Significant figures — rules for operations: Visual walkthrough

1.1.4 · D2 · Physics › Measurement, Vectors & Kinematics › Significant figures — rules for operations

Sirf ek naya idea chahiye, aur hum pehle use draw karte hain.

Step 1 — Ek measurement ek fuzzy edge wala number hai

KYA. Kisi bhi rule se pehle, hum redraw karte hain ki ek measured number hota kya hai. Jab tum ruler se " $4.6 cm$ " padhte ho, tum kisi exact spot par ungli nahi rakh rahe — tum number line par ek chhoti si fuzzy band par ungli rakh rahe ho. Sach value us band ke andar kahin hoti hai.

KYUN. Jo bhi rule hum derive karte hain woh sirf yeh bookkeeping hai ki jab hum numbers combine karte hain toh yeh fuzzy bands kaise behave karti hain. Agar hum fuzz kabhi draw hi nahi karte, toh rules arbitrary magic lagte hain. Toh fuzz hi puri kahani hai.

PICTURE. Figure dekho. Solid orange tick woh number hai jo tumne likha; pale band woh range hai jisme sach value ho sakti hai. Us band ki half-width ko hum ek naam dete hain:

$x = x_{0} \pm Δ x$

$x_{0}$ — woh number jo tumne actually likha (band ka centre).
$Δ x$ — fuzzy band ki half-width, jise absolute uncertainty kehte hain. Chhota $Δ x$ = sharp measurement.
" $\pm$ " — "sach kahin bhi $x_{0} - Δ x$ se lekar $x_{0} + Δ x$ tak ho sakta hai".

Figure — Significant figures — rules for operations

Step 2 — Significant figures = relative fuzz ki ek picture

KYA. Yahan hum "significant figures ki count" ko relative band width se jodte hain, taaki baad mein andaza na lagana pade.

KYUN. Parent ne claim kiya tha "ek 3-sig-fig number $0.1$ – $1%$ tak jaana jaata hai." Hume dekhna hai kyun, kyunki Rule 1 puri tarah isi par depend karta hai.

PICTURE. Same relative scale par do numbers: $4.6$ (2 sig figs) aur $4.63$ (3 sig figs). Last likha hua digit hamesha uncertain wala hota hai, toh uski band roughly $\pm$ half a unit of that last place hoti hai.

$4.6$ : last place hai tenths, band $\approx \pm 0.05$ , toh $\frac{0.05}{4.6} \approx 1%$ .
$4.63$ : last place hai hundredths, band $\approx \pm 0.005$ , toh $\frac{0.005}{4.63} \approx 0.1%$ .

Har extra significant figure relative band ko roughly das guna chhota kar deta hai. Yeh page ka key sentence hai:

Step 3 — Do bands multiply karo: relative fuzzes add ho jaate hain

KYA. $A = A_{0} \pm Δ A$ aur $B = B_{0} \pm Δ B$ lo aur product banao. Hum widest aur narrowest possible product track karte hain.

KYUN. Hum yahan multiplication use karte hain (addition nahi) kyunki Rule 1 multiplication/division rule hai — aur hum discover karna chahte hain ki $A$ aur $B$ ki kaunsi property $A \times B$ mein survive karti hai.

PICTURE. Rectangle picture. Area $= A \times B$ . $A$ ko horizontal side do aur $B$ ko vertical side. Nominal rectangle hai $A_{0} B_{0}$ . Har side ko outward uski band se nudge karo toh extra area woh shaded L-shaped strip hai.

$(A_{0} + Δ A) (B_{0} + Δ B) = A_{0} B_{0} + A_{0} Δ B + B_{0} Δ A + tiny — drop it Δ A Δ B$

$A_{0} B_{0}$ — woh rectangle jo tumne draw karna tha.
$A_{0} Δ B$ — upar along thin strip add hui.
$B_{0} Δ A$ — side along thin strip add hui.
$Δ A Δ B$ — chhota corner square. Do chhoti cheezein multiply huin → negligibly chhoti, toh hum ise discard karte hain. (Kyun allowed: agar har fuzz $1%$ hai, toh corner $0.01%$ hai.)

Step 4 — $A_{0} B_{0}$ se divide karo: sig figs bahar aa jaate hain

KYA. Poori growth $A_{0} Δ B + B_{0} Δ A$ ko nominal area $A_{0} B_{0}$ se divide karo taaki relative growth mile.

KYUN. Step 2 ne bataya ki significant figures relative band track karte hain. Toh sig figs se connect karne ke liye hume absolute growth ko relative growth mein convert karna hoga — exactly yahi $A_{0} B_{0}$ se divide karna hai.

PICTURE. Step 3 ki do strips, har ek ko poore rectangle ke fraction ke roop mein re-label karo.

$\frac{Δ ( A B )}{A B} = \frac{A _{0} Δ B + B _{0} Δ A}{A _{0} B _{0}} = \frac{Δ A}{A _{0}} + \frac{Δ B}{B _{0}}$

$\frac{Δ ( A B )}{A B}$ — answer ki relative fuzz.
$\frac{Δ A}{A _{0}}$ — $A$ ki relative fuzz (uski side-strip uski apni side par).
$\frac{Δ B}{B _{0}}$ — $B$ ki relative fuzz.

Relative fuzzes simply add ho jaate hain. Ab ise Step 2 ke lens se padho: answer ki relative fuzz kam se kam utni badi hai jitni worst input ki relative fuzz. Worst relative fuzz = fewest significant figures. Toh:

Step 5 — Do bands add karo: absolute fuzzes add ho jaate hain

KYA. Ab numbers ko multiply karne ki jagah stack karo. $A$ ki band aur $B$ ki band ko usi number line par slide karo aur combined band padho.

