BAB 1 Sistem Bilangan | Matematika Dasar | Alternatifa

Summary

The script discusses various number systems, focusing on natural numbers, integers, rational numbers, and real numbers. It explains that natural numbers are limited to addition and multiplication, integers include zero and negatives, rational numbers are expressed as fractions, and real numbers encompass all numbers on the number line. The concept of imaginary numbers and complex numbers is also briefly touched upon.

Takeaways

• 🔢 The number system is the foundation of mathematics.
• 🔢 There are various types of numbers, such as integers, natural numbers, real numbers, and rational numbers.
• 1️⃣ Natural numbers, also known as counting numbers, start from 1 and continue sequentially (1, 2, 3, etc.).
• ➕ Natural numbers can only be added or multiplied to remain natural numbers. Subtracting or dividing them can result in non-natural numbers.
• 0️⃣ Integers include natural numbers, zero, and negative numbers. Operations like addition, subtraction, and multiplication between integers result in integers.
• ➗ Dividing integers can result in fractions or decimal numbers, which are not integers.
• 🔢 Rational numbers can be expressed as fractions, with a numerator and a denominator.
• ✔️ Rational numbers can be operated on using basic arithmetic operations (addition, subtraction, multiplication, division) except for taking roots.
• 📐 Real numbers include all numbers on the number line, encompassing both rational and irrational numbers.
• 🔢 Imaginary numbers are defined when the square root of a negative number is taken, and combining real and imaginary numbers forms complex numbers.
• 🔢 Prime numbers are integers greater than 1 that are only divisible by 1 and themselves.

Q & A

What are the types of numbers discussed in the script?

The script discusses various types of numbers including natural numbers, integers, rational numbers, real numbers, complex numbers, and prime numbers.

What is the definition of natural numbers?

Natural numbers are the numbers used for counting, starting from 1, 2, 3, 4, 5, and so on. They are also referred to as counting numbers.

What operations can be performed on natural numbers to still result in natural numbers?

Natural numbers can be added and multiplied to each other to still result in natural numbers. Subtraction and division are not included as they can result in non-natural numbers like negatives or fractions.

What is the term used for the set of numbers that includes natural numbers, zero, and negative numbers?

The set of numbers that includes natural numbers, zero, and negative numbers is called integers.

How are integers different from natural numbers?

Integers include all natural numbers, zero, and negative numbers, whereas natural numbers only include the positive counting numbers starting from one.

What is a rational number?

A rational number is a number that can be expressed as a fraction where both the numerator and the denominator are integers, and the denominator is not zero.

What happens when you divide integers?

When you divide integers, the result is not necessarily an integer. It can be a fraction, a decimal, or an irrational number.

What is the definition of a real number?

A real number is any number that can be found on the number line, including integers, fractions, decimals, and irrational numbers.

What is a complex number?

A complex number is a number of the form a + bi, where a and b are real numbers, and i is the imaginary unit, which is defined as the square root of -1.

What are prime numbers?

Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. Examples include 2, 3, 5, 7, 11, 13, 17, and so on.

What is the difference between rational and irrational numbers?

Rational numbers can be expressed as a fraction of two integers, while irrational numbers cannot be expressed as a simple fraction and have non-repeating, non-terminating decimal expansions.

Outlines

00:00

📚 Introduction to Number Systems

This paragraph introduces the concept of number systems, focusing on the basic types of numbers such as natural numbers, whole numbers, rational numbers, and decimals. It explains that natural numbers are the starting point of counting, beginning with 1, 2, 3, etc., and that they can only be added or multiplied to maintain their natural state. The paragraph also touches on the limitations of other operations like subtraction and division, which can result in non-natural numbers such as negative numbers or fractions.

05:00

🔢 Understanding Whole Numbers

The second paragraph delves into the concept of whole numbers, which include natural numbers, zero, and negative numbers. It emphasizes that whole numbers can be operated on with any arithmetic operation, except division, to maintain their whole number status. Examples are given to illustrate how addition, subtraction, and multiplication of whole numbers result in other whole numbers, while division can lead to fractions, which are not considered whole numbers.

10:03

📈 Rational Numbers and Their Operations

This paragraph discusses rational numbers, which are numbers that can be expressed as fractions. It explains that rational numbers include both integers and proper fractions, and that they can be the result of basic arithmetic operations like addition, subtraction, and multiplication. However, division of rational numbers can lead to irrational numbers if the denominator is in the form of a square root. The paragraph also covers the process of rationalizing the denominator to ensure that the result remains a rational number.

