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NUMERIOAL ANALYSIS AND ERRORS 3 For eXample, the number 0.00345 is given upto five decimal Places. But the first two zeros after the decimal point are not Significant. It has only three significant digits. The absolute error of an approximate result or of a measurement ls the numerical difference between the true value of the quantity and the value computed or obtained by measurement. Since the absolute error is related to the number of decimal places, the : Sle NI cal nes nesta ; absolute error is y teh , : . , . ©fror is Dot very useful as an index of accuracy of the | approximate result. , 4 Let xX; and x, denote respectively the true and approximate values of a number. Then the absolute error E, is given by B=ir—X,. pes eee (1°}) The relative error E,, is defined as the absolute error divided ft by the true value. Thusif £, is the absolute error of an approxi- mation to a true value x, then EB X-— X ' ae : A coenencan . i A si F : 5. no (1,2) t is more meaningful if we express the error in the approximate value by the relative percentage error £, defined by [ x) he Gi ag Cs, Ue apd aha ete i erent gece ta nai Generally, when the exact value of a quantity is very small or very large, the relative error is meaningful. For example, let x,= 3463 10*" and E,=5x 10° then +x 10° relative error £,-= ca 010725), Thus even though the absolute error, 5x 10°, is very large the relative erfor is not large. However, if wo approximate = by 1°3333, the absolute error™ | i. ss ae Sum tf S598 ($ — 1°3333)—0,000: Scenes =O OOOOT, OU3 and the relative error = ——~ 3 Here the absolute error is of the same order as that relative error. 
