# 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*).

- is the degree to which the results of several measurements
- 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*

- is the degree to which the result of one measurement (or one computation based on several measurements)
- 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.)

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 Counting significant digits - Arithmetic using significant digits
- Addition & Subtraction
- The answer is only as precise as the
*least precise*measurement.

- The answer is only as precise as the
- Multiplication, Division, Powers & Roots
- The answer is only as accurate as the
*least accurate*measurement.

- The answer is only as accurate as the
- 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).

- Once the necessary number of significant digits is determined, identify the last digit to be recorded.

- Addition & Subtraction