Digits, rounded truncated

Typically, the error in equation (7.1) is called truncation error. This is a hypothetical definition of the error, because it excludes gross errors, which are caused by unpredictable mistakes (for example, human or mechanical mistakes in an experiment), and round off errors, which are a result of having only a finite number of digits. The truncation error is the error that is implied in an approximation. More in-depth definitions on error can be found in the literature, as presented by Achieser [2], Hildebrand [10] or Davis [8],... 

Machine errors - Errors such as discussed above due to the representation of numbers by a finite number of binary digits with a resulting finite number of significant digits. Also truncation and rounding errors in machine calculations. 

Precision all calculations are performed in so-called double precision , that is, to a precision of about 1 in 1015, even when only a few digits are shown. Truncation and round-off errors are therefore seldom a problem in spreadsheet calculations. 

As illustrated in the foregoing discussion, the numerical solution is expected to approach the exact solution as we use more refined discretizations. In reality, however, computer solutions may be somewhat limited in accuracy by the number of digits employed by the processor. This restriction associated with rounding to a finite number of digits results in round-off errors. These errors increase with the number of arithmetic operations required to produce a solution, hence with the number of grid points. The latter depends on the size of the problem and the degree to which the discretization is refined. The ultimate accuracy of a numerical solution can therefore be expected to be achieved in a trade-off between truncation errors and round-off errors. The effect of... 

In most states, a blood alcohol level of 0.08% is the cutoff for intoxication. How would a value of 0.0815 be interpreted What about 0.07999 0.0751 Are these values rounded off or are they truncated If they are rounded, to how many significant digits Of course, this is an artificial, but still telling, example. How numerical data are rounded depends on the instrumentation and devices used to obtain the data. Incorrect rounding might have consequences. 

Note that the value of sum has been truncated to three digits after the decimal, and the last digit is rounded up. The n at the end of the format string is the new line control character—any further output appears on a new line. 

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