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Diperbarui Contoh terverifikasi: 8
Rounded to 2 decimal places with the “half up (school rounding)” rule, the number becomes 1234.57 (it lay between 1234.56 and 1234.57); the rounding changed it by +0.0022.
| Rule | Hasil | Change |
|---|---|---|
| Half up (school rounding) | 1234.57 | 0.0022 |
| Half to even (banker's) | 1234.57 | 0.0022 |
| Half down | 1234.57 | 0.0022 |
| Half toward +∞ | 1234.57 | 0.0022 |
| Half toward −∞ | 1234.57 | 0.0022 |
| Floor (always down) | 1234.56 | −0.0078 |
| Ceiling (always up) | 1234.57 | 0.0022 |
| Truncate (toward zero) | 1234.56 | −0.0078 |
| Away from zero | 1234.57 | 0.0022 |
The number is above halfway, so it rounds to the nearer neighbour, 1234.57. (This rule only decides exact ties: ties go away from zero.)
Rounding replaces a number with the nearest value that has a chosen number of decimal places or significant figures. The calculator finds the two candidates either side of the number at that precision and applies one of nine rules: the five “half” rules (up, to even, down, toward +∞, toward −∞) differ only when the number sits exactly halfway, while floor, ceiling, truncate and away-from-zero always move in one direction.
Lab reports round measurements to significant figures, prices are rounded to two decimal places, and programmers check which rule their language uses. The default, 1234.5678 to 2 decimal places, gives 1234.57, a change of +0.0022; the table lists the result under all nine rules at once. Negative decimal places round to tens (−1), hundreds (−2) and so on.
The rounded result keeps significant trailing zeros, so 999.96 to one place shows 1000.0, not 1000. Scientific notation shows exactly the significant figures kept, which removes the ambiguity of trailing zeros in a number like 1200.
Sumber pemeriksaan: Python Decimal('1234.5678').quantize(Decimal('0.01'), ROUND_HALF_UP)
Sumber pemeriksaan: Python Decimal('2.345').quantize(Decimal('0.01'), ROUND_HALF_EVEN) = 2.34
Sumber pemeriksaan: Python Decimal quantize with ROUND_HALF_UP = 2.35
Sumber pemeriksaan: Python Context(prec=3, rounding=ROUND_HALF_UP).plus(Decimal('0.0012345')) = 0.00123
Count from the first non-zero digit, keep the number of digits you need, and round on the next one. For 0.0012345 to 3 significant figures, skip the leading zeros, keep 1, 2 and 3, and look at the 4: it is below 5, so the answer is 0.00123, or 1.23 × 10⁻³. Rounding 1234.5 to 2 significant figures gives 1200, where the two zeros only hold place value.
Banker's rounding, or round half to even, sends an exact tie to whichever candidate ends in an even digit: 2.345 to two places becomes 2.34, and 2.355 becomes 2.36. Ties then go up and down equally often, so long sums do not drift upward. NIST SP 811 (B.7.1) gives this as its rule for discarding a 5 followed by zeros, and it is the default in IEEE 754 arithmetic and Python's round(), where round(2.5) is 2.
Under school rounding (half up) an exact 5 rounds up, away from zero: 2.345 to two places is 2.35, and −2.5 to a whole number is −3. Under banker's rounding the same ties go to the even digit, giving 2.34 and −2. The rule matters only for exact ties; 2.3451 becomes 2.35 under both because it lies above halfway.
Floor always moves down toward −∞, ceiling always up toward +∞, and truncation drops the extra digits, which moves toward zero. Floor and truncate agree for positive numbers but split for negatives: −2.5 to a whole number floors to −3, truncates to −2, and has a ceiling of −2. None of them looks at which candidate is nearer, so 1.99 floors to 1.
Enter negative decimal places: −1 rounds to tens, −2 to hundreds and −3 to thousands. 1234.5678 to −2 places is 1200 under every half rule. 1250 sits exactly halfway between 1200 and 1300, so it becomes 1300 under half up but 1200 under banker's rounding, because 2 is the even digit.
Akurasi bergantung pada masukan dan asumsi metode. Aritmetika desimal memakai 50 digit signifikan, tetapi perkiraan, metode numerik dan data sumber bisa kurang presisi; pembulatan yang ditampilkan tidak menghilangkan batasan itu. Contoh penyelesaian yang diperiksa dengan sumber independen: 8. Misalnya, “1234.5678 to 2 decimal places” diperiksa dengan Python Decimal('1234.5678').quantize(Decimal('0.01'), ROUND_HALF_UP).
IEEE 754-2019 — rounding-direction attributes (roundTiesToEven, roundTiesToAway, toward ±∞, toward 0); NIST SP 811, Appendix B.7 — rounding numerical values; Python decimal module — rounding modes.
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