Class 7 Maths Part 2 Chapter 4: Another Peek Beyond the Point (Hinglish Notes)

Ye chapter kis baare mein hai?

Part 1 mein tumne decimals dekhe aur unhe jodna aur ghataana seekha. Ye chapter agla kadam hai: decimals ka guna aur bhaag. Dono wahi tareeke nikalte hain jo tum counting numbers ke liye pehle se use karte ho, bas thode aage badhe hue. Guna hamesha ki tarah karo aur point ginkar lagao, aur bhaag place value se karo, ones ke aage tenths aur hundredths tak chalte raho. Raaste mein tum aise bhaag dekhoge jo kabhi khatm nahi hote, ek anokha number 142857, aur leap years mein chhupa decimals ka ganit.

Decimals ka chhota recap

Decimal, Indian place value system ko ones ke daayein taraf tenths, hundredths, thousandths waghairah tak le jaata hai. Isliye 27.53 ka matlab hai 2 tens, 7 ones, 5 tenths aur 3 hundredths, aur 254/1000 jaisa fraction 2/10 + 5/100 + 4/1000 = 0.254 hai. Kilogram mein 50 g dalchini 50/1000 kg = 0.05 kg hai.

10, 100 ya 1000 se bhaag dene ke liye point ko utni jagah baayein le jao jitne divisor mein zero hain, aur digits kam padein to aage zero lagao: 123 ÷ 10 = 12.3 aur 12 ÷ 1000 = 0.012.

Decimals ka guna

Decimal ko counting number se guna karna baar baar jodna hai: ₹9.5 waale 5 pen ka daam 9.5 × 5 = ₹47.5. Do decimals ke liye har ek ko 10, 100 ya 1000 waale fraction mein likho aur guna karo: 12.5 × 7.5 = 125/10 × 75/10 = 9375/100 = 93.75. Numerators bina point waale numbers hain, aur denominator mein utne zero aate hain jitne dono denominators mein milakar the. Isse niyam milta hai: bina point ke guna karo, phir guna mein utne decimal places do jitne dono numbers mein milakar hain.

Decimals ka guna hamesha numbers se bada nahi hota. 2.25 × 8 = 18 dono se bada hai, 0.25 × 8 = 2 dono ke beech hai, aur 0.25 × 0.8 = 0.2 dono se chhota hai. Jab dono numbers 0 aur 1 ke beech hon, to guna dono mein se har ek se chhota hota hai. Aur padho decimals ka guna mein.

Decimals ka bhaag

Place value se bhaag bilkul counting numbers ki tarah chalta hai, bas ones pe rukna zaroori nahi. 1325 ÷ 4 mein 3 hundreds, 3 tens aur 1 one ke baad 1 one bachta hai. Use 10 tenths mein badlo, quotient mein decimal point lagao, aur aage badho: har hisse ko 2 tenths milte hain aur 2 tenths bachte hain, phir 20 hundredths se har hisse ko 5 hundredths. Isliye 1325 ÷ 4 = 331.25.

Decimal dividend bhi isi tarah hota hai: 9.5 kg cheeni ko 4 barabar thailon mein bharne pe har thaile mein 2.375 kg. Agar divisor decimal ho, to dono numbers ko 10, 100 ya 1000 se guna karo taaki divisor counting number ban jaaye: 2.5 ghante mein 126 km yaani 1260 ÷ 25 = 50.4 km prati ghanta. 0 aur 1 ke beech ke number se bhaag dene pe quotient dividend se bada hota hai, jaise 128 ÷ 0.4 = 320. Dekho decimals ka bhaag.

Kabhi na khatm hone waale quotient

10 ÷ 3 karke dekho. Har kadam mein 1 bachta hai, isliye 3 kabhi nahi rukte: 3.333…. 1 ÷ 7 mein remainders 1, 3, 2, 6, 4, 5 aur phir dobara 1 aate hain, isliye digits 142857 hamesha dohraate rehte hain.

142857 khaas hai. Ise 2, 3, 4, 5 ya 6 se guna karne pe wahi chhe digits usi gol kram mein aate hain, bas alag jagah se shuru hote hain, aur 142857 × 7 = 999999. Aise numbers ko cyclic numbers kehte hain, aur 1 ÷ 17 ek aur deta hai. Kya aise anant numbers hain, ye abhi tak pata nahi; Emil Artin ne 1927 mein andaaza lagaaya ki hain, par ab tak koi ise saabit nahi kar paaya.

Koodne se pehle dekho

Earth ko Sun ka ek chakkar lagaane mein 365.2422 din lagte hain, isliye 365 din ka har calendar saal 0.2422 din adhoora chhodta hai, jo 100 saal mein 24.22 din ho jaata hai. Har 4th saal ka leap day thoda zyaada sudhaar kar deta hai, isliye century saal leap year nahi hote, jab tak wo 400 se bhaag na hon. Decimals ka guna har kadam jaanchta hai: teeno niyamon ke saath 1000 calendar saalon mein 3,65,242 din hote hain, jabki Earth ko 3,65,242.2 din lagte hain.

Padho leap year aur decimals. Har sawaal ka jawab, Hidato puzzles samet, Part 2 Chapter 4 ke NCERT solutions mein hai.