why should I Study for Mathematics?

 

One of the most common misconceptions people have about mathematics is that it is a dull subject, pointless, and too complicated. Not only does mathematics serve an important role in society by providing solutions for things like bridge building and aeronautical engineering, but it also can be quite fun! It has no shortage of beautiful mathematical puzzles such as Fermat’s Last Theorem or the Goldbach Conjecture. Mathematics also gives people freedom from work. No matter how fast computers become, human mathematicians are still necessary as they can help to really push the limits of what is possible. There are modern applications of mathematics that children never even hear about. These include visualization methods in psychology and history, structural analysis in biology and physics, and even cryptography.

Mathematics is not all fun and games though! All students deserve a good old-fashioned challenge every now and then, but with the right study method, mathematics can be a great way to improve test scores while at the same time build up an excellent understanding of how numbers really work. The best way to learn mathematics is not through memorizing equations but by thinking critically about them. In this sense, mathematics is a true art form, as it provides visual representations of abstract concepts.

Not only can mathematics be fun, but it also has practical application in many aspects of life such as surveying and insurance. In fact, mathematics serves an important role as a teaching tool in the classroom at all ages, from kindergarten to higher education. While many people do not realize it, the common core that all states will begin to use in the fall of 2013 was developed through feedback and recommendations from mathematicians over a decade ago at The Mathematical Association of America. Of course there are no hard and fast rules when giving out grades for math classes—there are just guidelines that outline the kind of work that should go into each class. For example, English classes are graded on a curve, but math classes should have no curve. If a student is having trouble grasping something in class, that student should work with the teacher to help them find a solution, not simply give up and leave. That’s where the real power of mathematics lies, in the hands of teachers who can guide students along their paths to understanding—a difficult path, but not impossible to follow.

Mathematics is like a puzzle; once one fact has been discovered it becomes easier to discover another one. Once knowledge is gained about how numbers work together it is easy to understand and apply that knowledge at work or in school.

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TEORI PERBANDINGAN DAN KONVERSI SATUAN

TEORI PERBANDINGAN 

A. Pengertian

Maksudnya adalah membandingkan 2 besaran yang sejenis. 

Perbandingan dinyatakan dalam bentuk:

$a:b=\frac{a}{b}$

Perbandingan dapat disederhanakan dengan cara mengali atau membagi dengan bilangan yang sama.

$a:b=\frac{a}{b}=ma:mb=\frac{a}{m}:\frac{b}{m}$

Contoh: 

1.umur A: umur B = 16 tahun: 24 tahun = 2:3 

2.uang C: uang D = Rp 28.000: Rp 20.000 = 7:5

KONVERSI SATUAN

Satuan panjang

1 yard = 3 feet

1 feet = 12 inch

1 inch = 2,54 cm

1 ha = 10.000 m persegi

1 are = 100 m persegi

Satuan massa 

1 ton = 1000kg 

1 kwintal = 100 kg

1 kg  = 2 pon 

1 pon = 5 ons 

1 pon = 500g

Satuan Volume

1 L = 1 dm kubik

1 ml = 1cc = 1 centimeter cubic

Satuan waktu

1 abad = 100 tahun 

1 windu = 8 tahun

1 lustrum = 5 tahun

1 dasawarsa = 10 tahun 

1 tahun = 12 bulan 

1 tahun = 52 minggu

1 tahun = 365 hari

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Matriks

1. Definisi

Matriks adalah bilangan-bilangan atau angka-angka yang disusun berdasarkan baris dan kolom. banyak baris dan kolom dinamakan ordo.

Contoh:

$A=\begin{pmatrix}1 & 2\\ 2 & 3\end{pmatrix}$

Ordo matrik A adalah 2 x 2

2. Jenis-jenis Matriks

a. Matriks transpose 

Matriks yang elemen baris menjadi kolom dan sebaliknya. 

Contoh :

Misalkan $A=\begin{pmatrix}a & b\\ c & d\end{pmatrix}$ maka $A^{T}=\begin{pmatrix}a & c\\ b & d\end{pmatrix}$

b. Matriks singular

Matriks yang nilai determinannya sama dengan nol. 

c. Matriks identitas 

Matriks identitas adalah matriks bujur sangkar yang diagonalnya bernilai 1 dan yang lain bernilai nol. 

