Significant figures — rules for operations: Question bank

1.1.4 · D5 · Physics › Measurement, Vectors & Kinematics › Significant figures — rules for operations

True or false — justify

True or false:

0.00500

aur

5.00

mein significant figures ki sankhya same hai.

True — dono mein 3 hain.

0.00500

mein leading zeros sirf placeholder hain, isliye significant digits

5, 0, 0

hain; decimal ke baad ke trailing zeros real information hain.

True or false: ek bahut precise number ko ek bahut rough number mein add karne se rough number ki precision improve hoti hai.

False — absolute uncertainties add hoti hain, isliye sum zyada se zyada worst decimal place jaisi hi acchi hogi; precise term us tenths digit ko rescue nahi kar sakta jise rough term ne kabhi jaana hi nahi.

True or false:

4500

definitively 4 significant figures rakhta hai.

False — trailing zeros bina decimal point ke ambiguous hote hain. Yeh 2, 3, ya 4 sig figs ho sakta hai; sirf Scientific notation jaise

4.50 \times 1 0^{3}

ise resolve karta hai.

True or false: multiplication ka result legitimately zyada sig figs rakh sakta hai kisi bhi factor se.

False — answer sabse kam sig figs inherit karta hai factors mein se, kyunki uski relative error worst factor se dominate hoti hai. Yeh kabhi bhi apne worst ingredient se zyada precise nahi ho sakta.

True or false:

2 π r

mein "

2

" tera answer 1 sig fig tak limit karta hai.

False —

2

ek exact/defined number hai jiske paas infinite sig figs hain; sirf measured

r

(aur

π

, jise tum kaafi digits tak carry karte ho) result ko limit karte hain.

True or false: number

100.

(likh ke decimal point ke saath) ke 3 sig figs hain.

True — explicit decimal point signal karta hai ki trailing zeros meaningful hain, isliye teeno digits count hote hain.

True or false: har intermediate step par rounding karne se wahi final answer milta hai jaise end mein ek baar rounding karne se.

False — early rounding information discard karta hai aur rounding error accumulate karta hai; tumhe guard digits carry karne chahiye aur sirf final answer round karna chahiye.

True or false:

6.02 \times 1 0^{23}

aur

602000000000000000000000

same precision carry karte hain.

False as written — plain integer ke ambiguous trailing zeros hain, jabki

6.02 \times 1 0^{23}

unambiguously 3 sig figs state karta hai. Scientific notation hi precision ko explicit banata hai.

Spot the error

3.0 \times 1 0^{2} + 45 = 345

, 3 sig figs rakho → "

345

". Galti kahan hai?

Addition rule decimal places use karta hai, sig figs nahi.

3.0 \times 1 0^{2} = 300

sirf tens place tak jaana jaata hai, isliye sum tens mein uncertain hai: honest answer

3.4 \times 1 0^{2}

(

350

-ish) hai, confident

345

nahi.

12.11 + 0.3 + 1.012 = 13.422

, aur kyunki

1.012

ke 4 sig figs hain, 4 rakho →

13.42

." Galti dhundho.

Galat rule choose kiya. Addition fewest decimal places match karta hai; yahan

0.3

ka sirf 1 decimal place hai, isliye answer

13.4

hai, chahe kisi ka bhi sig-fig count kuch bhi ho.

0.6666 \dots

from

2.0/3.000

; decimal places kam rakho (

3.000

ke 3 hain) →

0.667

." Kya galat hua?

Yeh division hai, isliye sig figs use karo, decimal places nahi.

2.0

ke 2 sig figs hain, isliye answer

0.67

hai,

0.667

nahi.

Ek student likhta hai

5.0 = 2.2360679 \dots \to 2.24

. Reasoning error dhundho, agar koi ho.

Verdict theek hai lekin why check karo: square root ek power ki tarah behave karta hai (ek multiplicative operation), isliye yeh apne input ke sig figs rakhta hai —

5.0

ke 2 sig figs hain, isliye

2.2

2.24

nahi. Lambi digit tail se dhoka mat khao ki 3 rakho.

115.0 - 114.9 = 0.1

, aur

0.1

ke sirf 1 sig fig hain, isliye maine precision kho di — mera answer bura hai." Conceptual error kya hai?

"Answer is bad" ka verdict over-reads the situation. Subtraction 1 decimal place tak correct hai; sig figs ka drop (4 se 1 tak) real hai aur ise loss of significance kehte hain, lekin yeh ek arithmetic mistake nahi, balki near-equal numbers subtract karne ka honest consequence hai.

n = 6

trials, mean

= 4.83

; kyunki

6

ke 1 sig fig hain, mean ko

5

tak round karo." Flaw dhundho.

n = 6

ek counting number hai jiske paas infinite sig figs hain, isliye yeh kabhi precision limit nahi karta. Mean ke sig figs sirf measured data se aate hain, isliye

4.83

apne measurements ki allowance ke anusaar wahi rahega.

