Number Systems and Decimals

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Number Systems and Decimals — Quick Facts

The number system hierarchy: Natural numbers (1, 2, 3…) → Whole numbers (0, 1, 2…) → Integers (…-2, -1, 0, 1, 2…) → Rational numbers (p/q where q≠0) → Irrational numbers (like √2, π) → Real numbers (rational + irrational).

Key Decimal Types:

• Terminating decimals: 1/4 = 0.25 (finite digits after decimal)
• Recurring decimals: 1/3 = 0.333… (repeating pattern)
• Non-terminating, non-recurring: Irrational numbers cannot be expressed as a fraction

⚡ Exam tip for MAT: Always check if a fraction terminates — denominator in simplest form must have only 2 and/or 5 as prime factors. For example, 3/8 = 0.375 (8=2³) terminates, but 1/7 = 0.142857… recurs.

Conversions to remember:

• 1/8 = 0.125 | 3/8 = 0.375 | 5/8 = 0.625 | 7/8 = 0.875
• 1/16 = 0.0625 | 3/16 = 0.1875 | 9/16 = 0.5625

HCF and LCM relationship: For any two numbers a and b: a × b = HCF(a,b) × LCM(a,b)

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🟡 Standard — Regular Study (2d–2mo)

Standard content for students with a few days to months.

Number Systems and Decimals — Deep Dive

The Real Number Line

Every real number corresponds to exactly one point on the number line, and vice versa. This is fundamental to understanding inequalities and absolute value.

Types of Numbers with Examples:

Type	Definition	Example	Non-Example
Prime	Exactly 2 factors	2, 3, 5, 7	9 (3 factors)
Composite	More than 2 factors	4, 6, 8	5 (prime)
Even	Divisible by 2	0, 2, 4, 6	3, 5
Odd	Not divisible by 2	1, 3, 5, 7	2, 4
Rational	p/q, q≠0	3/7, 0.25, 0.333	√2, π

Decimal to Fraction Conversion

Terminating → Fraction method:

1. Count digits after decimal (n)
2. Remove decimal point → numerator
3. Denominator = 10ⁿ
4. Simplify

Example: 0.375 = 375/1000 = 3/8 ✓

Recurring → Fraction method:

1. Let x = 0.1666…
2. 10x = 1.666…
3. 10x - x = 1.5 → 9x = 1.5 → x = 1.5/9 = 1/6

Rational Number Properties:

• Sum/difference of two rationals = rational
• Product/dividing two rationals = rational ( divisor ≠ 0)
• √2 is irrational — proof by contradiction: assume √2 = p/q in lowest terms, then 2q² = p², so p is even, let p=2k, then 2q²=4k² → q²=2k², so q is even, but p/q not in lowest terms. Contradiction!

⚡ MAT Shortcut: When comparing fractions, cross-multiply. For 5/7 vs 6/8 → compare 5×8=40 vs 6×7=42, so 5/7 < 6/8.

Unit Digit Patterns (Key for MAT)

• Powers of 2 cycle: 2, 4, 8, 6 (period 4)
• Powers of 3 cycle: 3, 9, 7, 1 (period 4)
• Powers of 7 cycle: 7, 9, 3, 1 (period 4)
• Powers of 8 cycle: 8, 4, 2, 6 (period 4)

Example: Find unit digit of 7²³ → 23 mod 4 = 3 → unit digit = 7

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Number Systems and Decimals — Complete Theory

Euclid’s Division Lemma

For any integers a and b (b > 0), there exist unique integers q and r such that: a = bq + r, where 0 ≤ r < b

This lemma is the foundation for finding HCF using the Euclidean algorithm.

Euclidean Algorithm for HCF:

HCF(48, 18):
48 = 18×2 + 12
18 = 12×1 + 6
12 = 6×2 + 0
∴ HCF = 6

LCM × HCF = Product holds for any two positive integers.

Fundamental Theorem of Arithmetic

Every composite number can be expressed as a product of prime factors, and this factorisation is unique (up to the order of factors).

Example: 120 = 2³ × 3 × 5

This theorem helps in:

• Finding HCF/LGM
• Proving irrationality of numbers like √12 = 2√3
• Simplifying radicals

Surds and Indices

Laws of Indices:

• aᵐ × aⁿ = aᵐ⁺ⁿ
• aᵐ ÷ aⁿ = aᵐ⁻ⁿ
• (aᵐ)ⁿ = aᵐⁿ
• a⁰ = 1 (a ≠ 0)

Rationalising Denominators:

• For √a: multiply by √a/√a
• For a + √b: multiply by a - √b (conjugate pair)
• For √a + √b: multiply by √a - √b

Example: Rationalise 1/(3+√2) = (3-√2)/(9-2) = (3-√2)/7

Comparison of Surds: To compare √2 and √3: square both → 2 < 3, so √2 < √3.

⚡ MAT PYQ Pattern: Questions on number systems appear every year. Focus on: divisibility rules, unit digit cycles, HCF-LCM applications, and conversion between fractions and decimals. A typical MAT question: “If 2ⁿ leaves remainder 1 when divided by 3, find the smallest n” → Answer: 1 (2¹=2≡2, 2²=4≡1 mod 3)

Practice Problem Types for MAT

1. Divisibility: Find the largest 4-digit number divisible by 12, 15, and 20 → HCF(12,15,20)=60 → largest 4-digit = 9999, 9999 ÷ 60 = 166.65 → answer = 166×60 = 9960
2. Decimal sequences: Find the 50th digit after decimal in 1/7 → 1/7 = 0.142857 repeating (period 6) → 50 mod 6 = 2 → 2nd digit in cycle = 4
3. Unit digit: Find unit digit of 7⁴⁵ × 3²⁸ → 7⁴⁵ unit digit cycles (7,9,3,1), 45 mod 4 = 1 → 7; 3²⁸ cycles (3,9,7,1), 28 mod 4 = 0 → 1; product unit digit = 7×1 = 7
4. Digit displacement: If 2³⁷ is written, what digit is at the 20th place from right? (Use patterns)

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