15:07

🌐 Real Numbers and the Number Line

The fourth paragraph explores real numbers, which encompass all numbers that can be represented on the number line. It includes integers, fractions, and irrational numbers. The paragraph highlights that real numbers can be both rational and irrational, and it explains the difference between the two. It also introduces the concept of imaginary numbers, which are numbers that involve the square root of a negative number, and complex numbers, which combine real and imaginary parts.

20:11

🔑 Prime Numbers and Their Uniqueness

The final paragraph concludes with a discussion on prime numbers, which are whole numbers greater than 1 that have no divisors other than 1 and themselves. Examples of prime numbers include 2, 3, 5, 7, 11, 13, 17, and so on. The paragraph emphasizes the unique property of prime numbers in that they cannot be divided evenly by any other numbers, making them fundamental in various mathematical concepts and applications.

Keywords

💡Natural Numbers

Natural numbers are the set of positive integers starting from 1, 2, 3, and so on. They are fundamental in the script as they represent the basic counting numbers taught in early education. In the video, natural numbers are the starting point for discussing various number systems and their operations, such as addition and multiplication.

💡Integers

Integers include all whole numbers, both positive and negative, including zero. They are a broader set than natural numbers, encompassing all numbers without fractions. In the script, integers are introduced as a set that includes natural numbers, zero, and negative numbers, forming a complete set of whole numbers.

💡Rational Numbers

Rational numbers are numbers that can be expressed as a fraction where both the numerator and the denominator are integers, and the denominator is not zero. They can represent both finite and repeating decimals. In the video, rational numbers are explained as a set that includes all numbers that can be expressed as fractions, illustrating the concept with examples like 5/2 and -7/2.

💡Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction of integers. They have non-repeating, non-terminating decimal expansions. The script mentions irrational numbers in the context of square roots of non-perfect squares, such as the square root of 3, which cannot be simplified into a rational number.

💡Real Numbers

Real numbers include all the points on the number line, encompassing both rational and irrational numbers. They represent all possible values that can be measured on a continuous line. In the video, real numbers are described as including all numbers on the number line, highlighting their comprehensive nature.

💡Complex Numbers

Complex numbers are numbers that consist of a real part and an imaginary part, often represented in the form a + bi, where 'a' is the real part, 'b' is the imaginary part, and 'i' is the imaginary unit. The script introduces complex numbers as a combination of real numbers and imaginary numbers, such as 2 + 4√i.

💡Prime Numbers

Prime numbers are natural numbers greater than 1 that have no positive divisors other than 1 and themselves. They are fundamental in number theory and are mentioned in the script as numbers like 2, 3, 5, 7, 11, and 13, which can only be divided evenly by themselves and 1.

💡Imaginary Numbers

Imaginary numbers are numbers that, when squared, result in a negative number. They are represented by the square root of a negative number, such as √(-1), which is denoted as 'i'. In the script, imaginary numbers are introduced as part of complex numbers, where they represent the non-real component.

💡Operations on Numbers

The script discusses various operations that can be performed on numbers, such as addition, subtraction, multiplication, and division. These operations are crucial in understanding how different types of numbers interact and combine to form new numbers, such as rational or irrational results from division.

💡Number Systems

Number systems refer to the way numbers are represented and organized in mathematics. The script explores different number systems, such as natural numbers, integers, rational numbers, and real numbers, each with its own set of rules and properties for operations and representation.

💡Decimal Numbers

Decimal numbers are numbers that include a decimal point, allowing for the representation of fractions and non-whole numbers. The script mentions decimal numbers in the context of division, where the result of dividing two integers might not be a whole number, thus resulting in a decimal.

Highlights

Introduction to different types of numbers in the number system, focusing on natural numbers.

Natural numbers are the basic counting numbers starting from 1, 2, 3, etc.

Natural numbers can only be added or multiplied to remain as natural numbers.

Subtraction and division of natural numbers can result in non-natural numbers like negatives or fractions.

Introduction to whole numbers, which include natural numbers, zero, and negative numbers.

Whole numbers can be operated with any arithmetic operation to remain as whole numbers except division.

Division of whole numbers can result in non-whole numbers like fractions.

Rational numbers are numbers that can be expressed as fractions.

Rational numbers include both integers and fractions, and can be obtained from basic arithmetic operations.

Irrational numbers are numbers that cannot be expressed as fractions, often involving square roots of non-perfect squares.

To rationalize irrational numbers, multiply by the conjugate to eliminate the square root in the denominator.

