Leap year aur decimals

365 din kaafi nahi

Calendar saal mein 365 din hote hain, par Earth ko Sun ka ek chakkar lagaane mein 365.2422 din lagte hain. Isliye jab calendar saal khatm hota hai, Earth ko abhi 0.2422 din aur chalna hota hai, lagbhag 6 ghante (0.2422 × 24 = 5.8128 ghante). Ye chhota lagta hai, par badhta jaata hai: 100 calendar saal baad kami 0.2422 × 100 = 24.22 din ho jaati hai. Mausam is baat pe nirbhar hain ki Earth apne raaste pe kahan hai, isliye khisakta calendar dheere dheere garmi ko un mahinon mein le aata jinhe hum sardi ke mahine kehte hain.

Pehla sudhaar: har 4th saal ek leap day

4 se bhaag hone waale har saal ko ek extra din, 29 February, do. Tab chaar calendar saalon mein 4 × 365 + 1 = 1461 din hote hain, jabki chaar chakkar 4 × 365.2422 = 1460.9688 din lete hain, jo bahut paas hai. 100 saalon mein 25 saal 4 se bhaag hote hain, isliye calendar mein 100 × 365 + 25 = 36,525 din hote hain. Par Earth ko sirf 100 × 365.2422 = 36,524.22 din chahiye. Ab calendar har sadi 0.78 din aage hai: humne zaroorat se zyaada sudhaar kar diya.

Doosra sudhaar: century saal chhodo

Isliye 100 se bhaag hone waale saalon ko extra din nahi milta. 100 saalon mein ab sirf 25 − 1 = 24 leap years hain, jisse 24 × 366 + 76 × 365 = 36,524 din hote hain, 36,524.22 se bas 0.22 din kam. Ye chhota hai, par phir badhta hai: 1000 saalon mein calendar mein 36,524 × 10 = 3,65,240 din hain, jabki Earth ko 1000 × 365.2422 = 3,65,242.2 din chahiye. Ab calendar 2.2 din peeche hai.

Teesra sudhaar: har 400th saal rakho

Hazaar saal mein lagbhag do din waapas paane ke liye, 400 se bhaag hone waala har saal phir bhi leap year hai. 1 se 1000 tak ke saal gino:

Saal ka prakaar	Kitne
400 se bhaag (leap)	2
100 se bhaag, 400 se nahi (aam)	10 − 2 = 8
4 se bhaag, 100 se nahi (leap)	250 − 10 = 240
Baaki sab (aam)	1000 − 250 = 750

Isliye 242 saalon mein 366 din aur 758 mein 365: 242 × 366 + 758 × 365 = 3,65,242 din, 3,65,242.2 se sirf 0.2 din kam. Calendar banaane waale yahin ruk gaye, aur ye teen niyam Gregorian calendar banaate hain jo aaj zyaadatar duniya mein chalta hai.

10,000 saal mein kya?

10,000 saalon mein 2500 saal 4 se, 100 saal 100 se aur 25 saal 400 se bhaag hote hain, isliye 2500 − 100 + 25 = 2425 leap years hain. Calendar mein 10,000 × 365 + 2425 = 36,52,425 din hain, jabki Earth ko 10,000 × 365.2422 = 36,52,422 din chahiye, isliye calendar lagbhag 3 din aage nikal jaata hai. (Upar ke 1000 saalon mein ye thoda peeche tha, kyunki 1 se 1000 tak 400 ke sirf 2 multiples hain, jo average 2.5 se kam hai.) Ek sambhav hal: 4000 se bhaag hone waale saalon ko aam saal bana do. Isse 10,000 saal mein 2 leap days hat jaate hain, aur calendar sirf lagbhag 1 din aage rehta hai.

Ye kyun zaroori hai

0.2422 din ka fark bahut chhota lagta hai, par decimals ka guna dikhaata hai ki ye sadiyon mein kaise jud jaata hai, aur leap year ke niyam ko teen hisson ki zaroorat kyun hai. (Extra) Kai paaramparik Indian calendar Chaand aur Sun dono ke hisaab se chalte hain, aur lagbhag har teen saal mein ek extra mahina, adhik maas, jodkar tyohaaron ko unke mausam mein rakhte hain. Decimals ka guna aur decimals ka bhaag ka ganit dohrao. Textbook ke sawaalon ke jawab Part 2 Chapter 4 ke NCERT solutions mein hain; poore Part 2 Chapter 4 ke notes dekho.