Significant Digits

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Discussion

Measure what is measurable, and make measurable what is not so. (Galileo)

Summary

• Precision …
  • is the degree to which the results of several measurements agree with one another
  • is the exactness of a device
  • is determined by the place value of the last recordable digit (often referred to as the number of decimal places).
• Accuracy …
  • is the degree to which the result of one measurement (or one computation based on several measurements) agrees with its true value
  • is the exactness of a measurement (or computation based on measured values)
  • is determined by counting the number of significant digits
• The significant digits (or significant figures) in a measurement …
  • are those that are a part of the measurement
  • does not include placeholder zeroes
  • must be written even if the measurement …
    • is stated in standard form scientific notation
    • is converted to units that are decimal multiples or divisions of the original (for example, km, m, mm, etc.)

  Counting Significant Digits
  digit	location		significant?
  non-zero	anywhere		yes
  zero	initial		no
  "	medial		yes
  "	final	after the decimal point	yes
  "	"	before a written decimal point	yes
  "	"	before an unwritten decimal point	maybe

• Arithmetic using significant digits
  • Addition & Subtraction
    • The answer is only as precise as the least precise measurement.
  • Multiplication, Division, Powers & Roots
    • The answer is only as accurate as the least accurate measurement.
  • Numbers that are a part of a mathematical equation were never measured and therefore cannot affect the accuracy of a computation.
    • Rational numbers (1, 2, 3, ½, ⅔, etc.) are "perfect numbers" in theory and practice.
    • Irrational numbers (√2, π, e, etc.) …
      • cannot be written in decimal form using a finite number of digits
      • are only as accurate as the number of digits used for computational purposes
      • are effectively "perfect numbers" on a calculator since the number of digits returned is greater than that of nearly every measurement ever made.
        • The π button on calculator gives so many digits that it does not affect the accuracy of most computations.
  • The formally stated results of any computation based on measured values should be stated with an appropriate number of significant digits.
    • Once the necessary number of significant digits is determined, identify the last digit to be recorded.
      • Add one to this digit if the next digit is 5 or greater; that is, round up.
      • Do nothing to this digit if the next digit is 4 or less; that is, round down.
    • Any necessary rounding should be done only after computation is completely finished.
      • Never round the results of a partially calculated value.
      • If the results of one computation are to be used in another, use the unrounded value of the first computation (if it is available).

Problems

practice

1. Write something.
   • Answer it.
2. Write something.
   • Answer it.
3. Write something else.
   • Answer it.
4. Write something completely different.
   • Answer it.

numerical

1. Write something.

Resources

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