Real numbers include all numbers on the number line, encompassing both rational and irrational numbers.

Imaginary numbers are numbers that involve the square root of a negative number.

Imaginary numbers are represented as a real number multiplied by the imaginary unit i.

Complex numbers are a combination of real and imaginary numbers.

Prime numbers are whole numbers greater than 1 that have no divisors other than 1 and themselves.

Examples of prime numbers include 2, 3, 5, 7, 11, 13, 17, etc.

Transcripts

sistem bilangan ini awal dari segala

awal gitu ya yang

ada di materi PK ataupun PM tapi ini

lebih ke PK sih sebenarnya kita mengenal

jenis-jenis bilangan gitu ya atau sistem

bilangan Nah jadi pada sistem bilangan

yang kita kenal lebih awal adalah

jenis-jenis bilangannya mungkin

teman-teman pernah mendengar Apa itu

bilangan bulat bilangan asli bilangan

Ril bilangan rasional

gitu bilangan bulat dan sebagainya atau

bilangan desimal bilangan bentuk pecahan

gitu nah Sini gua akan membahas satu

persatu gitu ya pertama kita ke bilangan

asli

bilangan ya

bilangan asli n apa sih ee bilangan asli

itu guu ya Jadi ya bilangan asli itu Ya

bisa dibilang

sebagai bilangan hitung gitu ya yang

waktu ee waktu kecil kita diajarkan oleh

orang tua kita atau guru-guru TK kita

gitu ya bahwa sebenarnya

ee bilangan asli itu adalah bilangan

hitung

ya bilangan

hitung ya Yang mana bilangan hitung ini

dimulai dari apa dari

1 2 3 4 5 dan

seterusnya ya itu bilangan ngitung

bilangan hitung ya bukan bukan bilangan

ngitung ya bilangan Hitung dari 1 2 3 4

5 dan seterusnya gitu

nah apa yang terjadi kalau kita ee

melakukan sebuah operasi bilangan ya

pada bilangan asli ini jadi sesama

bilangan asli itu itu ee dioperasikan ya

dengan operasi bilangan tertentu akan

seperti Apa hasilnya Nah kalau bilangan

asli dengan bilangan asli itu hanya bisa

dijumlahkan dan juga dikalikan agar

hasilnya juga nanti bilangan asli contoh

gini ya jadi gua Tuliskan lagi

keterangannya ya

agar tetap

menghasilkan bilangan

asli ya agar tetap menghasilkan bilangan

asli operasi bilangannya Ya hanya

berlaku

Ya hanya

berlaku

operasi Bil

bilangan

penjumlahan

dan

perkalian Ya ini gua stabilo ya

penjumlahan dan

perkalian sor Emang kenapa kalau

misalkan bukan penjumlahan dan

perkalian contoh misalkan gini kita

ya

coba E mengurangi sama-sama bilangan

asli misalkan 1 -

2 hasilnya berapa -1 gitu Terus kalau

misalkan

e yang contoh yang kedua gitu ya

misalkan 3 Dib dengan 5 Nah ini kan

hasilnya kalau dijadiin ee dibagi gitu

Ya maksudnya 3 5 itu kan nanti hasilnya

0,6 gitu Yang mana keduanya itu bukan

bilangan asli gitu ya Jadi

bukan

bilangan

asli gitu jadi bilangan asli sesamanya

itu hanya terbatas pada operasi

penjumlahan dan perkalian dia enggak

bisa dikurang ataupun dibagi karena apa

ya dia akan berpotensi membentuk selain

bilangan asli ya seperti inilah pada

pengurangan dia berpotensi membentuk

bilangan negatif ya kalau 3/5 E kalau

yang pembagian dia berpotensi membentuk

bilangan pecahan ataupun bilangan

desimal

gitu

ya lanjut

ya yang kedua itu ada

bilangan

bulat

bilangan bulat gitu ya Nah Bil bilangan

bulat gitu ya atau integer sih biasanya

disebutnya tapi ini mungkin kita lebih

familiar dengan istilah bilangan bulat

gitu ya Nah bilangan bulat ini

ya adalah kumpulan dari bilangan asli ya

0 dan bilangan negatif ya terdiri

dari bilangan asli atau sekumpulan gitu

ya

atau terdiri dari ya gini a

ya

terdiri

dari bilangan asli

0

dan

bilangan

negatif gitu ya

nah pasangan bilangan bulat itu kalau

dioperasikan ya itu harus pada operasi

apa saja atau terbatas pada operasi

bilangan apa supaya nanti pasangan

bilangan bulat kalau dioperasikan dia

hasilnya tetap bilangan bulat gitu

nah bilangan bulat ya atau pasangan

bilangan bulat

ya ya pasangan bilangan

bulat pasangan bilangan

bulat

ya

hasilnya akan

tetap bulat

pada

operasi

selain pembagian gitu

ya pada operasi selain P pembagian ya

Operasi selain Sin pembagian gitu nah

selain pembagian itu ada apa aja Ada

penjumlahan pengurangan ya perkalian

gitu ya misalkan gini

e 3 Dik 5 berapa

15 ya atau -1 * 4 ya

-4 gitu Yang ketiga misalkan ee ber apa

nih ya 3 - 6 berarti -3 nah terus ee

berapa 7 +

-10

-3 nah hasil hasil ini itu kan merupakan

bilangan bulat ya kan ada bilangan bulat

positif ada bilangan bulat negatif gitu

ya jadi perkalian pengurangan

penjumlahan itu merupakan operasi yang

bisa menyebabkan pasangan bilangan bulat

hasilnya juga nanti bulat nah sedangkan

kalau misalkan

pembagian

ya pembagian Misalnya di sini 5 / 7

bagaimana kita menuliskan 5 / 7 karena

kan ya keduanya bilangan prima ya ya dan

juga ini pembagian nah bilangan prima

dibagi udah gitu beda lagi ya kan

jadinya kan paling kita bisa menuliskan

5/7 ya paling Banar kita Ubah menjadi

desim berarti apa bukan bilangan bulat

gitu

ya ya bukan bilangan

bulat

bukan

bilangan bulat gitu jadi pembagian itu

enggak bisa menghasilkan bilangan bulat

Jadi kalau misalkan pasangan bilangan

bulat dibagi ya itu hasilnya belum tentu

bulat juga karena bisa jadi dia

menghasilkan bilangan

pecahan lanjut bilangan

rasional

ya

[Musik]

