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Tuesday, August 2, 2011

MENGENAL NOMBOR - MTE 3101


































PENGENALAN


Mengenal nombor adalah satu
kaedah pengenalan kepada asas Matematik iaitu nombor. Kursus ini diambil oleh pelajar Program Ijazah Sarjana Perguruan Pegajian Matematik Rendah di Semester 01 .
1.0 Sejarah Sistem Pernomboran Awal
2.0 Teori Asas Nombor
3.0 Nombor Asli
4.0 Nombor Nisbah
5.0 Nombor Bukan Nisbah
6.0 Nombor Kompleks
7.0 Penganggaran Kuantiti


Tutorial 1

TOPIK : Sistem Awal Nombor

1. Anda dikehendaki membina satu jadual perbandingan antara

1) Sistem pernomboran Tally dan Egypt

2) Sistem pernomboran Mayan dan Roman,

3) Sistem pernomboran Greek dan Hindu –Arabic.

2. Tuliskan refleksi diri mengenai apa yang anda telah pelajari dan bagaimana anda dapat menggunakannya semasa pengajaran di sekolah.

Tutorial 2

TOPIK : Pelbagai Asas Sistem Nombor

1. Penukaran system nombor dari satu asas kepada asas yang lain

a) Asas sepuluh kepada asas dua, empat , lima dan tujuh

i) 21

ii) 357

b) Asas dua kepada asas sepuluh , empat , lapan dan enambelas

i) 1101

ii) 10111

iii) 11101011

c) Asas lapan kepada asas dua dan empat

i) 45

ii) 321

iii) 6017

2. Penambahan dan penolakan nombor asas dua

i) 11 + 10

ii) 1011 + 111

iii) 1101 + 1100

iv) 101 – 11

v) 11011 – 10110

3. Pendaraban nombor asas dua

i) 11 x 10

ii) 101 x 111

iii) 1011 x 1001


NOMBOR NYATA, R

1.4 Tertib Dalam ℜ
Garis nombor nyata
-3 -2 -1 1 2 3 4 5
!#####"#####$ 0 !#####"#####$
ℜ− ℜ+
Nombor nyata yang berpadanan dengan satu titik pada garis itu
dipanggil kordinat bagi titik itu. Titik itu dipanggil graf bagi
nombor itu.
ℜ+ = set bagi nombor nyata positif
ℜ− = set bagi nombor nyata negatif.
Hukum trikotomi
ℜ− ℜ+
0 a
Jika a satu nombor nyata, maka hanya satu daripada kenyataankenyataan
berikut ini benar:
a ialah positif, a ialah sifar atau a ialah negatif.
Kuliah 2 : Bab 1 (sambungan)
2
Hukum tutupan untuk nombor-nombor positif
Jika a dan b itu nombor-nombor nyata positif, maka a + b dan a.b
adalah positif.
Takrif 1.10 : Jika a,b ∈ ℜ maka a kurang daripada b jika dan
hanya jika wujud satu nombor nyata positif d
sedemikian hingga a+d=b
Simbol ketaksamaan
a < b dibaca a kurang daripada b.
a > d dibaca a lebih besar daripada d.
d a b
Teorem 1.12 : Untuk sebarang a,b ∈ ℜ
i) Jika a < b dan b < c, maka a < c
ii) Jika a < b, maka a + c < b + c
iii) Jika a < b dan c > 0 maka ac < bc
iv) Jika a < b dan c < 0 maka ac > bc
Contoh : a) 2 < 7 dan 5 < 7 maka 2 < 7
b) 3 < 8 maka 3 + 5 < 8 + 5 atau 8 < 13
c) 3 < 6 dan 2 > 0 maka 3.2 < 6.2 atau 6 < 12
d) 2 < 7 dan – 2 < 0, maka – 2(2) > - 2(7) atau - 4 > - 14
3
Simbol : ≤ dibaca kurang daripada atau sama dengan.
≥ dibaca lebih besar daripada atau sama dengan.
Contoh : x ≤ 7 dibaca x kurang daripada 7 atau x sama dengan 7.
Nilai mutlak
Tarkif 1.11 Jika a∈ℜ maka nilai mutlak bagi a diberi oleh
  
