Number Systems — Complete Lesson
The World of Numbers
All numbers belong to a number system. Let's understand the hierarchy:
Natural Numbers (N): 1, 2, 3, 4, ... (counting numbers)
Whole Numbers (W): 0, 1, 2, 3, ... (natural numbers + zero)
Integers (Z): ..., -3, -2, -1, 0, 1, 2, 3, ...
Rational Numbers (Q): Numbers expressible as p/q where q ? 0 (e.g., ½, -3/4, 0.75)
Irrational Numbers: Cannot be expressed as p/q (e.g., v2, p, v3)
Real Numbers (R): All rational + irrational numbers
Irrational Numbers
These are non-terminating, non-repeating decimals:
- v2 = 1.41421356... (never ends, never repeats)
- p = 3.14159265... (never ends, never repeats)
Decimal Expansions
| Type | Example | Rational/Irrational? |
|---|---|---|
| Terminating | 0.25, 1.5 | Rational |
| Non-terminating repeating | 0.333..., 1.272727... | Rational |
| Non-terminating non-repeating | v2, p | Irrational |
Laws of Exponents for Real Numbers
a? × an = a??n
(a?)n = a?n
a? / an = a??n
a? × b? = (ab)?
Number Systems — MCQ Practice
Number Systems — Mnemonic
Mnemonic 1: "RIN-IR" for Number System Types 📚🔢
Remember the main types of numbers in order:
- R - Real Numbers (सभी नंबर)
- I - Irrational Numbers (अपरिमेय)
- N - Natural Numbers (प्राकृतिक)
- I - Integers (पूर्णांक)
- R - Rational Numbers (परिमेय)
Pronounce as "RIN-IR" (रिन-इर) — sounds like a fun nickname!
Mnemonic 2: Hindi Rhyming Phrase for Number Types 🎶🇮🇳
“प्रकृति में पूर्ण, परिमेय की धुन, अपरिमेय है अनमोल, सब मिलकर बनाते हैं रियल गोल”
- प्रकृति में पूर्ण = Natural (प्राकृतिक) & Integers (पूर्णांक)
- परिमेय की धुन = Rational Numbers (परिमेय)
- अपरिमेय है अनमोल = Irrational Numbers (अपरिमेय)
- सब मिलकर बनाते हैं रियल गोल = All together form Real Numbers (वास्तविक संख्या)
Easy to remember and fun to recite before exams!
Mnemonic 3: Funny Acronym “NIRR” for Number System Hierarchy 😂📏
- N - Natural Numbers (1, 2, 3, …)
- I - Integers (… -2, -1, 0, 1, 2 …)
- R - Rational Numbers (fractions like 1/2, decimals)
- R - Real Numbers (includes rational + irrational)
Think: “Naughty Indians Read Regularly” 😄 to remember the order from smallest to biggest set!
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