difference between significant figures and exponential notation
|Significant figures||Exponential notation|
|All the reliable digits present in a number of important or significance are called significant figures.||The representation of a very large or very small number into the power of 10 is known as exponential notation.|
|In 1000 only one significant figure.||Exponential notation for 1000 is 103.|
What are Significant Figures?
Significant figures are those digits necessary to express the results of a measurement of the precision with which it was made. No measurement is ever absolutely correct since every measurement is limited by the accuracy or reliability of the measuring instrument used.
For example, if a thermometer is graduated in one-degree intervals and the temperature indicated by the mercury column is between S 5°C and 56°C. then the temperature can be read precisely only to the nearest degree (55°C or S6°C. whichever is closer). If the graduations are sufficiently spaced.
The fractional degrees between 55°C and 56°C can be estimated to the nearest tenth of a degree. If a more precise measurement is required, then a more precise measuring instrument (e.g.. a thermometer graduated in one-tenth degree intervals) can be used. This will increase the number of significant figures in the reported measurement.
In dealing with measurements and significant figures the following terms must be understood:
Precision tells the reproducibility of a particular measurement or how often a particular measurement will repeat itself in a series of measurements.
Accuracy tells how close the measured value is to a known or standard accepted value of the same measurement.
Measurements showing a high degree of precision do not always reflect a high degree of accuracy nor does a high degree of accuracy mean that a high degree of precision lies between obtained. It is quite possible for a single.
Random measurement to be very accurate as well as to have a series of highly precise measurements be inaccurate. Ideally. high degrees of accuracy and precision is desirable. and they usually occur together. but they are not always obtainable in scientific measurements.
Every measuring device has a series of markings or graduations on it that are used in making a measurement. The precision of any measurement depends on the size of the graduations. The smaller the interval represented by the graduation. the more precise the possible measurement.
However, depending on the size of the graduations, and, as a general rule. any measurement can only be precise to ± ½ of the smallest graduation on the measuring device used. provided that the graduations are sufficiently close together.
In the cases where the measurement intervals represented by the graduations are sufficiently large. you may be able to estimate the tenths of the graduation. then the uncertainty of that measurement can be considered to be ± one unit of the last digit in the recorded measurement. (See Figures 2 and 3)
The accuracy of a measuring device depends on how exact the graduations are marked or engraved on the device in reference to some standard measurements. For most measuring devices used in everyday work. the graduations on them are usually sufficiently accurate for general use. in the laboratory, it is not always advisable to accept a measuring device as accurate unless the instrument has been calibrated.
Calibration is the process of checking the graduations on a measuring device for accuracy. As an example. consider a thermometer which is graduated in Celsius degrees difference between significant figures and exponential notation.
When this thermometer is placed in an ice bath at 0°C, it reads -1°C and when placed in boiling water at 100°C, it reads 99°C. This thermometer has been roughly calibrated over a 100° temperature range and has been found to be 1° in error. As a correction factor, 1° must be added to all temperature readings in this temperature range.
It would be better. however, to check the thermometer at several different temperatures within this range to verify that the error is indeed linear before the uniform application of the correction factor at all temperatures.
Whether the information from a series of measurements is obtained first-hand or second hand through another source. the number of significant figures must be determined in order to keep all the results meaningful.
The rules for writing and identifying significant figures are:
- All nonzero digits (digits from 1 to 9) are significant.
254 contains three significant figures
4.55 contains three significant figures
129.454 contains six significant figures
- Zero digits that occur between nonzero digits are significant.
202 contains three significant figures
450.5 contains four significant figures
390.002 contains six significant figures
In these examples. the zeros are part of a measurement.
- Zeros at the beginning of a number (i.e.. on the left-hand side) are considered to be placeholders and are not significant.
0.00078 contains two significant figures
0.00205 contains three significant figures
0.0302 contains three significant figures
In these examples. the zeros ( on the left are placeholders). It is common practice to place a zero in front of the decimal point preceding a decimal fraction. It acts as a placeholder only. It is common practice to place a zero in front of the decimal point preceding a decimal fraction. It acts as a placeholder only.
- Zeros that occur at the end of a number (i.e., on the right-hand side) that include an expressed decimal point are significant. The presence of the decimal point is taken as an indication that the measurement is exact to the places indicated.
57500. contains five significant figures
2000. contains four significant figures
34.00 contains four significant figures
25.200 contains five significant figures.
0.002050 contains four significant figures
In these examples, the zeros on the right express part of a measurement.
- Zeros that occur at the end of a number (i.e., on the right-hand side) without an expressed decimal point are ambiguous (i.e.. we have no information on whether they are significant or not) and are not considered to be significant.
575000 contains three significant figures
2000 contains one significant figure
40620 contains four significant figures
In these examples, the zeros may only be placeholders. Do not count them unless a decimal point is present. One way of indicating that some or all the zeros are significant is to write the number in scientific notation form (See Section 6). For example, a number such as 2000 would be written as 2 x 103 if it contains one significant figure. and as 2.0 x 103 if it contains two significant figures. difference between significant figures and exponential notation
ROUNDING-OFF NUMBERS (the difference between significant figures and exponential notation)
When dealing with significant figures. it is often necessary to round-off numbers in order to keep the results of calculations significant. To round-off a number such as 64.82 to three significant figures means to express it as the nearest three-digit number. Since 64.82 is between 64.8 and 64.9. but closer to 64.8. then the result of the round-off is 64.8.
A number such as 64.85 is equally close to 64.8 and 64.9. In this case and in singular cases, the rule to observe is to round off to the nearest even number which is 64.82. This rule assumes that in a series of numbers that are to be rounded off, there will be approximately the same number of times that you would have to round- off upward to the nearest even number as you would have to round-off downward. difference between significant figures and exponential notation
- Round off 75.52 to three significant figures.
Answer: 75.52 is between 75.5 and 75.6. Since 75.52 is closer to 75.5. then the answer is 75.5
- Round off 9.08352 to two decimal places.
Answer: When expressed to two decimal places. 9.08352 falls between 9.08 and 9.09. Since it is
closer to 9.08. then the answer is 9.08
- Round off 1345.54 to a whole number.
Answer: 1345.54 is between 1345 and 1346. The number, 1345.54 is closer to 1346. thus, the
answer is 1346
- Round off 7962400 to three significant figures.
Answer: To round off 7962400. use the first three significant figures. Therefore. 7962400 is
between 7960000 and 7970000. but it is closer to 7960000. The zeros must be maintained as placeholders. The answer is 7960000
- Round off 0.00027 5 to two significant figures.
Answer: 0.000275 is between 0.00027 and 0.00028. Since it is equally close to both these numbers. then it will be rounded off to the nearest even number. Round off upward to 0.00028. difference between significant figures and exponential notation
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