points on a plane called the complex plane.

The real number a can be called by the complex number (a, 0). Through this way, the set of the real number R be the subset of the complex number set C.

In C, there is:

1) An additive identity (‘zero”) : (0,0)

2) A multiplicative identity (“one”) : (1,0)

3) An additive inverse (a, b) : (-a, -b)

4) A multiplicative inverse (reciprocal) nonzero (a, b) : [a/(a^2+b^2),-b/(a^2+b^2)]