1.1.1
Intuition Badi picture (WHY ye sets exist karte hain)Numbers problems solve karne ke liye banaye gaye the , aur har naya set pichle set ki ek limitation ko fix karta hai.
Aap cheezein count karte ho: 1 sheep, 2 sheep... → natural numbers .
Aap realise karte ho ki "koi sheep nahi" bhi ek state hai → zero invent karo → whole numbers .
Aap kisi ko 3 sheep dete ho (debt ) → negatives invent karo → integers .
Toh story yeh hai: har set pichla set plus jo cheez usme missing thi, woh hota hai . Is story ko yaad rakho aur definitions kabhi memorise nahi karni padegi.
Definition Natural numbersCounting numbers : N = { 1 , 2 , 3 , 4 , … } 1 se shuru hote hain aur hamesha chalte rehte hain (koi sabse bada number nahi hota).
WHY 1 se shuru? Kyunki aap existing objects count karte ho. Aap kabhi "zero apples" point karke count nahi karte — aap 1, 2, 3 count karte ho...
HOW ye behave karte hain:
Addition ke under closed hain (3 + 4 = 7 multiplication ke under bhi (3 × 4 = 12 Subtraction ke under closed nahi hain: 3 − 5 agle inventions ko force karta hai .
N Confusion kyun sahi lagti hai: kuch textbooks (specially set-theory / French tradition) 0 ko include karti hain.
School maths ke liye fix: Standard Indian/CBSE convention mein, ==N whole number hai, natural nahi). Agar pucha jaye toh hamesha apna convention batao.
Natural numbers 0 ke saath : W = { 0 , 1 , 2 , 3 , … } W = N ∪ { 0 }
WHY: Humein "kuch nahi / khali" ke liye ek symbol chahiye tha. 0 add karne se hum yeh keh sakte hain jaise "account mein 0 rupaye hain."
0 ki key property: yeh additive identity hai — a + 0 = a a 0 ka matlab hai: isse add karne se kuch nahi badalta.
Whole numbers plus unke negatives :
Z = { … , − 3 , − 2 , − 1 , 0 , 1 , 2 , 3 , … } Z Zahlen se aata hai = "numbers".)
WHY negatives? Kyunki 3 − 5 W − n define karte hain jo n n + ( − n ) = 0.
HOW ye bante hain (first principles): Har natural number n − n additive inverse ). Un partners ko whole numbers se jodo:
Z = { − n : n ∈ N } ∪ W . hamesha kaam karta hai: 3 − 5 = − 2 ∈ Z
Intuition Number line = numbers ko ek picture mein badalnaEk seedhi line jahan har number ko ek point milta hai. Right move karna = bada, left = chota. Isse "kaun bada hai?" ka sawaal "kaun zyada right hai?" mein badal jaata hai — ek ordering jo aap dekh sakte ho.
Ise kaise padhein:
0 beech mein hota hai (origin ).
Naturals (1 , 2 , 3 , … right ki taraf ke marks pe hote hain.
Wholes = naturals plus origin.
Integers = dono directions: negatives, 0 ke across positives ka mirror hote hain.
Order rule: a < b a b left mein hai. Toh − 5 < − 2 − 5 bada debt = chota number ).
− 5 − 2 Kyun sahi lagta hai: hum digits 5 2 Fix: number line par − 5 aur zyada left mein hai, isliye − 5 < − 2 chota . Debt socho: ₹5 dena (₹2 dene se) bura hai (paisa kam hai).
0 , 7 , − 4 , 2 3
7 ∈ N , W , Z Kyun? Yeh ek positive whole quantity hai.0 ∈ W , Z Kyun? 0 sirf W − 4 ∈ Z Kyun? Negatives pehli baar Z 2 3 inme se kuch nahi! Kyun? Yeh ek fraction hai; yeh rationals Q baad waala set hai.
− 3 , 2 , 0 , − 7 − 7 − 3 0 2 Answer: − 7 < − 3 < 0 < 2 Yeh step kyun? "Sabse chota = sabse zyada left" abstract order ko ek spatial order mein convert karta hai jo galat nahi ho sakta.
Worked example Ex 3 — Dikhao ki subtractionW fail hoti hai lekin Z kaam karti hai.
4 − 9 W Z − 5 4 − 9 = − 5 ∈ Z Yeh kyun matter karta hai? Yeh concrete failure hi woh reason hai jiske liye Z
Recall Feynman: 12-saal ke bachche ko samjhaoEk ruler imagine karo jo kabhi khatam nahi hota. Right jaate marks — 1, 2, 3 — cheezein count karne ke liye hain (natural numbers). Ab "bilkul kuch nahi" ke liye ek mark lagao — yeh 0 hai, aur counting numbers + 0 whole numbers hain. Lekin kya hoga agar aap candy rakhte nahi ho balki lete ho? Hum 0 ke left mein marks lagate hain: − 1 , − 2 , − 3 Dono directions ke saare marks milke integers hote hain. Ruler ka rule: jo zyada right mein hai woh bada hai. Toh − 1 − 100
Recall Quiz karo khud ko (answers cover karo)
Sabse chota natural number? Sabse chota whole number?
Kya har whole number ek integer hai? Kya har integer ek whole number hai?
N Line par − 8 < − 3
Natural numbers ka set kya hai? N = { 1 , 2 , 3 , … }
Whole numbers ka set kya hai? W = { 0 , 1 , 2 , 3 , … } = N ∪ { 0 }
Integers ka set kya hai? Z = { … , − 2 , − 1 , 0 , 1 , 2 , … }
Sabse chota natural number? 1.
Sabse chota whole number? 0.
Nesting (subset) chain batao. N ⊂ W ⊂ Z
Whole numbers banane ke liye 0 kyun add kiya gaya? "Kuch nahi" represent karne ke liye aur additive identity ke roop mein (a + 0 = a
Integers kyun invent ki gayi? Taaki subtraction ka hamesha answer ho (jaise 3 − 5 = − 2
n − n n + ( − n ) = 0
Number line par a < b Jab a b
− 5 − 2 − 2
Kya 2 3 Nahi — yeh ek fraction hai, rationals Q Z
N Subtraction (jaise 3 − 5 ∈ / N
Z German Zahlen , matlab "numbers".
Rational Numbers (Q) — agla set, division fix karne ke liye fractions add karta hai.Absolute Value — number line par 0 se distance.Ordering and Inequalities — line par left/right ko formally define karta hai.Closure Property — woh idea jo har naye number set ko drive karta hai.Additive Identity and Inverse — kyun 0 aur negatives define kiye gaye.
not closed under subtraction
makes subtraction always work
