Round values and describe their intervals

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Round values and describe their intervals

Decimal places count positions after the decimal point. Significant figures begin at the first nonzero digit and describe the meaningful size of a number. To round, inspect the first discarded digit: 5 or more increases the last retained digit under the usual positive-number convention.

Rounding removes information. A positive value reported as 2.4 to the nearest tenth could originally be any value from 2.35 inclusive up to 2.45 exclusive. The interval spans half a rounding step on each side. Use inequality notation to retain which endpoints belong.

Truncation removes digits without rounding. For a positive value truncated to 2.4 at one decimal place, the interval is 2.4 ≤ x < 2.5. This differs from the rounding interval. For negative values, truncation towards zero needs separate care; do not blindly reuse the positive interval.

Worked example: Round 0.006784 to two significant figures.

The first significant digit is 6, not either leading zero.

Keep 6 and 7; the next digit 8 rounds 7 upward.

Answer: 0.0068.

Check your understanding: A positive length rounds to 8 cm to the nearest centimetre. Give its interval.

Answer: 7.5 cm ≤ length < 8.5 cm.

Why: Half a centimetre on either side captures all values that round to 8 under the usual rounding rule.