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General Physics I:

Sunil Karna

Subsection 1.1.3 Significant Figures

Significant figures are the digits which give us useful information about the accuracy of a measurement. The result of a measurement includes digits that are known reliably and the last digit that is uncertain. For example, the length of an object measured to be \(287.5\) cm when measured with a meterstick marked with centimeter (0.1 resolution) and has four significant figures, the digits 2, 8, 7 are certain while the last digit 5 is uncertain. If someone measure the same object with another meterstick marked with millimeters (0.01 resolution) one could have measured the length to be 287.54 cm and that contains 5 significant figures, the digits 2, 8, 7, and 5 are certain while the last digit 4 is uncertain. Each smaller measurement allows observers to determine the length of the object with a bit more accuracy. Maybe someother can measure with great accuracy as 287.542 cm. No matter what you do, there must be some inaccuracy always come at every measurement. Scientists account for this unavoidable uncertainty in measurement by using significant digits. significant digits do not remove the uncertainty but warn readers to where uncertainty lies.
\(1000\) has one significant digit: only 1 is interesting and reliable. The zeroes may have been just the placeholders; they may have rounded something off to get this value.
1000.0 has five significant digits: the ".0" tells us about the accuracy of the measurement that the measurement is accurate to the tenths place.
In scientific notation if 200 has two significant figures then \(2.0\times10^{2}\) is used. If it has three sig.fig. then \(2.00\times10^{2}\) is used. If it had four then 200.0 is sufficient.
Here are some basic rules to determine significant figures:
  1. All nonzero digits are significant; in number 0.003004500 the digits 3, 4, and 5 are significant.
  2. All zeroes between significant digits are significant; in number 0.003004500 the 0, 0 between 3 and 4 are significant.
  3. Trailing zeros in the decimal portion are significant; in number 0.003004500 the 0,0 at end of 5 are significant.
  4. Trailing zeroes in a whole number with no decimal shown are not significant; in number "540" there are only two sig.fig. 5 and 4, the last 0 is not significant.
  5. Leading zeroes in the decimal portion to the first non-zero digit are not significant; in number 0.003004500 the 0 to the right of decimal point and digits 0, 0 to the left of decimal point before the digit 3 are not significant.
  6. Exponential digits in scientific notation are not significant; \(1.32\times 10^{6}\) has only three significant figures, 1, 3, and 2.

Subsubsection 1.1.3.1 Rounding off the Numbers

When rounding, we examine the digit following to the right of the digit that is to be the last digit in the rounded off number. The digit we are examining is the first digit to be dropped.
If digit to be dropped off is less than 5, then drop it off with all the other digits to the right of it.
If the digit to be dropped off is greater than 5, then increase the number by 1 to the preceeding digit to be rounded.
If the digit to be dropped is 5, then round the preceeding digit so that it will become even number. Zero is considered to be even when rounding off. This is the process of minimizing systematic error. Here are some examples:
  1. Round 742,396 to four, three, and two significant digits: To round to four sig. fig. start with the first significant digit, which is the 7. Then count to the right from there. The first four significant digits of 742,396 are the 7, the 4, the 2, and the 3. Just to the right of the 3 is a 9. Because this value is "greater than 5", round the 3 up to 4. Replace the remaining digits (the 9 and the 6) with zeroes. Then: 742,400 (four significant digits).
    To round 742,396 to three places, start again with the 7 and include the next two digits, being the 4 and the 2. Since the next digit is a 3, which is "less than 5", leave the 2 alone and don’t round up. Replace the three digits after the comma with zeroes. Then: 742,000 (three significant digits).
    To round 742,396 to two places, use only the first two digits, which are followed by a 2, so don’t round up. Instead, just replace the final four digits with zeroes, to get: 740,000 (two significant digits).
  2. Round 0.07284 to four, three, and two significant digits: To round 0.07284 to four significant digits, start with the first significant digit, which is the 7. (The zero between the decimal point and the 7 is not significant, as it serves only to "place" the 7 into the hundreds place.) There are only three more digits, so all of them will be included in answer. Since no digit follows the 4, there is no information about rounding, so just leave the 4 as it is. Hence, 0.07284 is four sig-digs.
    When rounding 0.07284 to three sig-digs, the final sig-dig is the 8, which is followed by the 4. Since 4 is less than 5, so simply drop it. Because these sig-digs are after the decimal point, not replace the 4 with a zero. Hence, 0.0728 is three sig-digs.
    To round 0.07284 to two sig-digs, use the 7 and the 2. Since the 2 is followed by an 8, round the 2 up to 3; drop everything that follows. Hence, 0.073 is two sig-digs.
  3. Round 425.35 to four and two significant digits: For four sig.fig. this number is 425.4, because the value to be rounded off (3) is "odd" and followed by a "5", then round the 3 up to 4 and remove the remaining digits. For two sig.fig. it would be 420, because the value to be rounded off (2) is "even" and followed by a "5", then leave 2 as it is and replace the other with zeroes.

Subsubsection 1.1.3.2 Mathematical Operations with Significant Figures

  1. Addition or subtraction: The final result should retain as many decimal places as there are in the number with the least decimal places in the data. For example, the sum of the numbers 436.32 g, 227.2 g and 0.301 g is 663.821 g by mere arithmetic addition but the least precise measurement (227.2 g) is correct to only one decimal place. Therefore, the final result should be rounded off to 663.8 g. Similarly, the difference in lengths of 66.25 m and 23.3055 m is \(66.25 \,m - 23.3055 \,m = 42.9445 \,m\text{.}\) Which can be expressed as 42.94 only.
  2. Multiplication or division: The final result should retain as many significant figures as are there in the original number with the least significant figures. For example, if mass of an object is measured to be, say, 4.237 g (four significant figures) and its volume is measured to be \(2.51 \,cm^{3}\text{,}\) then its density, by mere arithmetic division, is 1.68804780876 \(g/cm^{3}\) upto 11 decimal places. But density should be reported to three significant figures.
    \begin{equation*} Density =\frac{4.237 \,g}{2.51 \,cm^{3}}= 1.69\,g/cm^{3}. \end{equation*}
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