Contoh:

$I=\begin{pmatrix}1 & 0\\ 0 & 1\end{pmatrix}$; $ I=\begin{pmatrix}1 & 0 & 0\\ 0 & 0 & 1\\ 0 & 0 & 1\end{pmatrix}$

3. Kesamaan Matriks 

Dua buah matriks atau lebih dikatakan sama apabila nilai dari elemen-elemen yang seletak bernilai sama dan matriks tersebut harus berordo sama 

4. Operasi Penjumlahan dan Pengurangan Matriks

Syarat: ordo sama

$\begin{pmatrix}a & b\\ c & d\end{pmatrix} \pm \begin{pmatrix}e & f\\ g & h \end{pmatrix}=\begin{pmatrix}a\pm e  & b\pm f\\ c\pm g & d \pm h\end{pmatrix}$

5. Perkalian Skalar dengan Matriks

$k\begin{pmatrix}a & b\\ c & d\end{pmatrix}= \begin{pmatrix}ka & kb\\ kc & kd\end{pmatrix}$

6. Perkalian Dua Matriks 

Dua buah matriks dapat dikalikan apabila matriks pertama memiliki jumlah kolom sama dengan jumlah baris pada matriks kedua. 

Misalkan:

$A=\begin{pmatrix}a & b\\ c & d\end{pmatrix}$ dan $B=\begin{pmatrix}p & q\\ r & s\end{pmatrix}$ maka:

$AB=\begin{pmatrix}a & b\\ c & d\end{pmatrix}\begin{pmatrix}p & q\\ r & s\end{pmatrix}=\begin{pmatrix} ap+br & cp+dr\\ aq+bs & cq+ds\end{pmatrix}$

7. Determinan (det) 

a. Matriks ordo 2x2 

Misalkan $A=\begin{pmatrix}1 & 2\\ 2 & 3\end{pmatrix}$; $det A = ad-bc$

b. Matriks ordo 3 x 3

Misalkan $A=\begin{pmatrix}a & b & c\\ d & e & f\\ g & h & i \end{pmatrix}$

$det A = a.e.i + b.f.g + c.d.h-(c.e.g + a.f.h + b.d.i) $

8. Persamaan Matriks 

Misalkan persamaan matriks: AB = C, maka:

a. $A=CB^{-1}$

b. $B=A^{-1}C$

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LOGIKA MATEMATIKA

A. OPERASI LOGIKA

Misal:

p: hari ini hujan

q: siswa hadir ke sekolah

1. Disjungsi

$p\vee q:$ p atau q

$p\vee q:$ hari ini hujan atau siswa hadir ke sekolah

2. Konjungsi

$p\wedge q:$ p dan q

$p\wedge q:$ hari ini hujan dan siswa hadir ke sekolah

3. Implikasi

$\Rightarrow  q:$ jika p maka q

$\Rightarrow  q:$ jika hari ini hujan maka siswa hadir ke sekolah

4. Biimplikasi

$p\Leftrightarrow q:$ p jika dan hanya jika q

$p\Leftrightarrow q:$ hari ini hujan jika dan hanya jika siswa hadir ke sekolah

5. Negasi

$\sim p:$ negasi p/ ingkaran p/ bukan p/ tidak p

$\sim p:$ hari ini tidak hujan

$\sim \left (\sim p  \right )=p$

B. KUANTOR

🕀$\exists:$ ada, beberapa

🕀$\forall:$ untuk setiap, untuk semua

C. NEGASI PERNYATAAN MAJEMUK

\begin{array}{rcl} \sim \left ( \forall p \right )&\equiv& \exists \left ( \sim p \right )\\ \sim \left (\exists p \right )&\equiv& \forall \left ( \sim p \right )\\ \sim \left ( p\vee  q\right )&\equiv& \sim p\wedge \sim q\\ \sim \left ( p\wedge  q\right )&\equiv& \sim p\vee \sim q\\ \sim\left (p\Rightarrow q  \right )&\equiv&\sim \left ( \sim p\vee q \right )\equiv p\wedge \sim q\end{array}

