📑 Table of Contents

1. Number System कितना Important है? — Exam Weightage
2. Number System — Core Concepts (Foundation + Intermediate + Advanced)
3. Exam में यह पूछा जाता है — Pedagogy & NCF/NEP Angle
4. Current Updates — NCF 2023 & NEP 2020 Impact
5. Deep Theory + Flowcharts — Number System का पूरा Map
6. Comparison & Differentiation — Confusing Pairs
7. 🧠 Number System Mnemonics — Memory Architecture
8. Conclusion + Next-Topic Roadmap
9. Frequently Asked Questions — Number System CTET Notes 2026

Exam	Paper	Questions (Last 5 Yrs)	Weightage
CTET	I & II	4–5	12–15%
KVS	PRT/TGT	3–4	10–12%
DSSSB	PRT/TGT	4–6	14–18%
UPTET	I & II	3–5	12–14%
BPSC TRE	1.0–3.0	3–4	10–12%

• Natural Numbers (N): 1, 2, 3, 4, 5… — Counting numbers (गिनती की संख्याएँ)
• Whole Numbers (W): 0, 1, 2, 3, 4… — Natural Numbers + Zero
• Integers (Z): …, -3, -2, -1, 0, 1, 2, 3… — Whole Numbers + Negative Numbers
• Even Numbers: 2, 4, 6, 8… (divisible by 2) — Odd Numbers: 1, 3, 5, 7… (not divisible by 2)
• Prime Numbers: 2, 3, 5, 7, 11… (only 1 and itself as factors) — Composite Numbers: 4, 6, 8, 9… (more than 2 factors)

CTET Paper I (2022): “निम्नलिखित में से कौन-सी संख्या Prime है?” — 1, 2, 9, 15
✅ Answer: 2 (1 is neither prime nor composite; 9 and 15 are composite)
📊 📊 सभी CTET Paper I PYQs का Data देखें

• Rational Numbers (Q): Numbers that can be expressed as p/q where p and q are integers and q ≠ 0 — e.g., ½, -3/4, 0.75, 0.333…, 4 (as 4/1)
• Irrational Numbers: Numbers that CANNOT be expressed as p/q — e.g., √2, √3, π (22/7 is approximation, NOT exact), e
• Real Numbers (R): All Rational + All Irrational Numbers — वास्तविक संख्याएँ
• Non-Real Numbers: √(-1) = i — CTET में नहीं पूछा जाता

Feature	Rational Numbers	Irrational Numbers
p/q Form	✅ Always possible	❌ Never possible
Decimal Form	Terminating or Repeating	Non-terminating, Non-repeating
Examples	0.75, 0.333…, 5/2	√2, π, √3
CTET में पूछा	✅ बहुत बार	✅ बहुत बार

CTET Paper II (2023): “Which of the following is an irrational number?” — (A) √4 (B) 0.575757… (C) π (D) 22/7
✅ Answer: (C) π — because it’s non-terminating non-repeating; √4=2 (rational), 0.575757… is repeating (rational), 22/7 is rational

Advanced Number Theory — CTET PGT / KVS PGT Level:
• Density property: Rational numbers are dense — between any two rationals, there exists another rational.
• Countability: Rational numbers are countable (can be listed), but irrational numbers are uncountable — यह concept उच्च स्तर के pedagogy questions में पूछा जाता है जब “how would you teach infinite numbers” पूछा जाता है।
• NCF 2023 connection: “Number sense” को foundational literacy के रूप में develop करने पर जोर — CTET में “How would you introduce irrational numbers to Class 8 students?” जैसा pedagogy question आ सकता है।

• Remember: Define Natural, Whole, Integers, Rational, Irrational — definitions याद करें
• Understand: Explain why 0.333… is rational but √2 is irrational — decimal form से समझें
• Apply: Given a list of numbers, classify them into rational/irrational — PYQs में यही आता है
• Analyze: Why is 1 neither prime nor composite? — CTET 2022 में पूछा गया
• Evaluate: Which teaching method is best for introducing irrational numbers to Class 8? — Pedagogy questions

KVS PRT (2024): “A teacher wants to introduce ‘Irrational Numbers’ to Class 8 students. Which of the following approaches is most effective according to Constructivism?”
✅ Answer: Start with rational numbers (already known), show their decimal expansions, then show √2’s non-terminating non-repeating expansion — discover the difference through examples (Vygotsky’s ZPD).

