Significant figures are the digits in a measurement that are known reliably, plus the first uncertain one. They are how a number honestly advertises its own precision: writing a length as 2.50 m claims more than 2.5 m, because the trailing zero says the hundredths place was actually measured. Two skills live here — counting the significant figures a number already carries, and not manufacturing precision when you compute with them.
1 — The least count sets the precision. Every instrument can resolve only down to its least count (LC) — the smallest division it can read. A metre scale marked in millimetres has LC = 1 mm; a vernier calliper has LC = 1 MSD − 1 VSD (equivalently 1 MSD / n when n vernier divisions span n−1 main divisions), typically 0.1 mm; a screw gauge / micrometer has LC = pitch / (number of circular-scale divisions), typically 0.01 mm. A measurement is read as main-scale reading + (coinciding division × LC), corrected for any zero error (a non-zero reading when the jaws are closed: a positive zero error is subtracted, a negative one is added).
2 — Counting significant figures. The rules: (i) every non-zero digit is significant; (ii) zeros between non-zero digits are significant (3.05 → 3 s.f.); (iii) leading zeros are never significant — they only fix the decimal point (0.0047 → 2 s.f.); (iv) trailing zeros are significant only if there is a decimal point (4.50 → 3 s.f., but 4500 is ambiguous); (v) scientific notation removes the ambiguity — 4.5 × 10³ shows 2 s.f., 4.50 × 10³ shows 3. A change of unit never changes the count: 5.60 cm and 0.0560 m both have 3 s.f.
3 — Rounding. To round to a required number of significant figures or decimal places: if the first dropped digit is > 5 round up, < 5 round down, and for exactly 5 the common convention rounds up (some texts round to the nearest even digit — state which you use). Rounding to N significant figures and to N decimal places are different operations — don't confuse them.
4 — Arithmetic doesn't create precision. The result of a calculation can be no more precise than its least-precise input. Addition and subtraction: the result keeps the least number of decimal places among the operands (12.3 + 4.56 = 16.9, one decimal). Multiplication and division: the result keeps the least number of significant figures (2.5 × 3.42 = 8.6, two s.f.). The classic trap is applying the wrong rule — using least-significant-figures on a sum, or least-decimals on a product. …