Round-off error
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A round-off error is the difference between the calculated approximation of a number and its exact mathematical value. Numerical analysis specifically tries to estimate it when using approximation equations and/or algorithms, especially when using finite digits to represent infinite digits of real numbers.
Example
| Notation | Represent | Approximate | Error |
|---|---|---|---|
| 1/7 | 0.142857 | 0.142857 | 1/7000000 |
| ln 2 | 0.69314718055994530941... | 0.693147 | 0.00000018055994530941... |
| log10 2 | 0.30102999566398119521... | 0.3010 | 0.00002999566398119521... |
| ∛ 2 | 1.25992104989487316476... | 1.25992 | 0.00000104989487316476... |
| √ 2 | 1.41421356237309504880... | 1.41421 | 0.00000356237309504880... |
| e | 2.71828182845904523536... | 2.718281828459045 | 0.00000000000000023536... |
| π | 3.14159265358979323846... | 3.141592653589793 | 0.00000000000000023846... |
There are, at least, two ways of performing the termination at the limited digit place:
- Chopping: It simply chops off the remaining digits.
- 0.142857 ≈ 0.142 (chopping at the 5th digits.)
- Rounding: It adds 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 round up because the next digit, 8, is >= 5)
- 0.142857 ≈ 0.14 (rounding at the 4th digits. This is round down because the next digit, 2, is < 5)