KYUN. Addition/subtraction bilkul alag geometric operation hai — sliding, scaling nahi. Hume check karna hoga ki kya wahi "relative" logic survive karta hai. Spoiler: nahi karta.

PICTURE. Do bands end-to-end rakhe. Widest possible sum dono upper edges use karta hai; narrowest dono lower edges. Combined half-width do half-widths ko add karna hai:

$Δ (A + B) = Δ A + Δ B$

$Δ (A + B)$ — sum ki band ki half-width.
$Δ A, Δ B$ — do input half-widths, same units mein, tip-to-tip rakhe.

Notice karo: kahin bhi humne divide nahi kiya. Jo cheez add hoti hai woh physical units mein absolute width hai — jo number line par last trustworthy digit ka decimal place hai. Koi sig figs nazar nahi aate.

Step 6 — Kyun sums ke liye decimal places, sig figs nahi

KYA. "Sabse badi absolute band wins" ko ek aisi rule mein translate karo jo tum aankhon se apply kar sako.

KYUN. Hum ek shortcut chahte hain jisme koi $\pm$ arithmetic nahi lagti — bas likhe hue numbers dekho.

PICTURE. Decimal point par aligned ek vertical stack (jaise parent ki addition figure). Jis term ka last digit sabse zyada baaye baitha hai (fewest decimals) uski fattest absolute band hai. Us column ke daaye ki sab cheez already noise hai, toh answer wahan rukna chahiye.

Step 7 — Degenerate cases (kabhi ambush mein mat aana)

KYA. Char edge situations jinhe do rules survive karne chahiye.

KYUN. Contract yeh hai: reader ko koi aisa scenario nahi milna chahiye jo humne nahi dikhaya.

PICTURE. Char mini-panels, har ek mein ek band-picture.

Exact / counting numbers. " $2$ " jo $2 π r$ mein hai, ya " $n = 6$ trials". Uski band ki zero width hai ( $Δ = 0$ ). Step 4 mein woh relative sum mein $0$ contribute karta hai; Step 5 mein absolute sum mein $0$ contribute karta hai. Toh woh kabhi answer ko limit nahi karta — infinite sig figs.
Near-equal numbers ka subtraction ("catastrophic cancellation"). $5.03 - 5.01 = 0.02$ . Absolute bands ( $\pm 0.005$ dono) $\pm 0.01$ tak add ho jaati hain — answer ki size ki aadhi! Rule 2 rakhta hai 2 decimals ( $0.02$ ) lekin relative precision collapse ho gayi. Flag karo: yahan tum real error propagation par switch karo.
Kisi $\approx 0$ cheez se multiply karna. Agar $A_{0} \to 0$ , toh relative fuzz $Δ A / A_{0}$ blow up ho jaata hai. Product essentially sab noise hai; tidy sig-fig count nahi, $0$ report karo warning ke saath.
Sum result mein trailing-zero ambiguity. Agar sum $115.0$ par land kare, toh trailing zero significant hai (woh last honest decimal par baitha hai). Agar move karna ho toh Scientific notation use karo.

Ek-picture summary

Do operations, do geometries, do "what adds":

Multiply → rectangle scale karo → relative fuzzes add → SIG FIGS gino.
Add → line par slide karo → absolute fuzzes add → DECIMAL PLACES gino.

Recall Feynman: poora walkthrough simple words mein

Har measured number actually ruler par ek chhota sa smudge hai, perfect dot nahi — ek centre jiske around fuzzy band hai. Jab tum do smudges multiply karte ho tum actually do fuzzy sides se ek rectangle bana rahe ho, aur har side ka percentage fuzz area mein pile up ho jaata hai. Percentage fuzz exactly wahi hai jo "significant figures" count karta hai, toh answer woh sig-fig count inherit karta hai jo sabse sloppy side ka hai. Jab tum do smudges add karte ho tum unhe same line par slide karte ho, toh real units mein widths instead pile up hoti hain — aur real-units mein width ka matlab hai "decimal places", toh answer fattest smudge ke last decimal par ruk jaata hai. Same fuzz, do alag games: scaling versus sliding. Exact counting numbers ki zero width hoti hai, toh woh dono games se bahar baith jaate hain.

Recall Quick self-check

×/÷ sig figs kyun use karta hai lekin +/− decimal places? ::: ×/÷ relative fuzzes ko add karata hai (rectangle area), aur sig figs relative fuzz track karte hain; +/− absolute fuzzes ko add karata hai (line par sliding), aur decimal places absolute fuzz track karte hain. Exact number jaise $2 π r$ mein "2" ka kya hota hai? ::: Uski band ki zero width hai, toh woh kisi bhi fuzz sum mein kuch contribute nahi karta — infinite sig figs, kabhi answer ko limit nahi karta. $+ / -$ precision silently kab barbad kar deta hai? ::: Near-equal numbers subtract karte waqt: absolute bands same size rehti hain lekin answer shrink ho jaata hai, toh relative precision collapse ho jaati hai (catastrophic cancellation).

Connections

Significant figures — rules for operations — parent note jiske do rules hum ne abhi build kiye.
Error propagation — relative vs absolute — yahan har band picture ke peeche exact $\pm$ arithmetic.
Measurement & uncertainty — jahan se fuzzy band $x_{0} \pm Δ x$ aati hai.
Scientific notation — ambiguous trailing zeros ka clean fix (Step 7, case 4).
Orders of magnitude & estimation — jab sirf sabse badi band hi matter karti hai.
Dimensional analysis — check karta hai ki tumne kya compute kiya; sig figs check karte hain kitne precisely.