In a number with or without a decimal point, trailing zeros those to the right of the last non-zero digit are significant provided they are justified by the precision of their derivation: ,; 2. Rules for Arithmetic Operations with Significant Figures In multiplication or division, the final result should retain as many significant figures as are there in the original number with the least significant figures. In addition or subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places.
Dimensions of Physical Quantities The dimensions of a physical quantity are the powers or exponents to which the base quantities are raised to represent that quantity. The exponential term is never considered as a significant figures.
There are five 5 significant figures in m. This is because zero is present between the nonzero digit. Since the solution to 45P from 2 chapter was answered, more than students have viewed the full step-by-step answer. This full solution covers the following key subjects: figures, measured, quantity, significant.
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Elite Notetakers. Referral Program. Campus Marketing Coordinators. In a correctly reported measurement, the final digit is significant but not certain. Insignificant digits are not reported. The 3 is not significant and would not be reported. Some error or uncertainty always exists in any measurement. The amount of uncertainty depends both upon the skill of the measurer and upon the quality of the measuring tool. Many measuring tools such as rulers and graduated cylinders have small lines which need to be carefully read in order to make a measurement.
The bottom ruler contains no millimeter markings. While the 2 is known for certain, the value of the tenths digit is uncertain. The top ruler contains marks for tenths of a centimeter millimeters. The measurer is capable of estimating the hundredths digit because he can be certain that the tenths digit is a 5. In this case, there are two certain digits the 2 and the 5 , with the hundredths digit being uncertain. Clearly, the top ruler is a superior ruler for measuring lengths as precisely as possible.
When you look at a reported measurement, it is necessary to be able to count the number of significant figures. The rules below details how to determining the number of significant figures in a reported measurement.
This indicates a low-precision, high-accuracy measuring system. However, in Figure b , the GPS measurements are concentrated quite closely to one another, but they are far away from the target location. This indicates a high-precision, low-accuracy measuring system. The precision of a measuring system is related to the uncertainty in the measurements whereas the accuracy is related to the discrepancy from the accepted reference value. Uncertainty is a quantitative measure of how much your measured values deviate from one another.
There are many different methods of calculating uncertainty, each of which is appropriate to different situations. Some examples include taking the range that is, the biggest less the smallest or finding the standard deviation of the measurements. If the measurements are not very precise, then the uncertainty of the values is high. If the measurements are not very accurate, then the discrepancy of the values is high.
Recall our example of measuring paper length; we obtained measurements of We might average the three measurements to say our best guess is We might calculate the uncertainty in our best guess by using the range of our measured values: 0.
Then we would say the length of the paper is Returning to our paper example, the measured length of the paper could be expressed as Since the discrepancy of 0.
Some factors that contribute to uncertainty in a measurement include the following:. At any rate, the uncertainty in a measurement must be calculated to quantify its precision. If a reference value is known, it makes sense to calculate the discrepancy as well to quantify its accuracy.
Another method of expressing uncertainty is as a percent of the measured value. A grocery store sells 5-lb bags of apples. We obtain the following measurements:. We then determine the average weight of the 5-lb bag of apples is 5. The uncertainty in this value,. We can use the following equation to determine the percent uncertainty of the weight:.
SignificanceWe can conclude the average weight of a bag of apples from this store is 5. Notice the percent uncertainty is dimensionless because the units of weight in. A high school track coach has just purchased a new stopwatch. Why or why not? The uncertainty in the stopwatch is too great to differentiate between the sprint times effectively.
Uncertainty exists in anything calculated from measured quantities. For example, the area of a floor calculated from measurements of its length and width has an uncertainty because the length and width have uncertainties. How big is the uncertainty in something you calculate by multiplication or division? If the measurements going into the calculation have small uncertainties a few percent or less , then the method of adding percents can be used for multiplication or division.
This method states the percent uncertainty in a quantity calculated by multiplication or division is the sum of the percent uncertainties in the items used to make the calculation. For example, if a floor has a length of 4. Expressed as an area, this is 0. An important factor in the precision of measurements involves the precision of the measuring tool. In general, a precise measuring tool is one that can measure values in very small increments. For example, a standard ruler can measure length to the nearest millimeter whereas a caliper can measure length to the nearest 0.
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