Round-off error
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A round-off error, also called rounding error, is the difference between the calculated approximation of a number and its exact mathematical value. Numerical analysis specifically tries to estimate this error when using approximation equations and/or algorithms, especially when using finite digits to represent infinite digits of real numbers.
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Example
| Notation | Represent | Approximate | Error |
|---|---|---|---|
| 1/7 | 0.142 857 | 0.142 857 | 1/7000 000 |
| ln 2 | 0.693 147 180 559 945 309 41... | 0.693 147 | 0.000 000 180 559 945 309 41... |
| log10 2 | 0.301 029 995 663 981 195 21... | 0.3010 | 0.000 029 995 663 981 195 21... |
| ∛ 2 | 1.259 921 049 894 873 164 76... | 1.25992 | 0.000 001 049 894 873 164 76... |
| √ 2 | 1.414 213 562 373 095 048 80... | 1.41421 | 0.000 003 562 373 095 048 80... |
| e | 2.718 281 828 459 045 235 36... | 2.718 281 828 459 045 | 0.000 000 000 000 000 235 36... |
| π | 3.141 592 653 589 793 238 46... | 3.141 592 653 589 793 | 0.000 000 000 000 000 238 46... |
There are, at least, two ways of performing the termination at the limited digit place:
- chopping or truncation: simply chop off the remaining digits.
- 0.142857 ≈ 0.142 (chopping at the 5th digits.)
- rounding: add 5 to the next digit and then chop it. The result may round up or round down.
- 0.142857 ≈ 0.143 (rounding at the 5th digits. This is rounded up because the next digit, 8, is >= 5)
- 0.142857 ≈ 0.14 (rounding at the 4th digits. This is rounded down because the next digit, 2, is < 5)
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