The Physics
Hypertextbook
Opus in profectus

# Significant Digits

## 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).