Numbers & Number Systems | Criticalyx Mathematics

🎯 Learning Objectives🎯 Objektif Pembelajaran

Classify and operate with rational numbersMengelas dan mengoperasi nombor nisbah

Find HCF and LCM using prime factorisationCari FKT dan FKG menggunakan pemfaktoran perdana

Apply index laws correctlyAplikasi hukum indeks dengan betul

Convert numbers to and from standard formTukar nombor ke dan daripada bentuk piawai

Convert between number bases (base 2, 5, 8, 10)Tukar antara asas nombor (asas 2, 5, 8, 10)

📖 Key Concepts📖 Konsep Utama

Rational Numbers (Form 1)Nombor Nisbah (Tingkatan 1)

Rational numbers include integers, fractions, and terminating/recurring decimals.Nombor nisbah termasuk integer, pecahan, dan perpuluhan tertentu/berulang.

½ = 0.5 (terminating) | ⅓ = 0.333... (recurring) | √2 is IRRATIONAL

Factors & Multiples (Form 1)Faktor & Gandaan (Tingkatan 1)

HCF (Highest Common Factor) and LCM (Lowest Common Multiple) found via prime factorisation.FKT dan FKG ditemui melalui pemfaktoran perdana.

60 = 2² × 3 × 5 → prime factors60 = 2² × 3 × 5 → faktor perdana

12 = 2² × 3 | 18 = 2 × 3²
HCF = 2 × 3 = 6 | LCM = 2² × 3² = 36

Indices (Form 3)Indeks (Tingkatan 3)

Index laws allow us to simplify expressions with powers.Hukum indeks membolehkan kita permudahkan ungkapan dengan kuasa.

xᵐ × xⁿ = xᵐ⁺ⁿ | xᵐ ÷ xⁿ = xᵐ⁻ⁿ | (xᵐ)ⁿ = xᵐⁿ
x⁻ⁿ = 1/xⁿ | x^(1/n) = ⁿ√x | x⁰ = 1

Standard Form (Form 3)Bentuk Piawai (Tingkatan 3)

A number in standard form: A × 10ⁿ where 1 ≤ A < 10 and n is an integer.Nombor dalam bentuk piawai: A × 10ⁿ di mana 1 ≤ A < 10 dan n ialah integer.

34 500 = 3.45 × 10⁴ | 0.0021 = 2.1 × 10⁻³

Number Bases (Form 4)Asas Nombor (Tingkatan 4)

Converting between base 10 (denary) and bases 2 (binary), 5, and 8 (octal).Menukar antara asas 10 dan asas 2 (binari), 5, dan 8 (oktal).

1011₂ = 1(8)+0(4)+1(2)+1(1) = 11₁₀
25₈ = 2(8)+5(1) = 21₁₀

✏️ Worked Examples✏️ Contoh Penyelesaian

Simplify 2³ × 2⁵ ÷ 2⁴Permudahkan 2³ × 2⁵ ÷ 2⁴

Step 1: Multiply first — 2³ × 2⁵ = 2³⁺⁵ = 2⁸Langkah 1: Darab dulu — 2³ × 2⁵ = 2³⁺⁵ = 2⁸

Step 2: Divide — 2⁸ ÷ 2⁴ = 2⁸⁻⁴ = 2⁴Langkah 2: Bahagi — 2⁸ ÷ 2⁴ = 2⁸⁻⁴ = 2⁴

Step 3: Evaluate — 2⁴ = 16Langkah 3: Nilaikan — 2⁴ = 16

Express 0.00047 in standard formUngkapkan 0.00047 dalam bentuk piawai

Step 1: Move decimal point 4 places right → A = 4.7Langkah 1: Alih titik perpuluhan 4 tempat ke kanan → A = 4.7

Step 2: 4 places right = negative power (10⁻⁴)Langkah 2: 4 tempat ke kanan = kuasa negatif (10⁻⁴)

4.7 × 10⁻⁴

Convert 11011₂ to base 10Tukarkan 11011₂ ke asas 10

Step 1: Write place values from right: 2⁰, 2¹, 2², 2³, 2⁴Langkah 1: Tulis nilai tempat dari kanan: 2⁰, 2¹, 2², 2³, 2⁴

1(16) + 1(8) + 0(4) + 1(2) + 1(1)

= 16 + 8 + 0 + 2 + 1

27₁₀

Evaluate (3²)³ ÷ 3⁴Nilaikan (3²)³ ÷ 3⁴

Step 1: Power of a power — (3²)³ = 3²ˣ³ = 3⁶Langkah 1: Kuasa kuasa — (3²)³ = 3²ˣ³ = 3⁶

