Significant Figures Notes

Leading zeros and significance

  • Rule from transcript: leading zeros are not significant. If there is a zero before or after the decimal point and there are zeros before a nonzero digit, those zeros are not significant.
  • Statement 2 from transcript: anything that's not a zero is significant (i.e., digits 1–9 are significant).
  • Statement 3 from transcript: any digit after a decimal point is significant (i.e., digits to the right of the decimal are significant; zeros there count as significant in this context).
  • Quick takeaway from the transcript: use the digits that carry meaning; leading zeros do not count toward significant figures; nonzero digits always count; digits to the right of the decimal point count as significant.
  • Examples (based on transcript rules to illustrate):
    • 0.00456 has significant digits 4, 5, and 6 (3 significant figures).
    • 0.070 has significant digits 7 and 0 (2 significant figures) because the zeros to the left of the first nonzero digit are not significant, while digits to the right of the decimal point are significant.
    • 1.2300 has significant digits 1, 2, 3, 0, 0 (depending on interpretation of trailing zeros; under the transcript rules, zeros after the decimal are significant).

Addition and rounding rules

  • Core rule from transcript: when you add, you use both numbers, and you look to see which one has the least number of decimal places; you round to that number of decimal places.
  • This means the precision of the result cannot exceed the least precise addend in terms of decimal places.
  • Formula reference (based on transcript):
    • Let $da$ be the number of digits to the right of the decimal in $a$, and $db$ be the number for $b$. Then the result $c = a + b$ should be rounded to $ ext{min}(da, db)$ decimal places.
  • Worked example (illustrative):
    • Consider $a = 12.4$ and $b = 7.89$.
    • $da = 1$, $db = 2$, so we round to 1 decimal place.
    • $12.4 + 7.89 = 20.29 \,\to\, \,\text{rounded to 1 decimal place} \,=\, 20.3$.
    • This can be shown as: 12.4+7.89=20.29rounded to 1 decimal place: 20.3.12.4 + 7.89 = 20.29 \quad \text{rounded to } 1 \text{ decimal place: } 20.3.