bilangan

rasional apa sih bilangan rasional gitu

ya jadi bilangan rasional itu adalah

sekumpulan bilangan yang bisa

diekspresikan atau yang di apa ya

ditampilkan gitu ya dalam bentuk apa

pecahan gitu ya ya jadi bilangan e

sekumpulan

bilangan

ya ya sekumpulan

bilangan ya

Yang

bisa ya

diekspresikan ya

diekspresikan

dengan

bentuk

pecahan gitu kalau dia enggak bisa di

dalam e di E menjadi bentuk pecahan ya

dia berarti bukan bilangan rasional atau

bilangan irasional gitu Nah misalkan

gini

ya contoh misalkan ya misalkan ee

bilangan bulatnya berapa nih 5 / 2

berarti kan di sini kalau kita

Ekspresikan menjadi 5/

2 misalkan ya - 7 / 2 ya berarti di sini

-

7/2

nah istilahnya dalam bilangan rasional

ya ya atau pecahan Nah ada dua bilangan

di sini ya atas dan bawah di mana atas

ini disebut sebagai

pembilang nah berikutnya yang bawah

adalah

penyebut

gitu ya Nah terus operasi bilangan apa

yang bisa menghasilkan pasangan bilangan

rasional nanti hasilnya juga bilangan

rasional sebenarnya semua operasi basic

gitu ya kayak penjumlahan perkalian

pengurangan pembagian itu bisa semua

kecuali Apa akar kuadrat gitu

ya kecuali bentuk akar gitu jadi pasang

ee kalau

ya

penyebut

bilangan atau penyebut berupa

bilangan bentuk

akar

maka

disebut

Ira ya

irasional

gitu Nah kalau misalkan dia irasional

maka dia harus dirasionalkan yaah gimana

caranya

merasionalkan ya misalkan gini ada

bilangan e 2 Dib dengan ak3 itu kan

berarti jadinya

2/ √3 Nah merasionalkannya berarti apa

mengalikan dengan penyebutnya karena

kita mau menjadikan penyebut ini bukan

bentuk akar gitu kita ingat prinsipnya

bahwa bilangan bentuk akar kalau sukunya

ya suku di dalam akarnya itu sama ya Nah

maka nanti nanti akarnya jadi hilang

gitu jadi dikali dengan

√3 per dikali ak√3 gitu jadi

2√3/ 3 maka penyebutnya bukan lagi

bilangan rasional tetapi bilangan bulat

jadilah di sini sebagai bilangan rasio

rasional ini namanya ee istilahnya

merasionalkan bentuk

akar

gitu nah terus apaagi bilangan Ril

ya di sini Siapa yang apa pernah

berpikir bahwa bilangan asli dengan

bilangan R itu sama beda ya ya Jadi ini

bilangan ya

bilangan Ril ya bilangan R Itu apa semua

bilangan Ya

semua

bilangan yang

ada

yang

ada

di

garis bilangan gitu semua yang ada di

garis bilangan itu termasuk

pada bilangan r gitu ya Ya semua itu

termasuk bilangan Ril gitu bahkan e

termasuk bilangan yang irasional pun

masuk dalam bilangan r gitu Jadi kalau

dibuat garis bilangan di ini

ya ya Nah ini

0 ya let's say ini

1 Ini du ini 3 ini 4 ini -1 ini -2 ini

-3 gitu ya Nah terus di tengah-tengahnya

0 misalkan di sini ada nilai