Step 2: Division — 3⁶ ÷ 3⁴ = 3⁶⁻⁴ = 3²Langkah 2: Bahagi — 3⁶ ÷ 3⁴ = 3⁶⁻⁴ = 3²

Find HCF and LCM of 24 and 36Cari FKT dan FKG bagi 24 dan 36

24 = 2³ × 3 | 36 = 2² × 3²

HCF = lowest powers = 2² × 3 = 12FKT = kuasa terendah = 2² × 3 = 12

LCM = highest powers = 2³ × 3² = 72FKG = kuasa tertinggi = 2³ × 3² = 72

HCF = 12, LCM = 72FKT = 12, FKG = 72

⚠️ Common Mistakes⚠️ Kesilapan Biasa

❌ Mixing up index rulesMengelirukan hukum indeks

xᵐ × xⁿ = xᵐ⁺ⁿ (ADD powers, NOT multiply). (xᵐ)ⁿ = xᵐˣⁿ (MULTIPLY powers, NOT add). Remember: same base multiplying → add; power of power → multiply.xᵐ × xⁿ = xᵐ⁺ⁿ (TAMBAH kuasa, BUKAN darab). (xᵐ)ⁿ = xᵐˣⁿ (DARAB kuasa, BUKAN tambah). Ingat: asas sama darab → tambah; kuasa kuasa → darab.

❌ Wrong A value in standard formNilai A salah dalam bentuk piawai

A must satisfy 1 ≤ A < 10. So 23 × 10³ is WRONG → must be 2.3 × 10⁴. And 0.5 × 10² is WRONG → must be 5 × 10¹.A mesti memenuhi 1 ≤ A < 10. Jadi 23 × 10³ SALAH → mesti 2.3 × 10⁴. Dan 0.5 × 10² SALAH → mesti 5 × 10¹.

❌ Number base conversion direction errorsRalat arah penukaran asas nombor

TO base 10: multiply each digit by its place value and add. FROM base 10: repeatedly divide by the new base and collect remainders (read bottom to top).KE asas 10: darab setiap digit dengan nilai tempatnya dan tambah. DARIPADA asas 10: bahagi berulang kali dengan asas baharu dan kumpulkan baki (baca dari bawah ke atas).

📝 SPM Practice📝 Amalan SPM

Paper 1 — Multiple ChoiceKertas 1 — Pilihan Berganda

Evaluate 2⁻³Nilaikan 2⁻³

−8

⅛

−6

Express 56 000 in standard formUngkapkan 56 000 dalam bentuk piawai

56 × 10³

5.6 × 10³

5.6 × 10⁴

0.56 × 10⁵

HCF of 24 and 36 isFKT bagi 24 dan 36 ialah

101₂ + 110₂ = (in base 2)101₂ + 110₂ = (dalam asas 2)

1011₂

211₂

1001₂

1111₂

Simplify (2⁵ × 2³) ÷ 2⁴Permudahkan (2⁵ × 2³) ÷ 2⁴

2⁴

2⁸

2²

2¹²

Paper 2 — Structured QuestionsKertas 2 — Soalan Berstruktur

6 marks6 markah

(a) Express 0.0000723 in standard form. [4 marks]
(b) Hence, evaluate (7.23 × 10⁻⁵) × 10³. Give your answer in standard form. [2 marks](a) Ungkapkan 0.0000723 dalam bentuk piawai. [4 markah]
(b) Seterusnya, nilaikan (7.23 × 10⁻⁵) × 10³. Berikan jawapan dalam bentuk piawai. [2 markah]

Solution (a)Penyelesaian (a)

Move decimal 5 places right to get AAlih titik perpuluhan 5 tempat ke kanan untuk dapatkan A

A = 7.23 ✓ (1 ≤ 7.23 < 10)

5 places right = power of −55 tempat ke kanan = kuasa −5

7.23 × 10⁻⁵ [4 marks4 markah]

Solution (b)Penyelesaian (b)

7.23 × 10⁻⁵ × 10³ = 7.23 × 10⁻⁵⁺³

= 7.23 × 10⁻²

7.23 × 10⁻² [2 marks2 markah]

6 marks6 markah

(a) Convert 47₁₀ to base 5. [3 marks]
(b) Convert 11010₂ to base 10. [3 marks](a) Tukarkan 47₁₀ ke asas 5. [3 markah]
(b) Tukarkan 11010₂ ke asas 10. [3 markah]

Solution (a)Penyelesaian (a)

47 ÷ 5 = 9 remainder 2

9 ÷ 5 = 1 remainder 4

1 ÷ 5 = 0 remainder 1

Read remainders bottom to topBaca baki dari bawah ke atas

142₅ [3 marks3 markah]

Solution (b)Penyelesaian (b)

1(16) + 1(8) + 0(4) + 1(2) + 0(1)

= 16 + 8 + 0 + 2 + 0

26₁₀ [3 marks3 markah]

📋 Formula Summary📋 Ringkasan Formula

Index Laws:Hukum Indeks:
xᵐ × xⁿ = xᵐ⁺ⁿ | xᵐ ÷ xⁿ = xᵐ⁻ⁿ | (xᵐ)ⁿ = xᵐⁿ
x⁻ⁿ = 1/xⁿ | x^(1/n) = ⁿ√x | x⁰ = 1

Standard Form:Bentuk Piawai: A × 10ⁿ where 1 ≤ A < 10

Base Conversion:Penukaran Asas:
To base 10 → multiply place values and addKe asas 10 → darab nilai tempat dan tambah
From base 10 → divide repeatedly, collect remaindersDaripada asas 10 → bahagi berulang, kumpul baki

HCF/LCM:FKT/FG: Use prime factorisationGuna pemfaktoran perdana