Significant figures — rules for operations
WHY do we even need rules?
WHY: Suppose you measure a length as with a ruler marked in mm. The "2" and "3" are reliable, but you guessed the 3 by eye — it's the first uncertain digit. Writing would be a lie: it claims you know the length to a hundred-thousandth of a cm. So when we calculate with such numbers, the result must honestly reflect this limited knowledge.
Counting sig figs (quick refresher)
- Non-zero digits are always significant. ()
- Zeros between non-zeros count. ()
- Leading zeros never count (just place-holders). ()
- Trailing zeros after a decimal count. ()
- Trailing zeros with no decimal are ambiguous → use scientific notation. ( unclear; )
Rule 1 — Multiplication & Division
WHY (derived from relative error): When you multiply and , the relative uncertainties add: The number of sig figs roughly measures relative precision (a 3-sig-fig number is known to ~0.1–1%). So the answer's relative error is dominated by the worst factor → it inherits that factor's sig-fig count.
Rule 2 — Addition & Subtraction
WHY (derived from absolute error): For sums, the absolute uncertainties add: A number known to 1 decimal place () is uncertain in the tenths. Adding it to a number known to 3 decimals can't make the tenths digit reliable — the worst absolute precision wins. Hence we count decimal places here, not sig figs.
Rounding convention
- If the digit after your cutoff is < 5 → drop it (round down).
- If > 5 → round up.
- If exactly 5 (with nothing after) → round to the nearest even digit ("banker's rounding"): , . This avoids systematic upward bias.
Flashcards
Rule for multiplication/division sig figs
Rule for addition/subtraction sig figs
Why does ×/÷ use sig figs but +/− use decimal places?
When should you round in a multi-step calculation?
Sig figs in
How many sig figs do exact/counting numbers have?
Round to 1 decimal place
(correct sig figs)
Banker's rounding of and
Recall Feynman: explain to a 12-year-old
Imagine you measure a table with a cheap ruler — you can only trust it down to the centimetre. Your friend measures a pencil with a fancy tool down to the millimetre. If you add the two lengths together, the answer can only be as good as the worst tool — you can't suddenly know the table to the millimetre! For adding, we match the messiest decimal place. For multiplying, we match the number that has the fewest trustworthy digits. We never pretend to know more than our worst measurement.
Connections
- Measurement & uncertainty — sig figs are a shorthand for error propagation.
- Error propagation — relative vs absolute — the rigorous justification of both rules.
- Scientific notation — resolves ambiguous trailing zeros.
- Orders of magnitude & estimation — when only 1 sig fig matters (80/20 thinking).
- Dimensional analysis — checks what you compute; sig figs check how precisely.
Concept Map
Hinglish (regional understanding)
Intuition Hinglish mein samjho
Dekho, significant figures ka pura funda ek hi baat par tika hai: tumhara answer utna hi precise ho sakta hai jitna tumhara sabse kamzor measurement hai. Tum jhooth nahi bol sakte ki tumne table ki length millimetre tak naap li, jab ki ruler sirf centimetre tak hi sahi tha. Isliye jab numbers ko combine karte ho, rules follow karne padte hain taaki extra fake digits na aa jaayein.
Do main rules hain. Multiply ya divide karte ho to answer mein utne hi significant figures rakho jitne us factor mein hain jiske sabse kam sig figs hain. Jaise : yahan ke sirf 2 sig figs hain, to answer bhi 2 sig figs ka — . Add ya subtract karte ho to ab sig figs nahi, balki decimal places count karte hain — jiske sabse kam decimal places hain wahi answer decide karega. Jaise (kyunki mein sirf 1 decimal place hai).
Yaad rakhne ka trick: "Multiply → Sig figs, Add → decimal Aligns." Reason simple hai — multiply/divide mein relative error add hota hai (sig figs relative precision dikhati hain), aur add/subtract mein absolute error add hota hai (decimal places absolute precision dikhati hain).
Ek bada mistake jo students karte hain: beech-beech mein round kar dete hain. Mat karo! Calculation poori hone tak ek-do extra guard digit le ke chalo, aur sirf last answer ko round karo. Aur exact numbers (jaise ginti ke numbers ya ka 2) ke infinite sig figs hote hain — wo kabhi tumhare answer ki precision kam nahi karte. Exams mein yeh chhoti baatein hi marks bachati hain!