1/2 nah 1/2 juga bilangan Ril

gitu dia bentuknya pecahan nah terus ada

Katakanlah ada ak√3 √3 itu nilainya 1,7

misalnya di sini ya ak3 jadi bilangan

bentuk akar juga termasuk

bilangan r gitu Jadi ini kalau kita

petakan gitu teman-teman ya ya

E ini gua kasih stabilo ya 0 1 2 3 4 ya

ini ya dari 0 sampai 4 ya termasuk

bilangan

apa

bilangan cacah

ya bilangan cacah jadi 0 itu termasuk

bilangan cacah jadi 0 1 2 3 4 5 dan

seterusnya itu bilangan Ca Caca tapi

kalau cuman dimulai dari

S

ya dan seterusnya itu termasuk

bilangan ya bilangan

asli gitu bilangan as asli Nah terus

kalau misalkan ini gua highlight lagi ya

-3 -2 -1 gitu

ya Nah ini namanya

bilangan bulat

negatif ya

bilangan bulat

negatif gitu nah Sedangkan ini ya yang

setengah bilangan

rasional

ya

bilangan

rasional

nah a wall of number di sini atau semua

ada di sini termasuk bilangan R Adakah

yang tidak R ada ya atau disebut sebagai

bilangan Kompleks bilangan imajiner sih

sebenarnya bilangan imaginer gitu ya ya

bilangan imaginer dulu gitu jadi

bilangan imajiner

tuh

bilangan

imaginer atau disebut sebagai akar I

gitu ya ya sebuah bilangan disebut

imaginer itu ketika ya ak ee di dalam

akar itu ada tanda negatif misalkan akar

-4 memang ada hasil perkalian ya Ee

angka kembar itu e hasilnya negatif

padahal kan apa negatif kalali negatif

itu kan positif gitu Jadi kalau akar

negatif itu enggak ada

hasilnya imaji gitu ya atau imajiner

gitu kan jadi kalau ini dipisah jadinya

4 ya dikali dengan

ak--1 √4nya 2 tapi ak-1nya enggak bisa

jadi di sini disebutnya bilangan

imajiner Nah kalau bilangan imaginer itu

ditambah dengan bilangan bulat

ya atau ditambah dengan bilangan R sor

ya ya bilangan imaginer ditambah dengan

bilangan R Jadinya apa

bilangan

Kompleks ya bilangan Kompleks itu

berarti a d+ e let's say di sini

eh b di* i di mana a ini nanti bilangan

R eh bilangan ya bilangan R nah I adalah

bilangan imajiner jadi misalkan

bentuknya gini 2 +ah ad ee 4 √i ya

berarti di sini Kompleks bilangannya ya

gitu atau ini disebut bilangan Kompleks

ada lagi bilangan

prima

ya ya

bilangan prima Apa itu Prima di sini

Prima itu bilangan

yang bilangan

yang atau bilangan

bulat

yang

habis atau

hanya

habis

dibagi dirinya

[Musik]

sendiri gitu misalnya berapa hanya

dibagi dengan dirinya sendiri misalkan

du 2 3 5

7 berapa lagi Jangan bilang 9 9 itu 3 *

3 ya habis 7 11

13 ya 17 dan lain-lain itulah bilangan

prima Ya hanya habis dibagi dengan

dirinya sendiri ya 2 habis dibagi 2

enggak ada sisa hasilnya berapa sat gu 3

juga sama 5 juga sama dan seterusnya

gitu ya Oke ini dia pengetahuan tentang

sistem bilangan