Measurement

Scientific Notation

    What is Scientific Notation?

  • Scientific notation is a convenient way to represent very large or very small numbers concisely
  • In scientific notation, numbers are expressed as the product of a number between 1 and 10 multiplied by some power of 10
  • Examples:
    • 8,340 → 8.34 · 103
    • 0.0000000080 → 8.0 · 10–9

    Steps for Scientific Notation

  • Count the places that you need to move the decimal (so that there is one digit followed by the decimal point), and that number becomes the exponent of 10
  • Large numbers: move the decimal to the left → exponent is positive
  • Small numbers: move the decimal to the right → exponent is negative

Significant Digits

    What are Significant Digits?

  • Significant digits are the numbers in a quantity that are actually measured, and are an indication of the degree of accuracy or precision in a measurement
  • Significant digits are used to know where to round off a calculation, as they indicate the limits of a number's precision

    Rules for Significant Digits

  • All counted quantities (e.g. 3 apples) are exact
  • All non-zero digits are significant
  • All zeros to the right of a decimal place (after a significant digit) are significant
  • All zeros between digits are significant
  • Trailing zeros in whole numbers (no decimals) may or may not be significant
  • Leading zeros are not significant

    Examples of Counting Significant Digits

  • 5.00 → 3 significant digits
  • 12000 → usually 2 significant digits (if it is likely rounded)
  • 0.0000205 → 3 significant digits

    Calculations with Significant Digits

  • Note: numbers should only be rounded after a calculation, and never round measurements
  • Addition/Subtraction: the answer should be expressed to the same number of decimal places as the least precise number in the calculation
    • Example: 15.35 + 236.4 + 0.645 = 252.395 → 252.4 (1 decimal place)
  • Multiplication/Division: the answer should have the same number of significant digits as the least precise number in the calculation
    • Example: 452.6 ÷ 37.2 x 0.0171 = 0.20805 → 0.208 (3 significant digits)