Significant Figures
Not all digits tell the truth. Learn which digits in a measurement actually matter, and how rounding preserves the honesty of scientific data.
Printable Worksheets
Print or save as PDF — or build a custom worksheet from any module's questions.
Worksheet
Download or print the worksheet to work through this lesson.
Q1: A student measures a table as 120.5 cm long. Another student measures the same table as 120.50 cm. Are these measurements the same? What does the extra zero in the second measurement tell you?
Q2: You measure the length of your classroom as 8.24 m and your friend measures it as 8.2 m. Whose measurement is more precise? How many significant figures does each measurement have?
Learning Intentions
Know
- The rules for counting significant figures in a number.
- How to round a number to a given number of significant figures.
Understand
- That significant figures represent the precision of a measurement.
- Why trailing zeros in a decimal are significant but leading zeros are not.
Can Do
- Count significant figures in any number.
- Round to a specified number of significant figures.
- Apply significant figures in calculations.
Success Criteria
- I can count the significant figures in any given number.
- I can round a number to 1, 2 or 3 significant figures.
- I can explain why $0.0045$ has 2 significant figures but $450$ has 2 or 3 depending on context.
- I can give answers to calculations with appropriate precision.
Key Terms
Common Mistakes to Avoid
Wrong: “$450$ has 3 significant figures.” Without a decimal point, trailing zeros in a whole number are ambiguous. $450$ could have 2 or 3 sig figs.
Right: $450$ has at least 2 sig figs. To show 3 sig figs, write $450.$ or $4.50 \times 10^2$. To show 2 sig figs, write $4.5 \times 10^2$.
Wrong: “$0.00450$ has 5 significant figures.” Leading zeros are never significant.
Right: $0.00450$ has 3 significant figures: the 4, the 5, and the trailing zero after the decimal.
Significant figures tell us how precisely a number was measured. Every non-zero digit counts, and some zeros count too.
Rule 1: All non-zero digits are significant.
$3.45$ has 3 significant figures.
Rule 2: Zeros between non-zero digits (sandwiched zeros) are significant.
$2.07$ has 3 significant figures. $5008$ has 4 significant figures.
Rule 3: Leading zeros are NEVER significant.
$0.0045$ has 2 significant figures. The zeros are just placeholders.
Rule 4: Trailing zeros ARE significant if there is a decimal point.
$3.400$ has 4 significant figures. $1200$ has ambiguous trailing zeros (could be 2, 3 or 4).
Dealing with ambiguity: When trailing zeros in a whole number might be significant, use scientific notation. $4.50 \times 10^3$ clearly has 3 sig figs. $4.5 \times 10^3$ clearly has 2.
What to write in your book
- All non-zero digits are significant
- Zeros between non-zero digits (sandwiched) are significant
- Leading zeros are never significant
- Trailing zeros ARE significant if a decimal point is present
Correct! The significant figures are 4, 0, 3, 0. Leading zeros are not significant. The trailing zero after the decimal IS significant.
Not quite. The significant figures are 4, 0, 3, 0. Leading zeros are not significant. The trailing zero after the decimal IS significant.
When we round to significant figures, we keep only the digits that carry meaningful precision and replace the rest with appropriate placeholders.
Steps to round to $n$ significant figures:
- Count $n$ digits from the first non-zero digit.
- Look at the next digit. If it is 5 or more, round up the last kept digit.
- Replace remaining digits with zeros (or remove them if after the decimal).
What to write in your book
- Count $n$ digits from the first non-zero digit
- Look at the next digit: 5 or more rounds up, 4 or less stays the same
- Replace remaining digits with zeros or remove if after the decimal
Correct! The first 2 sig figs are 8 and 5. The next digit is 4, which is less than 5, so we round down (leave the 5 unchanged).
Not quite. The first 2 sig figs are 8 and 5. The next digit is 4, which is less than 5, so we round down (leave the 5 unchanged).
Interactive: Significant Figures Calculator
Your Turn
Question 1: How many significant figures in $0.007020$?
Question 2: Round $6.549$ to 2 significant figures.
Question 3: Round $128{,}700$ to 3 significant figures.
Revisit Your Thinking
Look back at your Think First answer about $120.5$ cm vs $120.50$ cm. How many significant figures does each measurement have? What does the extra zero in $120.50$ tell you about the precision of the measuring instrument used?
Earlier you were asked: What was your first thought on this topic?
Now that you've worked through the lesson, write a fuller answer. What changed in your thinking?
How many significant figures does $0.004030$ have?
Round $8.549$ to 2 significant figures.
Which of the following has exactly 3 significant figures?
Round $96{,}450$ to 3 significant figures.
A measurement is recorded as $4.50 \times 10^3$ metres. How many significant figures does this have?
State the number of significant figures in each of the following.
(a) $0.00560$ (1 mark)
(b) $20{,}300$ (1 mark)
(c) $100.$ (1 mark)
Round each number to the number of significant figures specified.
(a) $7.856$ to 2 sig figs (1 mark)
(b) $345{,}900$ to 3 sig figs (1 mark)
(c) $0.009876$ to 3 sig figs (1 mark)
(d) $2.995$ to 3 sig figs (1 mark)
A rectangular room is measured as $5.2$ m by $3.8$ m.
(a) Calculate the area of the room. (1 mark)
(b) Both measurements are given to 2 significant figures. Explain why the area should also be given to 2 significant figures. (2 marks)
(c) Write the area correct to 2 significant figures. (1 mark)
(d) A student writes the area as $19.76$ m$^2$. Explain why this is misleading. (1 mark)
Non-zero digits
Always significant
Sandwiched zeros
Always significant
Leading zeros
Never significant
Trailing zeros
Significant if decimal present
Rounding
Next digit 5+ rounds up
Calculations
Answer precision matches input precision
Real-Life Link
In engineering and manufacturing, significant figures directly affect safety and cost. When Boeing builds aircraft, every rivet hole is drilled to precise tolerances — too many significant figures in specifications means unnecessary expense; too few means potential failure. Australian building standards specify concrete strength to 2 or 3 significant figures because that is what quality control testing can reliably verify. Understanding significant figures means understanding the difference between mathematical exactness and practical reality.
Game Time!
Test your significant figures skills in an interactive challenge.
Play Significant Figures Challenge