Significant figures — rules for operations

1.1.4 · Physics › Measurement, Vectors & Kinematics

Rules ki ZAROORAT kyun hai?

KYUN: Maan lo tumne ek length $2.3\text{ cm}$ measure ki ek ruler se jo mm mein marked hai. "2" aur "3" reliable hain, lekin "3" tumne aankhon se guess kiya hai — ye pehla uncertain digit hai. $2.30000\text{ cm}$ likhna ek jhooth hoga: iska matlab ye hoga ki tum length ko ek lakh-waan centimetre tak jaante ho. Toh jab hum aisi numbers se calculate karte hain, result ko honestly is limited knowledge ko reflect karna chahiye.

Sig figs count karna (quick refresher)

• Non-zero digits hamesha significant hote hain. ($234\to 3$)
• Non-zeros ke beech wale zeros count hote hain. ($2.05\to 3$)
• Leading zeros kabhi count nahi hote (sirf place-holders hain). ($0.0042\to 2$)
• Decimal ke baad wale trailing zeros count hote hain. ($2.300\to 4$)
• Bina decimal ke trailing zeros ambiguous hote hain → scientific notation use karo. ($4500$ unclear; $4.50\times 10^3\to 3$)

Rule 1 — Multiplication & Division

KYUN (relative error se derived): Jab tum $A$ aur $B$ multiply karte ho, relative uncertainties add hoti hain: $\frac{\Delta (AB)}{AB}=\frac{\Delta A}{A}+\frac{\Delta B}{B}.$ Sig figs roughly relative precision measure karte hain (ek 3-sig-fig number ~0.1–1% tak known hota hai). Toh answer ki relative error sabse kharab factor se dominate hoti hai → usse us factor ka sig-fig count milta hai.

Rule 2 — Addition & Subtraction

KYUN (absolute error se derived): Sums ke liye, absolute uncertainties add hoti hain: $\Delta (A+B)=\Delta A+\Delta B.$ Ek number jo 1 decimal place tak known hai ($x.x$) woh tenths mein uncertain hai. Use 3 decimals tak known number ke saath add karne se tenths digit reliable nahi ho jaata — sabse kharab absolute precision jeetti hai. Isliye yahan decimal places count karte hain, sig figs nahi.

Rounding convention

• Agar cutoff ke baad wala digit < 5 hai → use drop karo (round down).
• Agar > 5 hai → round up karo.
• Agar exactly 5 hai (baad mein kuch nahi) → nearest even digit tak round karo ("banker's rounding"): $2.5\to 2$, $3.5\to 4$. Isse systematic upward bias avoid hota hai.

Flashcards

Connections

• Measurement & uncertainty — sig figs error propagation ka shorthand hain.
• Error propagation — relative vs absolute — dono rules ka rigorous justification.
• Scientific notation — ambiguous trailing zeros resolve karta hai.
• Orders of magnitude & estimation — jab sirf 1 sig fig matter karta hai (80/20 thinking).
• Dimensional analysis — check karta hai kya compute kar rahe ho; sig figs check karte hain kitni precisely.

Concept Map

Deep Dive

• D1 · Foundations — har symbol zero se
• D2 · Visual walkthrough — derivation pictures mein
• D3 · Worked examples — har scenario
• D4 · Exercises — graded, full solutions
• D5 · Question bank — concept traps