• NCF 2005: “Mathematics should be taught as a way of thinking” — Number System को rote learning से नहीं, बल्कि pattern recognition और logical reasoning के through समझाना चाहिए।
• NEP 2020: Foundational Literacy & Numeracy (FLN) — Number System को Class 1–5 में concrete materials (गिनती की छड़ें, abacus) के through समझाना चाहिए। CTET में NEP 2020 based questions आ सकते हैं — “NEP 2020 के अनुसार number sense कैसे develop करें?”
• NIPUN Bharat: Class 3 तक number recognition, place value, and basic operations पर focus — Number System CTET Notes 2026 में NIPUN Bharat के learning outcomes को pedagogy questions में शामिल करें।
• RTE Act 2009: CCE (Continuous & Comprehensive Evaluation) — Number System में students की समझ का assessment “paper-pencil test” के बजाय “oral or activity-based” होना चाहिए।

1. NCF 2023 (August 2023) — School Mathematics Framework: “Number System should be introduced through real-life contexts — money transactions, measurement, and data representation.”
→ Exam Impact: CTET 2026 में “Real-life example से Number System कैसे पढ़ाएँ?” पर pedagogy question आ सकता है।

2. NCERT Textbook Revision (2024): Class 6–8 Mathematics books में Number System का presentation more activity-based किया गया है — emphasis on “discovery of irrational numbers through square roots.”
→ Exam Impact: “How would you help students discover that √2 is irrational?” — जैसे questions CTET 2025–26 में आ सकते हैं।

3. NIPUN Bharat (2025 Update): Class 3 तक “Number Sense” के 5 learning outcomes को revision किया गया — place value, comparing numbers, basic operations पर focus बढ़ा।
→ Exam Impact: CTET Paper I में “Number Sense development के लिए best activity?” — जैसे questions आ सकते हैं।

4. CBSE Circular (February 2025): “Competency-based questions in Mathematics from 2025–26” — Number System से अब application-based questions (not just definition-based) पर जोर होगा।
→ Exam Impact: CTET 2026 में “Given this number, identify its type and justify” — जैसे higher-order questions आ सकते हैं।

1. Euclid’s Division Lemma: For any two positive integers a and b, there exist unique integers q and r such that a = bq + r, where 0 ≤ r < b. CTET में यह NOT directly पूछा जाता, लेकिन HCF/LCM के pedagogy questions में इसकी understanding जरूरी है।

2. Fundamental Theorem of Arithmetic: Every integer greater than 1 is either a prime number or can be expressed as a product of primes (unique up to order). CTET 2024 में “Why is prime factorization important?” — जैसा pedagogy question आया था।

3. Rational Numbers की Decimal Expansion की Condition: A rational number in p/q form (in lowest terms) has terminating decimal expansion IF AND ONLY IF q = 2^m × 5^n (where m, n are non-negative integers). अन्यथा repeating decimal expansion. CTET में यह directly MCQ में पूछा जाता है — “0.375 का q का form क्या होगा?”

4. Irrational Numbers की Proof (√2): Proof by contradiction — assume √2 is rational → p/q in lowest terms → 2q² = p² → p is even → q is even → contradiction. CTET में यह “How would you prove √2 is irrational?” — pedagogy question में पूछा जा सकता है।

Feature	Natural Numbers	Whole Numbers
Set	{1, 2, 3, 4…}	{0, 1, 2, 3…}
Smallest Number	1	0
CTET में पूछा	“Smallest natural number?”	“Smallest whole number?”

Feature	Prime Numbers	Composite Numbers
Factors	Exactly 2 (1 and itself)	More than 2
Examples	2, 3, 5, 7, 11, 13	4, 6, 8, 9, 10, 12
1 की status	❌ Neither prime nor composite	❌ Neither prime nor composite

Feature	Rational	Irrational
p/q Form	✅ Yes (q≠0)	❌ No
Decimal	Terminating or Repeating	Non-terminating Non-repeating
Examples	½, 0.75, 0.333…, 4	√2, √3, π, e
Density	Dense (between any two, another exists)	Also dense (more abundant!)

Number System Classification Tree ├── Real Numbers (R) │ ├── Rational (Q) — p/q, q≠0 │ │ ├── Integers (Z) │ │ │ ├── Whole (W) — 0,1,2… │ │ │ │ └── Natural (N) — 1,2,3… │ │ │ └── Negative Integers — -1,-2,-3… │ │ └── Fractions (non-integer rational) │ │ ├── Terminating Decimals — 0.5, 0.75 │ │ └── Repeating Decimals — 0.333…, 0.1212… │ └── Irrational — non-terminating, non-repeating │ ├── Algebraic (roots) — √2, √3, √5 │ └── Transcendental — π, e └── Non-Real (Imaginary) — √(-1) = i

