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