1.3: significant figures in measurements

significant figures

• significant figures: the digits in a measurement that were actually measured
• all known numbers + one estimated number
• communicate the uncertainty of a measurement
• reflect the precision of the instrument with which a measurement is made

determining which digits are significant

• all non-zero digits
  • eg. 1245 (4 significant figures)
  • 2.3 (2 significant figures)
• all captive zeros (zeros between non-zero digits)
  • 2021 (4 significant figures)
  • 5000.1 (5 significant figures)
• leading zeros are never significant
  • 0.0012 (2 significant figures) → placeholders
• zeros at the end of a number (trailing zeros) are
  • not significant if there is no decimal point in the number:
  • 230 (2 significant figures)
  • 7000 (1 significant figure)
• significant if there is a decimal point in the number
  • 3.230 (4 significant figures)
  • 7.0 (2 significant figures)

scientific notation

• used to describe very large or very small numbers because
  • it’s time-consuming to read/write
  • easy to lose track of zeros and mess up
• numbers are expressed with
  • one digit to the left of the decimal, multiplied by a power of ten
  • eg. 4.59x10^5
  • 4.59 is the significand, 10 is the base, and 5 is the exponent.
  • if x > 1, the exponent is positive, and if x < 1, the exponent is negative
• scientific notation and significant figures
  • significand determines number of significant figures
  • eg. 4.590 x 10^5 => 5 sigfigs (4.590)
• significant figures in calculations
  • addition/subtraction—answer is rounded to the smallest significant decimal place
  • multiplication/division—answer is rounded to the number of digits that corresponds with the least number of significant figures in any of the numbers used in the calculation

dimensional analysis

• used to convert a measurement from one unit to another using conversion factors
  • conversion factor: a ratio of equivalent measures (numerator = denominator)
• when a measurement is multiplied by a conversion factor, the expression of measurement changes (because the units change) but the actual quantity measured remains the same