Rounding and estimation

Decimal places count from the point

To round 8.746 to two decimal places, keep the hundredths digit 4 and inspect the next digit 6. Round up to 8.75. To one decimal place it is 8.7, because the next digit is 4. Round from the original value each time; rounding an already rounded value can change the result.

For a negative value such as −2.678 to two decimal places, locate the two nearest hundredth values: −2.67 and −2.68. It is nearer −2.68. Do not confuse increasing a retained digit’s magnitude with moving right on a number line.

Significant figures start at the first non-zero digit

In 0.004072, the leading zeros locate the value but do not count as significant figures. The significant digits begin 4, 0, 7, 2. To three significant figures the value is 0.00407. The zero between 4 and 7 is significant.

To three significant figures, 58,746 is 58,700. Standard form can make the intended precision clearer: 5.87 × 10⁴. Carrying across a place boundary, 9.997 to three significant figures is 10.0, or 1.00 × 10¹.

An estimate is a deliberate rough calculation

For (19.8 × 4.06)/0.51, use (20 × 4)/0.5 = 160. A calculator gives about 157.624. A result near 16 or 1600 would be implausible. The estimate checks scale; it does not establish that every digit is correct.

Keep extra digits during intermediate calculations and round the requested final result. Use ≈ for an approximation rather than claiming exact equality.

Check your understanding 1 Read the requested precision

Round 0.04568 to two decimal places and to two significant figures. Why are the answers different?

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Two decimal places gives 0.05. Two significant figures gives 0.046. Decimal places count positions after the point; significant figures begin at the first non-zero digit.

Choose a useful estimate

An answer to 48.7 × 0.203 is recorded as 98.861. Check its scale without repeating the full multiplication.