Significant figures and rounding rules
WHY do significant figures exist?
WHAT is the problem? Suppose you measure a length with a ruler marked in mm. You can confidently read to the mm, and estimate one digit beyond. If you report cm, you are lying — you claimed you know the length to a millionth of a cm. If you report cm, you threw away real information.
WHY does it matter in chemistry? A calculated answer can never be more precise than the measurements it came from. If you divide a mass known to 3 digits by a volume known to 2 digits, the density is only good to 2 digits. Sig figs protect you from fake precision leaking into your final result.
WHAT counts as a significant figure?
The rules (with the WHY behind each):
- All non-zero digits are significant. ( are always "real" measured digits.)
- Zeros between non-zeros are significant — "sandwiched zeros." has 3 sig figs. Why: you can't measure the 5 without passing through that 0.
- Leading zeros are NOT significant. has 2 sig figs. Why: those zeros only mark the decimal point's place; they are placeholders, not measurements.
- Trailing zeros AFTER a decimal point ARE significant. has 3 sig figs. Why: you would not bother writing that final 0 unless you measured it.
- Trailing zeros in a whole number (no decimal) are ambiguous. could be 2, 3, or 4 sig figs. Fix: use scientific notation.
Exact numbers have INFINITE sig figs
Why: there is no uncertainty in "exactly 12." Counting 12 objects isn't an estimate.
Rounding rules — the honest way to shrink a number
To round to a chosen number of sig figs, look at the first digit you are dropping (the "decider"):
WHY round-half-to-even? If you always rounded 5 up, then over many calculations you'd systematically push numbers upward, creating a bias. Rounding half the time up and half down (by choosing even) keeps the average error near zero. This is the steel-manned, unbiased rule.
Sig figs in CALCULATIONS
WHY are they different?
- Multiplying/dividing scales relative uncertainty (percent error), so we track significant figures (a relative measure).
- Adding/subtracting stacks absolute uncertainty (in the same units), so we track decimal places (an absolute position measure).
Worked examples
Common mistakes (Steel-manned)
Recall Feynman: explain to a 12-year-old
Imagine you tell a friend, "I walked about 3 kilometers." You don't really know it was 3.000000 km — you just know it's roughly 3. Significant figures are the "how sure am I?" digits. If your steps only tell you "about 3," it's cheating to say "3.0000." And when you do maths with fuzzy numbers, your answer can't be sharper than the fuzziest number you started with — mud plus clean water is still a bit muddy. Rounding is trimming your answer back to how sure you honestly are, and when you're right on the edge (a 5), you flip fairly between up and down so you don't cheat in one direction.
Active-recall flashcards
How many significant figures in ?
Why are leading zeros not significant?
Rule for sig figs in multiplication/division?
Rule for sig figs in addition/subtraction?
Why do +/- and ×/÷ use different rules?
What is banker's rounding and why use it?
How many sig figs does an exact/counted number have?
Round to a whole number using round-half-to-even.
Convert (3 sig figs) to scientific notation.
Why round only at the end of a calculation?
Connections
- Scientific notation and orders of magnitude
- Accuracy vs precision
- Uncertainty and error propagation
- Units and dimensional analysis
- The mole and molar mass calculations
- Density and derived quantities
Concept Map
Hinglish (regional understanding)
Intuition Hinglish mein samjho
Dekho, significant figures ka matlab simple hai: har measurement ek promise hai ki tumhe kitna pata hai. Jab tum likhte ho mL, tum keh rahe ho ki aur pakke hain, aur thoda uncertain hai. Agar tum likh do to yeh jhooth hai — tum aisi precision claim kar rahe ho jo tumhare paas hai hi nahi. Isliye extra zeros mat lagao jo tumne actually measure nahi kiye.
Rules yaad rakhne ke liye: non-zero digits hamesha count hote hain. Beech ke zeros (jaise ka ) count hote hain. Shuru ke zeros (jaise ) count nahi hote — woh sirf decimal ki jagah batate hain, placeholder hain. Aur decimal ke baad wale trailing zeros ( ka last ) count hote hain, kyunki tumne unhe jaan-boojhkar likha. Confusion se bachne ke liye scientific notation best hai — usme har digit significant hota hai.
Calculation me do alag rules hain, aur yeh log ka common galti hai. Multiply ya divide karo to answer utne hi sig figs ka jitna sabse kam wale number me tha. Add ya subtract karo to sig figs nahi, balki decimal places dekho — jiske sabse kam decimal places, utne answer me. Yeh isliye alag hain kyunki multiplication me relative error judta hai aur addition me absolute error.
Rounding me last rule interesting hai: jab exactly ho aur aage kuch na ho, to nearest even pe jao (banker's rounding). Iska reason yeh hai ki agar hamesha 5 ko upar round karoge to numbers dheere-dheere bade hone lagenge — bias aa jayega. Aur ek important tip: beech me har step pe round mat karo, warna chhoti-chhoti errors jama ho jaati hain — sirf final answer ko round karo. Bas itna dhyaan rakho, exam me sig figs ke marks kabhi nahi katenge!