What are significant figures?

Significant figures (sig figs) tell us how precise a measured number is — they show which digits are meaningful. When you report results, you must not claim more precision than your measurements allow.

Basic rules (step by step)

• Nonzero digits are always significant. Example: 123 has 3 sig figs.
• Leading zeros (zeros before the first nonzero digit) are NOT significant. They only show the decimal place. Example: 0.0045 has 2 sig figs (4 and 5).
• Captive (or trapped) zeros (zeros between nonzero digits) ARE significant. Example: 1002 has 4 sig figs.
• Trailing zeros (zeros at the end) ARE significant only if the number has a decimal point. Example: 12.300 has 5 sig figs; 1200 (no decimal) usually has 2 sig figs unless more information is given.
• Exact numbers (counts or defined constants) have unlimited sig figs. Example: 12 eggs = exactly 12 (not 2 sig figs; it's exact).

Why trailing zeros can be ambiguous

1200 could mean 1.200×10^3 (4 sig figs) or 1.2×10^3 (2 sig figs) or just 1200 with 2 sig figs. To avoid confusion, use scientific notation when you want to show the exact number of sig figs: 1.20×10^3 (3 sig figs), 1.200×10^3 (4 sig figs), etc.

How to round

1. Look at the digit right after the last digit you want to keep.
2. If that digit is 0–4, round down (drop it).
3. If it is 5–9, round up the last kept digit by 1.

Example: Round 3.276 to 3 sig figs → keep 3.27 and look at 6 → round up → 3.28.

Rules for calculations

• Multiplication and division: The final answer should have the same number of sig figs as the factor with the fewest sig figs.
  Example: 4.56 (3 sig figs) × 1.4 (2 sig figs) = 6.384 → round to 2 sig figs → 6.4.
• Addition and subtraction: The final answer should have the same number of decimal places as the quantity with the fewest decimal places.
  Example: 12.11 (2 decimal places) + 0.3 (1 decimal place) = 12.41 → round to 1 decimal place → 12.4.

Using scientific notation to show sig figs

Scientific notation makes sig figs clear. Write the number so there is one nonzero digit before the decimal point, then count digits in the coefficient (the part before ×10^...). Example:

• 0.00450 = 4.50×10^-3 → coefficient 4.50 has 3 sig figs (4, 5, 0).
• 1200 (as 1.200×10³) has 4 sig figs.

Quick cheat sheet

• Nonzero digits: always significant.
• Leading zeros: never significant.
• Captive zeros: always significant.
• Trailing zeros: significant if decimal point is shown.
• Exact numbers: infinite sig figs.
• Multiply/divide: limit = fewest sig figs.
• Add/subtract: limit = fewest decimal places.

Practice problems (with answers)

1. How many sig figs in 0.00720?
   Answer: 3 (7, 2, 0).
2. How many sig figs in 1200?
   Answer: Ambiguous. If written as 1200 (plain) assume 2 sig figs; to show 4 sig figs write 1.200×10³.
3. Multiply: 2.5 × 3.42.
   Work: 2.5 (2 sig figs) × 3.42 (3 sig figs) = 8.55 → round to 2 sig figs → 8.6.
4. Add: 45.67 + 0.239.
   Work: 45.67 (2 decimal places) + 0.239 (3 decimal places) = 45.909 → round to 2 decimal places → 45.91.
5. Divide: 0.00450 ÷ 2.0.
   Work: 0.00450 (3 sig figs) ÷ 2.0 (2 sig figs) = 0.00225 → round to 2 sig figs → 0.0023 (in ordinary form) or 2.3×10^-3.
6. Express 120000 with 3 sig figs.
   Answer: 1.20×10⁵.

Final tips

• If you are unsure about trailing zeros, use scientific notation — it’s unambiguous.
• When doing multi-step calculations, keep extra digits during intermediate steps, and only round the final answer to the correct number of sig figs.
• Remember the difference between "decimal places" (for addition/subtraction) and "significant figures" (for multiplication/division).

If you want, I can give you more practice problems (easy to hard) or check answers to problems you already have — tell me which you prefer.