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1.1.6 · HinglishMatter, Measurement & the Mole

Significant figures and rounding rules

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1.1.6 · Chemistry › Matter, Measurement & the Mole

Significant figures KYUN exist karte hain?

Problem KYA hai? Maan lo aap ek ruler se length measure karte ho jo mm mein marked hai. Aap confidently mm tak read kar sakte ho, aur uss se ek digit aage estimate kar sakte ho. Agar aap cm report karo, toh aap jhooth bol rahe ho — aapne claim kiya ki aapko length ek millionth of a cm tak pata hai. Agar aap cm report karo, toh aapne real information throw away kar di.

Chemistry mein KYU matter karta hai? Ek calculated answer kabhi bhi us measurements se zyada precise nahi ho sakta jisse wo aaya hai. Agar aap ek mass ko jo 3 digits tak known hai, ek volume se divide karo jo 2 digits tak known hai, toh density sirf 2 digits tak hi sahi hogi. Sig figs aapko fake precision se bachate hain jo aapke final result mein leak ho sakti hai.

KYA cheez significant figure count hoti hai?

Rules (har ek ke peeche KYU ke saath):

  1. Saare non-zero digits significant hote hain. ( hamesha "real" measured digits hote hain.)
  2. Non-zeros ke beech wale zeros significant hote hain — "sandwiched zeros." mein 3 sig figs hain. Kyun: aap 5 ko measure nahi kar sakte us 0 se guzre bina.
  3. Leading zeros significant NAHI hote. mein 2 sig figs hain. Kyun: wo zeros sirf decimal point ki jagah mark karte hain; ye placeholders hain, measurements nahi.
  4. Decimal point ke BAAD trailing zeros significant HOTE HAIN. mein 3 sig figs hain. Kyun: aap wo final 0 likhne ki takleef nahi uthate jab tak aapne usse measure na kiya ho.
  5. Whole number mein trailing zeros (bina decimal ke) ambiguous hote hain. mein 2, 3, ya 4 sig figs ho sakte hain. Fix: scientific notation use karo.
Figure — Significant figures and rounding rules

Exact numbers mein INFINITE sig figs hote hain

Kyun: "exactly 12" mein koi uncertainty nahi hai. 12 objects count karna koi estimate nahi hai.

Rounding rules — number ko shrink karne ka honest tarika

Kisi chosen number of sig figs tak round karne ke liye, pehli digit dekho jo aap drop kar rahe ho (the "decider"):

Round-half-to-even KYU? Agar aap hamesha 5 ko round up karo, toh kaafi calculations ke baad aap numbers ko upar ki taraf systematically push kar rahe hote, jo ek bias create karta hai. Adha time upar aur adha time neeche round karna (even choose karke) average error ko zero ke paas rakhta hai. Ye steel-manned, unbiased rule hai.

CALCULATIONS mein Sig figs

Ye alag KYU hain?

  • Multiply/divide karna relative uncertainty ko scale karta hai (percent error), isliye hum significant figures track karte hain (ek relative measure).
  • Add/subtract karna absolute uncertainty stack karta hai (same units mein), isliye hum decimal places track karte hain (ek absolute position measure).

Worked examples

Common mistakes (Steel-manned)

Recall Feynman: 12-saal ke bachche ko explain karo

Imagine karo tum apne dost ko bolo, "Main lagbhag 3 kilometer chala." Tumhe sachchi mein nahi pata tha ki wo 3.000000 km tha — tum bas itna jaante the ki roughly 3 hai. Significant figures "main kitna sure hoon?" wali digits hain. Agar tumhare steps sirf "lagbhag 3" batate hain, toh "3.0000" kehna cheating hai. Aur jab tum fuzzy numbers ke saath maths karte ho, tumhara answer us fuzziest number se zyada sharp nahi ho sakta jisse tumne shuru kiya tha — keechad aur saaf paani milao toh thoda keechda wala hi rahega. Rounding apne answer ko trim karna hai waapas us hisaab se ki tum actually kitne sure ho, aur jab tum bilkul edge pe ho (ek 5), tum fairly upar-neeche jaate ho taaki ek direction mein cheat na karo.

Active-recall flashcards

mein kitne significant figures hain?
4 (sandwiched 0 aur decimal ke baad trailing 0 count hote hain; leading zeros nahi).
Leading zeros significant kyun nahi hote?
Ye sirf decimal point ki position mark karte hain; ye placeholders hain, measured digits nahi.
Multiplication/division mein sig figs ka rule kya hai?
Answer mein kisi bhi factor ke sabse kam significant figures rakhe jaate hain.
Addition/subtraction mein sig figs ka rule kya hai?
Answer mein kisi bhi term ke sabse kam decimal places rakhe jaate hain.
+/- aur ×/÷ alag rules kyun use karte hain?
Addition absolute error stack karta hai (→ decimal places); multiplication relative error stack karta hai (→ sig figs).
Banker's rounding kya hai aur ise kyun use karte hain?
Akele trailing 5 ko nearest even digit tak round karo; ye hamesha 5 ko upar round karne ke upward bias ko khatam karta hai.
Ek exact/counted number mein kitne sig figs hote hain?
Infinite — ye kabhi bhi calculation ki precision ko limit nahi karta.
Round-half-to-even use karke ko whole number tak round karo.
2 (nearest even).
(3 sig figs) ko scientific notation mein convert karo.
.
Calculation ke end mein hi round kyun karna chahiye?
Intermediate rounding error accumulate karta hai; guard digits carry karo aur ek baar round karo.

Connections

Concept Map

bookkeeping via

rule 1

rule 2

rule 3

rule 4

rule 5

fixed by

N ki har digit count hoti hai

infinite sig figs, kabhi limit nahi

honest tak shrink karta hai

se bachata hai

Measurement ek precision promise hai

Significant figures

Non-zero digits significant

Sandwiched zeros significant

Leading zeros significant nahi

Decimal ke baad trailing zeros significant

Whole-number trailing zeros ambiguous

Scientific notation

Exact numbers

Rounding rules

Result inputs se zyada precise nahi