Measurement

and

Significant Figures

 

 

A measurement is a comparison of an unknown quality with a precisely specified quantity called a standard unit. Measurement should be made and recorded as accurately as possible. The term accuracy refers to how close the measurement comes to the exact value. Accuracy depends on the careful use of measuring instruments and the ability of the individual taking the measurement. Errors will be made in every measurement, but the magnitude of the errors can be kept small when the observer is conscientious that is, to the maximum  possible error of the measurement.

When a measurement is made, the number recorded should contain all known digits of the measurement plus one doubtful digit. The number of digits recorded for a measurement is referred to as the number of significant figures in the measurement. By definition, a significant figure is a known digit or a doubtful digit that has been determined and recorded when the measurement was taken. Remember, a proper measurement always contains all known digits that were taken utilizing the entire last count of the measuring device that was used, plus one doubtful digit that had been estimated by the person taking the reading.

Instruments for taking measurements are constructed with a calibrated scale for obtaining numerical values concerning the property being measured. The smallest division on the calibrated scale that can be read by the observer, without guessing, is known as the least count of the instrument. For example: A meter stick is divided up into 100 equal divisions and marked on the stick as centimeters. Each centimeter is further divided into 10 equal divisions. Thus, the least count of the meter stick is one tenth of one centimeter, or one-thousandth of one meter.

When the meter stick is used to take a measurement of an unknown quantity (for example, length), the observer can always obtain a value within one-thousandth (0.001) of a meter of the exact value of the unknown length, without guessing. But an additional step can increase the accuracy by one doubtful digit, because the observer can estimate a fractional part of the smallest division on the meter stick. For example, if one end of the meter stick is placed at one end of the object to be measured, and the other end of the object falls between two of the smallest divisions, the observer estimates this additional value and adds it to the known scale reading. This estimated digit is significant and is the last digit recorded when taking a measurement.

Recorded numbers may contain the digit zero (0), which may or may not be significant. When a zero digit is used to locate the decimal point, it is not significant. For example, the numbers 0.048, 0.0032, and 0.00057 each have two significant digits. When a zero appears between two nonzero digits in a number, it is significant. For example, 2.04 has three significant digits; 8.002 has four significant digits. Zeros appearing at the end of a number may or may not be significant. For example, in the number 5480 if the digit eight is a doubtful digit, then the zero is not significant. If the eight is a known digit and the zero is a doubtful digit, then the zero is significant.

The powers-of-10 notation can be used to remove the ambiguity concerning a number like 5480. When using the powers-of-10 notation, we first write all the significant digits of the number. This is followed by 10 to the correct power to locate the decimal point. For example, for three significant digits we write 5.48 X 103 for four significant digits we write 5.480 X 103

The following procedure is usually used to round off significant figures to fewer digits. If the last significant digit on the right is less than 5, drop it and insert zero instead. If the last significant digit on the right is 5 or greater, drop it and increase the preceding digit by one.

Examples

Round off the following numbers to two significant digits.

247    Because the last digit on the right is greater than 5, drop it and increase the preceding digit by one, for the result 250.

 

243     Because the last digit on the right is less than 5, drop it and insert zero instead, for the result 240.

 

245     Because the last digit on the right is 5 or greater, drop it and increase the preceding digit by one, for the result 250.

 

As a general rule, the number of significant digits of the product or the division of two or more measurements should be no greater than that of the measurement with the least number of significant digits. For example, suppose the area of a table is to be determined. This is accomplished by measuring the length and width of the table, then multiplying one value by the other. In this procedure, it is inaccurate and meaningless to calculate and give an answer indicating greater accuracy than justified by the original data. For example: the length of a table is measured with a meter stick as 1.8245 m (5 significant figures) and the width as 0.3672 m (4 significant  figures). The area A = 1.8245 m X 0.3672 m = 0.06703213 m2 as shown on a calculator. The correct value for the area is 0.06703 m2. This value has four significant digits corresponding to the least number of significant digits in the two numbers making up the original data.

When adding or subtracting significant figures use the following two rules:

For example:                    2.34      the 4 is doubtful

                                     + 16.5        the 5 is doubtful

                                         18.84     the 8 and the 4 are doubtful

 

From Rule 1, the known digit 3 plus the doubtful digit 5 equals a doubtful digit 8. Therefore, the correct answer is 18.8. This is a significant figure with only one doubtful number, which satisfies Rule 2.