− <
= ≥
jika 0
jika 0
a a
a a
a
Contoh : −3 = −(3) = 3.
7 = 7
0 = 0
Tafsiran tertib secara geometri
1. a positif a di kanan asalan.
0 a
2. a negatif a dikiri asalan.
a 0
3. a > b graf a terletak di sebelah kanan graf b.
b a
4. a < b graf a terletak disebelah kiri graf b.
a b
4
a c b
5. a < c < b graf c ke kanan graf a dan
kekiri graf b.
6. a < c graf a kurang daripada c
- a 0 a c unit dari asalan.
Tatanda selang Graf
1. (a,b) kurungan terbuka
a b
2. [a,b) kurungan separuh
terbuka/tertutup a b
3. (a,b]
a b
4. [a,b] selang tertutup
a b
5. (-∞,a)
a
6. (-∞,a]
a
7. (a,∞)
a
8. [a,∞)
a
1.5 Nombor Kompleks
Simbol C ialah set bagi nombor kompleks.
Takrif 1: Satu nombor kompleks ialah satu nombor yang berbentuk
Z=a+bi, di mana a,b ∈ℜ.
5
Takrif 2: Jika Z1=a+bi dan Z2=c+di, maka Z1=Z2 jika dan hanya jika
a=c dan b=d.

Takrif 3: Jika Z1=a+bi dan Z2=c+di, maka
i) Z1+Z2=(a+c)+(b+d)i
ii) Z1.Z2=(ac-bd)+(ad+bc)i
Contoh : a) (2+3i)+(6+1i) = (2+6)+(3+1)i
=8+4i
b) (2+3i)(6+1i) = (2.6 – 3.1)+(2.1+3.6)i
= 9+20i
Nombor kompleks a+bi dengan b≠ 0 dipanggil nombor khayal. Jika
a= 0, bi dipanggil nombor khayal tulen.
6
i2 = (0+i)(0+i)
= (0.0 – 1.1)+(0.1+1.0)i
= - 1
dan jika b > 0 nombor nyata, maka
( bi ) = (−1)b = −b
2
Oleh kerana i2 = - 1, i juga di tulis sebagai −1 dan untuk b > 0,
−b = bi2 = b i = i b
Contoh : − 2. −3 = i 2 . i 3
6
2 6
= −
= i
Perhatian: − 2. −3 ≠ (−2)(−3)
Teorem 1 : Jika Z1, Z2 dan Z3 ∈ C dan Z2 , Z3 ≠ 0, maka
2 3
1 3
2
1
Z Z
Z Z
Z
Z
=
Takrif 4 : Konjugat untuk Z= a+bi diberi symbol
_
Z ialah a – bi
Contoh : a) Konjugat untuk 2 + 3i ialah 2 – 3i
b) Konjugat untuk – 3 – i ialah – 3 + i
7
Hasilbahagi nombor kompleks,
c di
a bi
+
+ juga boleh ditulis dalam bentuk
x + yi dengan mendarabkan pembilang dan penyebut dengan konjugat bagi
penyebut iaitu c – di .
Contoh :
i
i

+
3
4
i
i
i
i i i
i
i
i
i
10
7
10
11
10
11 7
9 2
12 4 3 2
3
3
.
3
4
= +
= +

= + + +
+
+

= +
Nombor kompleks adalah suatu medan kerana semua aksiom bagi ℜ
dipenuhi.
Identiti penambahan dalam C ialah 0 + 0i= 0
Identiti pendaraban dalam C ialah 1 + 0i = 1
Songsangan penambahan bagi Z = a + bi, Z ≠ 0 dalam C ialah
i
a b
b
a b
a
Z 2 2 2 2
1
+

+
=
Beza nombor kompleks
Z1 – Z2 = Z1+( – Z2)

Nombor Kompleks

All supplementary chapters contain materials that are part of the standard high school mathematics curriculum, therefore the material is only provided for completeness and should mostly serve as revision.

Although the real numbers can, in some sense, represent any natural quantity, they are in another sense incomplete. We can write certain types of equations with real number coefficients which we desire to solve, but which have no real number solutions. The simplest example of this is the equation:

\begin{matrix} x^2 + 1 &=& 0 \\ x^2 &=& -1 \\ x &=& \sqrt {-1} \end{matrix}

Your high school math teacher may have told you that there is no solution to the above equation. He/she may have even emphasised that there is no real solution. But we can, in fact, extend our system of numbers to include the complex numbers by declaring the solution to that equation to exist, and giving it a name: the imaginary unit, i.

Let's imagine for this chapter that i = \sqrt{-1}exists. Hence x = i is a solution to the above question, and i2 = - 1.

A valid question that one may ask is "Why?". Why is it important that we be able to solve these quadratics with this seemingly artificial construction? It is interesting delve a little further into the reason why this imaginary number was introduced in the first place - it turns out that there was a valid reason why mathematicians realized that such a construct was useful, and could provide deeper insight.

