Rounding Numbers to Decimal Places and Significant Figures

Before you begin

You should be comfortable with place value (units, tenths, hundredths) and reading decimals.

The intuition

Rounding gives a number that is close to the original but easier to use. The rule depends on a single "deciding digit": if it is 5 or more, round up; if it is 4 or less, keep the rounding digit unchanged.

Definitions

• Decimal place (d.p.): the position of a digit after the decimal point. The 1st d.p. is tenths, the 2nd is hundredths, and so on.
• Significant figure (s.f.): a digit that contributes to the precision of a number. Rules:
  1. Non-zero digits are always significant.
  2. Zeros between significant digits are significant.
  3. Leading zeros (before the first non-zero digit) are not significant.
  4. Trailing zeros after a decimal point are significant.

Step-by-step method

Rounding to decimal places:

1. Find the digit in the required decimal place — this is the rounding digit.
2. Look at the next digit to the right (the deciding digit).
3. If the deciding digit is 5 or more, add 1 to the rounding digit. If it is 4 or less, leave the rounding digit unchanged.
4. Remove all digits to the right of the rounding digit.

Rounding to significant figures:

1. Identify the first significant digit (the rounding digit) counting from the left.
2. Look at the next digit (the deciding digit).
3. Apply the same 5-or-more rule.
4. Keep the place value correct — fill with zeros if needed.

Worked examples

Foundation example

Round $3.462$ to 1 decimal place. The 1st d.p. is the tenths digit: $4$. The deciding digit is $6$ (hundredths). Since $6 \geq 5$, round up. $3.462 \to 3.5$ Answer: $3.5$.

Developing example

Round $0.03875$ to 2 significant figures. The first significant digit is $3$ (the leading zeros are not significant). The second significant digit is $8$. The deciding digit is $7$. Since $7 \geq 5$, round the $8$ up to $9$. $0.03875 \to 0.039$ Answer: $0.039$.

Advanced example

A circle has area $A = 56.3\text{ cm}^2$ and radius $r$. Using $A = \pi r^2$, calculate $r$ to 3 significant figures. $r^2 = \frac{A}{\pi} = \frac{56.3}{\pi} = 17.922\ldots$ $r = \sqrt{17.922\ldots} = 4.2335\ldots$ The first three significant figures are $4, 2, 3$. The deciding digit is $3$ (the fourth significant figure), so we leave the $3$ unchanged. $r = 4.23\text{ cm (3 s.f.)}$

Common mistakes

• Counting leading zeros as significant: $0.0045$ has 2 significant figures, not 4.
• Losing place value: rounding $4996$ to 1 s.f. should give $5000$, not $5$.
• Rounding in the middle of a calculation: keep full accuracy until the final step, then round the answer.

Concept checklist

• I can identify the rounding digit and the deciding digit.
• I know the 5-or-more rule.
• I can count significant figures, including leading and trailing zeros.
• I preserve place value with zeros when needed.

Practice questions

1. Round $7.845$ to 2 d.p.
2. Round $0.00637$ to 1 s.f.
3. Round $12\,450$ to 2 s.f.
4. Round $0.9995$ to 3 d.p.
5. Round $3.14159$ to 4 s.f.
6. A calculation gives $x = 2.449489\ldots$. Round $x$ to 2 d.p.

Challenge questions

1. A number rounded to 2 s.f. is $0.030$. What is the smallest and largest value the original number could have taken?
2. Show that rounding $9.999$ to 1 d.p. gives $10.0$, and explain why the number of significant figures changes.

Frequently asked questions

What is the difference between decimal places and significant figures? Decimal places count positions after the decimal point. Significant figures count the meaningful digits of the number, starting from the first non-zero digit.

Do I round before or after the final calculation? Round only the final answer. Keep full precision during intermediate steps to avoid accumulating errors.

Related concepts

• Standard form and very large or small numbers
• Upper bounds, lower bounds and error intervals
• Understanding fractions, decimals and percentages