Why questions

\times / \div

sig figs kyun count karta hai jabki

+ / -

decimal places count karta hai?

Kyunki multiplication relative errors ko add karta hai (sig figs ≈ relative precision) jabki addition absolute errors ko add karta hai (decimal places ≈ absolute precision) — do alag tarah ki precision, do alag tarah ki bookkeeping.

Leading zeros kabhi bhi significant kyun nahi count hote?

Woh sirf decimal point ki location fix karte hain (woh placeholders hain);

0.0042

wahi information carry karta hai jaise

4.2 \times 1 0^{- 3}

, jo plainly sirf 2 significant digits dikhata hai.

Banker's rounding (round-half-to-even) kyun exist karta hai instead of hamesha

5

ko upar round karne ke?

Hamesha upar round karne se bahut saare results systematically upar ki taraf bias hote hain; halves ko nearest even digit tak round karne se roughly aadhe upar aur aadhe neeche jaate hain, jo large batches of data ko average par unbiased rakhta hai.

Do close numbers subtract karna dangerous kyun ho sakta hai, chahe har ek bahut precise ho?

Unki absolute errors individually chhoti hain, lekin true digits cancel hote hain jabki errors nahi, isliye chhote difference ki relative error balloon ho jaati hai — ise catastrophic cancellation kehte hain.

2.30000 cm

likhna ek ruler-and-eye measurement ke liye "jhooth" kyun hai?

Yeh claim karta hai ki length ek hundred-thousandth of a cm tak jaani jaati hai, lekin measurer ne sirf last visible digit eye se guess kiya tha. Extra trailing zeros aisa knowledge assert karte hain jo kabhi measure hi nahi hua.

Sig-fig rules true error propagation ko sirf approximately kyun capture karti hain?

Woh precision ko whole digits mein quantise karti hain, isliye woh express nahi kar saktiñ, jaise "0.3% tak jaana jaata hai"; careful work ke liye tumhe Error propagation — relative vs absolute mein actual relative/absolute uncertainty formulas par wapas jaana hoga.

Order-of-magnitude estimate mein often sirf 1 sig fig kyun chahiye hota hai?

Estimation sirf poochti hai "kaunsi power of ten?" — leading digit answer ki size dominate karta hai, isliye extra digits cost add karte hain bina conclusion change kiye, jaise Orders of magnitude & estimation mein explore kiya gaya hai.

Edge cases

Case:

0.0

(ek measured zero to one decimal). Kya iske significant figures hain?

Haan — ek measured zero to 1 decimal place ka matlab hai "tenths ke andar zero jaana jaata hai", jo genuine information hai; iski precision 1 decimal place hai, chahe ek lone zero ke sig figs count karna ek special/ambiguous case hai jo additive contexts ke liye decimal places se handle karna best hai.

Case: ek measured

3.2

ko ek exact conversion factor jaise

1000

(m to mm) se multiply karna. Kitne sig figs survive karte hain?

2 — exact factor ke paas infinite sig figs hain aur woh result ko shrink nahi kar sakta, isliye

3.2 m = 3.2 \times 1 0^{3} mm

apne 2 sig figs rakhta hai.

Case:

lo g_{10} (2.0 \times 1 0^{4}) = 4.30103 \dots

. Answer mein kitne decimals rakhne chahiye?

Logs ke liye, mantissa (decimal ke baad ke digits, yahan

30103

) input ke sig figs carry karta hai, jabki integer part (

4

) sirf exponent hai. 2 sig figs in hain, toh mantissa mein 2 rakho →

4.30

Case:

7 \times 1 0^{0}

7.000 \times 1 0^{0}

ke roop mein likha gaya ek value. Same physical quantity, alag notes — kya woh same hain?

Nahi —

7 \times 1 0^{0}

1 sig fig claim karta hai (units tak jaana jaata hai),

7.000

4 claim karta hai (thousandths tak jaana jaata hai). Number line par same position, bahut alag precision claims.

Case: alag units of magnitude mein numbers add karna, jaise

1.2 \times 1 0^{3} + 3.45 \times 1 0^{1}

. Pehle kya karna chahiye?

Inhe ek common power of ten par align karo (

1200 + 34.5

) uske baad decimal-place rule apply karo, kyunki "fewest decimal places" tabhi meaningful hai jab numbers same place-value grid share karte hain.

Case: exactly

0

input (ek defined value, jaise reference point). Kya yeh precision limit karta hai?

Nahi — ek defined/exact zero mein kisi bhi exact number ki tarah infinite precision hoti hai, isliye yeh kabhi bhi result ke sig figs ko cap nahi karta.

Connections

Significant figures — rules for operations — parent rules jinhe yeh traps stress-test karti hain.
Error propagation — relative vs absolute — har "why" ka rigorous source.
Scientific notation — trailing-zero ambiguity ka ilaaj.
Orders of magnitude & estimation — jahan 1 sig fig enough hota hai.
Measurement & uncertainty — ek "reliable digit" actually kya hota hai.