✅ Number System CTET Notes 2026 में हमने सीखा: Natural → Whole → Integers → Rational → Irrational — पूरा hierarchy और हर category की definition.
✅ Rational numbers को p/q में express किया जा सकता है, Irrational numbers को नहीं — decimal expansion (terminating/repeating vs non-terminating non-repeating) से पहचानें.
✅ CTET Pedagogy में Number System को Constructivist approach (Piaget/Vygotsky) और Bloom’s Taxonomy के through पढ़ाने पर focus रहेगा — NCF 2023 और NEP 2020 के अनुसार real-life examples से number sense develop करना.
✅ Prime vs Composite, Natural vs Whole, Rational vs Irrational — सभी confusing pairs clear कर लिए हैं।
✅ 2026 के CTET में Number System से 4–5 questions (Content + Pedagogy) आने की संभावना है — हर MCQ 30 seconds में solve करने की strategy तैयार है।

• Natural: 1,2,3… • Whole: 0,1,2… • Integers: …,-1,0,1… • Rational: p/q, q≠0 — Terminating/Repeating decimal • Irrational: NOT p/q — Non-terminating Non-repeating • 1: Neither prime nor composite • 2: Only even prime • π: Irrational (22/7 is Rational!) • √2: Irrational

• 3. Factors Affecting Development & Individual Differences — Number System की समझ individual differences (जैसे कुछ students को numbers में difficulty होती है) से connect होती है — pedagogy questions में यह pair आता है। Study Time: 2 Hours
• 5. Learning Process & Principles — Number System जैसे foundational topics को कैसे सीखा जाता है, इसका pedagogy connection — CTET में “how do students learn number concepts?” पर questions आते हैं। Study Time: 2.5 Hours
• 15. Assessment & Evaluation — Number System में students की समझ का assessment कैसे करें? CCE और formative assessment techniques — CTET Pedagogy में जरूरी है। Study Time: 2 Hours

Number System को और गहराई से समझने के लिए NCERT Class 6 Mathematics Chapter 1 — Knowing Our Numbers और NCERT Class 7 Mathematics Chapter 9 — Rational Numbers के official resources देखें।

Recommended Study Path: Number System → Factors Affecting Development → Learning Process & Principles.

What is Number System in CTET Maths Pedagogy?

Number System CTET Notes 2026 covers classification of numbers — Natural, Whole, Integers, Rational, and Irrational — along with their properties, decimal expansions, and teaching strategies as per NCF 2005 and NEP 2020.

How many questions from Number System come in CTET?

CTET Paper I & II में Number System से 4–5 questions (Content + Pedagogy) आते हैं — जो कुल marks का 12–15% होता है।

What is the difference between rational and irrational numbers for CTET?

Rational numbers can be expressed as p/q (q≠0) and have terminating or repeating decimals. Irrational numbers cannot be expressed as p/q and have non-terminating non-repeating decimals — e.g., √2 and π are irrational.

Is 1 a prime number in Number System CTET Notes?

No — 1 is neither prime nor composite. CTET 2022 में यह directly पूछा गया था — 1 के factors sirf 1 (itself) हैं, इसलिए prime नहीं माना जाता (prime requires exactly 2 factors).

How to teach Number System as per NEP 2020?

NEP 2020 का FLN (Foundational Literacy & Numeracy) framework Number System को concrete materials (abacus, number lines, real-life examples) के through Class 1–5 में सिखाने पर focus करता है — CTET Pedagogy questions में इसी approach को सही माना जाता है।

What is the best teaching method for irrational numbers in CTET?

Constructivist approach (Vygotsky’s ZPD) — rational numbers से शुरू करें, उनकी decimal expansions दिखाएँ, फिर √2 के decimal expansion को compare करके “discovery” के through irrational numbers introduce करें।

Why is 22/7 rational but π irrational in CTET?

22/7 is rational because it’s in p/q form (q≠0). π is irrational because it cannot be expressed as p/q — its decimal expansion is non-terminating and non-repeating. CTET में यह सबसे common confusion है — याद रखें: 22/7 is an APPROXIMATION of π, not equal to π.

What are the 5 main types of numbers in Number System for CTET?

Natural Numbers (N), Whole Numbers (W), Integers (Z), Rational Numbers (Q), and Irrational Numbers — together they form Real Numbers (R).

How does Bloom’s Taxonomy apply to Number System questions?

Remember (definitions) → Understand (why 0.333… is rational) → Apply (classify given numbers) → Analyze (why 1 is neither prime nor composite) → Evaluate (best teaching method for irrational numbers) — CTET questions cover all these levels.