Calculus of errors, adjustment of data, approximation theory

Abstract

The calculus of errors is concerned with the precision of numerical data and the results of calculations. Errors that rest upon false mathematical reasoning, failure to pay attention to the laws of calculation, haste and lack of care in calculating, are not the subject of the calculus of errors. It does not excuse the calculator in any way from exercising extreme care in carrying out the operations of calculation. From the sides a= 7.49, b =5.32 of a triangle and the included angle y =30° a schoolboy calculates the length of the third side c from the formula c= y (a²+ b²- 2ab cos y) (see the accompanying calculation). The teacher finds the result too inaccurate; he had expected the solution c =3.92. In looking up cos λ the boy has clearly taken too few decimal places into consideration. The questions arise, what error occurs as the result of the neglected decimal places, and how many places of cos λwould have had to be retained in order to give the required accuracy (see Accuracy of the result obtained by calculating with approximate values, Example 4).