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Measurement**

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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

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.

- Precision may be expressed as a plus or minus correction.
- All measurements are approximate.
- It is impossible to know the "exact" length, mass, or amount of anything.
- The observer and the measuring instrument place a limit on the accuracy of all measurements.

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 10^{3} for four significant digits we
write 5.480 X 10^{3}

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 m^{2} as shown on a
calculator. The correct value for the area is 0.06703 m^{2}. 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:

- Rule 1. A known digit plus or minus a doubtful digit will give a doubtful digit.
- Rule 2. Only one doubtful digit is allowed in a significant figure.

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.