The answer to the question lies not in the solution of quadratics, but rather in the solution of the intersection of a cubic and a line. The mathematician Cardano managed to come up with an ingenious method of solving cubics - much like the quadratic formula, there is also a formula that gives us the roots of cubic equations, although it is far more complicated. Essentially, we can express the solution of a cubic x3 = 3px + 2q in the form

x=\sqrt[3]{q + \sqrt{q^2 - p^3}} + \sqrt[3]{q - \sqrt{q^2-p^3}}

An unsightly expression, indeed!

You should be able to convince yourself that the line y = 3px + 2q must always hit the cubic y = x3. But try solving some equation where q2 < p3, and you run into a problem - the problem is that we are forced to deal with the square root of a negative number. But, we know that in fact there is a solution for x; for example, x3 = 15x + 4 has the solution x = 4.

It became apparent to the mathematician Bombelli that there was some piece of the puzzle that was missing - something that explained how this seemingly perverse operation of taking a square root of a negative number would somehow simplify to a simple answer like 4. This was in fact the motivation for considering imaginary numbers, and opened up a fascinating area of mathematics.

The topic of Complex numbers is very much concerned with this number i. Since this number doesn't exist in this real world, and only lives in our imagination, we call it the imaginary unit. (Note that i is not typically chosen as a variable name for this reason.)

[edit] The imaginary unit

As mentioned above

\begin{matrix} i^2 & = & -1 \end{matrix}.

Let's compute a few more powers of i:

\begin{matrix} i^1 & = & i \\ i^2 & = & -1 \\ i^3 & = & -i\\ i^4 & = & 1\\ i^5 & = & i\\ i^6 & = & -1\\ &\mbox{...}& \end{matrix}

As you may see, there is a pattern to be found in this.

[edit] Exercises

  1. Compute i25
  2. Compute i100
  3. Compute i1000

Exercise Solutions

[edit] Complex numbers as solutions to quadratic equations

Consider the quadratic equation:

\begin{matrix} x^2 - 6x + 13 & =& 0 \\ x & = & \frac{6 \pm \sqrt{36 - 4 \times 13}}{2}  \\ x & = & \frac{6 \pm \sqrt{-16}}{2} \\ x & = & \frac{6 \pm \sqrt{-1}\sqrt{16}}{2} \\ x & = & \frac{6 \pm 4i}{2} \\ x & = & 3 + 2i \ , \ 3 - 2i\\  \end{matrix}

The x we get as a solution is what we call a complex number. (To be nitpicky, the solution set of this equation actually has two complex numbers in it; either is a valid value for x.) It consists of two parts: a real part of 3 and an imaginary part of \pm 2. Let's call the real part a and the imaginary part b; then the sum a+bi = 3 \pm 2iis a complex number.

Notice that by merely defining the square root of negative one, we have already given ourselves the ability to assign a value to a much more complicated, and previously unsolvable, quadratic equation. It turns out that 'any' polynomial equation of degree n has exactly n zeroes if we allow complex numbers; this is called the Fundamental Theorem of Algebra.

We denote the real part by Re. E.g.:

Re(x) = 3

and the imaginary part by Im. E.g.:

\mathrm{Im}(x) = \pm 2

Let's check to see whether x = 3 + 2i really is solution to the equation:

\begin{matrix} x & = &3 + 2i &\\ x^2 & = & (3)^2 + 2(3)(2i) + (2i)^2 \\ & = & 5 + 12i\\ x^2 - 6x + 13 &=& 5 + 12i - 6(3+2i) + 13\\ &=& 0\\  \end{matrix}

[edit] Exercises

  1. Convince yourself that x = 3 - 2i is also a solution to the equation.
  2. Plot the points A(3, 2) and B(3, -2) on a XY plane. Draw a line for each point joining them to the origin.
  3. Compute the length of AO (the distance from point A to the Origin) and BO. Denote them by r1 and r2 respectively. What do you observe?
  4. Compute the angle between each line and the x-axis and denote them by φ1 and φ2. What do you observe?
  5. Consider the complex numbers:

1 comment:

  1. Fundamental of complex number
    (1) A complex number is a number of the form a + bi where a, b are real numbers and i^2=-1.
    (2) The set C of all complex numbers is defined by C = {a+bi;a,b subset of R,and i^2 = -1}
    where a is called the real part of z and Re(Z) and b is called the imaginary part of z and Im(Z) .
    (3) is said to be purely imaginary if and only if Re(z)=0 and Im(Z) = 0 .

    (4) When Im(Z)=0 , the complex number z is real.

    ReplyDelete