D. IMPLIKASI, KONVERSE, INVERS, DAN KONTRAPOSISI

Dari suatu implikasi $p\Rightarrow q$, diperoleh:

1. $q\Rightarrow p:$ konvers dari $p\Rightarrow q$

2. $\sim p\Rightarrow \sim q:$ invers dari $p\Rightarrow q$

3. $\sim q\Rightarrow \sim p:$ kontraposisi dari $p\Rightarrow q$

Akan diperoleh $p\Rightarrow q$ sama dengan tabel kebenaran  $\sim q\Rightarrow \sim p$ dan tabel kebenaran $q\Rightarrow p$ sama dengan tabel kebenaran $\sim p\Rightarrow \sim q$.

E. EKUIVALENSI

\begin{array}{rcl} p\Rightarrow q &\equiv&\sim p\vee q\\ p\Rightarrow q &\equiv& \sim q \Rightarrow \sim p \end{array}

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GRADIEN DAN PERSAMAAN GARIS LURUS

A. GRADIEN

🔍 Bentuk umum persamaan garis lurus yaitu:

$y = mx + c$

di mana m = gradien

Jika persamaan garis lurus dalam bentuk $ax + by + c = 0$

Maka mencari gradiennya :

\begin{array}{rcl} ax+by+c &=& 0\\ by &=& -ax-c\\ y &=& -\frac{a}{b}x-\frac{c}{b} \end{array}

🔍  Gradien garis yang melalui dua titik $P\left ( x_{1},y_{1} \right )$ dan $Q\left ( x_{2},y_{2} \right )$ adalah: 

$m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}$

🔍  Hubungan dua garis lurus:

Jika ada dua garis $k:y=m_{1}x+c$ dan $l:y=m_{2}x+c$

$m_{1}=m_{2}$, untuk garis $k\parallel l$.

$m_{1}.m_{2}=-1$, untuk garis $k\perp l$

B. PERSAMAAN GARIS LURUS

 🔍 Persamaan  garis  yang memiliki gradien m dan melalui satu titik $\left ( x_{1},y_{1} \right )$ adalah:

\begin{array}{rcl} y-y_{1}=m(x-x_{1}) \end{array}

Contoh:

Persamaan garis dengan gradien $m = -2$ dan melalui titik $A(-3,4)$ adalah…

Jawab:

Titik $A(-3, 4)$, berarti $x_{1}=-3$, $y_{1}=4$, dan bergradien -2 berarti $m=-2$

\begin{array}{rcl}y-y_{1}&=& m\left ( x-x_{1} \right )\\y-4 &=& -2\left ( x+3 \right )\\y-4 &=& -2x-6\\y &=& -2x-6+4\\y &=& -2x-2\end{array}

  🔍 Persamaan  garis  yang melalui dua titik $P\left ( x_{1},y_{1} \right )$ dan $Q\left ( x_{2},y_{2} \right )$ adalah: 

\begin{array}{rcl}\frac{\left ( y-y_{1} \right )}{\left ( y_{2}-y_{1} \right )}=\frac{\left ( x-x_{1} \right )}{\left ( x_{2}-x_{1} \right )}\end{array}

Contoh:

Tentukan persamaan garis yang melalui titik $A(3,4)$ dan $B(5,8)$.

Jawab:

\begin{array}{rcl} \frac{\left ( y-y_{1} \right )}{\left ( y_{2}-y_{1} \right )}&=& \frac{\left ( x-x_{1} \right )}{\left ( x_{2}-x_{1} \right )}\\ \frac{\left ( y-4 \right )}{\left ( 8-4 \right )}&=& \frac{\left ( x-3 \right )}{\left ( 5-3 \right )}\\ \frac{\left ( y-4 \right )}{4 }&=& \frac{\left ( x-3 \right )}{2}\\ 2\left ( y-4 \right ) &=& 4\left ( x-3 \right )\\ 2y-8 &=& 4x-12\\ 2y-4x &=& -4\\ y-2x &=& -2 \end{array}

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