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MetrologyMetro's Brief Measurement Tutorial for Quizlet Takers
Topics:
 | Approximate Versus Exact Numbers |
| Calculations With Approximate Numbers (+, -, x, ÷) |
| General Rules For Rounding Off |
| Significant Digit Rule |
Approximate
Versus Exact Numbers
An exact number is a number that has been
determined as a result of counting, such as 5 toes on my
right foot (we won't discuss my left foot), or by a
definition, such as 1 kg = 0.0685 slug or 1 N = 0.2248
lb, a conversion definition agreed to by the world
governments' bureaus of standards.
However, nearly all data of a technical nature involve approximate numbers: they have been determined as a result of a measurement process---as with a ruler. You must realize that no measurement can be found exactly; the better the measuring device used, the better the measurement.
Accuracy
A measurement may be expressed in terms of its accuracy
or its precision. The accuracy of a measurement
refers to the number of digits, called significant
digits (see below), which indicate the number of
units that we are reasonably sure of having counted when
making a measurement. The greater number of significant
digits given in a measurement, the better the accuracy,
and vice versa.
| example 1 | A measurement of 246,000 km indicates measuring 246 thousands of kilometres; its accuracy is indicated by three significant digits. |
| example 2 | A measurement of 0.087 s indicates measuring 87 thousandths of a second; its accuracy is indicated by two significant digits. |
| example 3 | A measurement of 0.04300 cm indicates measuring 43 hundred-thousandths of a centimetre; its accuracy is indicated by four significant digits. |
Notice that a zero is sometimes significant and sometimes not. To clarify this, we give the following rules for significant digits:
- All nonzero digits are significant: 562.78 mg has five significant digits.
- All zeros between significant digits are significant: 50.01 cm has four significant digits.
- A zero in a number greater than 1 which is specially tagged, such as by a bar above it, is significant: 150 g has 3 significant digits.
- All zeros to the right of a significant digit and a decimal point are significant: 210.90 m has five significant digits.
- Zeros at the right in whole-number measurements which are not tagged are not significant: 100 cm has one significant digit.
- Zeros at the left in measurements less than 1 are not significant: 0.0000418 L has three significant digits.
When a number is written in scientific notation, the decimal part indicates the number of significant digits. For example, 960,000 m would be written in scientific notation as 9.600 x 105 m.
Precision
The precision of a measurement refers
to the smallest unit with which a measurement is made,
that is, the position of the last significant digit.
| example 1 | The precision of the measurement 765,000 m is 1000 m. (The position of the last significant digit is in the thousands place.) |
| example 2 | The precision of the measurement 0.048 kg is 0.001 kg. (The position of the last significant digit is in the thousandths place.) |
| example 3 | The precision of the measurement 0.0300 cm is 0.0001 cm. (The position of the last significant digit is in the ten-thousandths place.) |
A measurement of 0.00002 cm has good precision and poor accuracy when compared with the measurement 7334.0 cm, which has much better accuracy (one versus five significant digits) and poorer precision (0.00001 cm versus 0.1 cm).
Calculations
With Approximate Numbers
The sum or difference of measurements can be no
more precise than the least
precise measurement. That is:
To add or subtract measurements:
- Make certain that all of the measurements are expressed in the same units. If they are not, then convert them all to the same units.
- Next, round each measurement to the same precision as the least precise measurement.
- Then, add or subtract.
The product or quotient of measurements can be no more accurate than the least accurate measurement. That is:
To multiply or divide measurements:
- First, multiply or divide the measurements as given.
- Then, round the result to the same number of significant digits as the measurement with the least number of significant digits.
General
Rules For Rounding Off
When a value is to be reduced in number of decimal places
(precision), one of the following three rules is
followed:
- When the digit to be
dropped is less than 5, there is no change in the
preceding figures.
Examples:
0.360414 to 0.36041 to 0.3604 to 0.360 to 0.36 - When the digit to be
dropped is greater than 5, the preceding digit is
increased by 1.
Examples:
0.026857 to 0.02686 to 0.0269 to 0.027 to 0.03 - When
the digit to be dropped is exactly 5,
round off to the nearest even number.
Examples:
0.08475 to 0.0848 but 0.08485 to 0.0848
Significant Digit
Rule
The Significant Digit Rule states that in any
calculation, the number of significant digits in the
result should not exceed the number of significant digits
in the least accurate number used in the calculation.