Section 1.1 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 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 first digit 5 is uncertain. If someone measure the same object with another meterstick marked with millimeters (0.01 resolution) he/she 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 first 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 how accuratly you measure things, there is always some inaccuracy in 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. : notice that ”point” after the zero! This has four significant digits, because the decimal point tells us that the measurement was made to the nearest unit, so the zero is not just a placeholder.
- 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 \(2.0 \times 10^2\) is used. If it has three sig.fig. then \(2.00 \times 10^2. \) If it had four sig. fig. then 200.0 is sufficient.
Here are some basic rules to determine significant figures:
- All nonzero digits are significant; in number 0.003004500 the digits 3, 4, and 5 are significant.
- All zeroes between significant digits are significant; in number 0.003004500 the 0, 0 between 3 and 4 are significant.
- Trailing zeros in the decimal portion are significant; in number 0.003004500 the 0,0 at end of 5 are significant.
- 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.
- 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.
- Exponential digits in scientific notation are not significant; \(1.32 \times 10^6\) has only three significant figures, 1, 3, and 2.
