Significant Figures — Core Principles

Core Principles

Significant figures are the meaningful digits in a number, representing the precision of a measurement. They include all certain digits plus one estimated digit. The rules for identifying them are crucial: non-zero digits are always significant.

Zeros between non-zero digits are significant. Leading zeros (e.g., $0.005$) are never significant; they just place the decimal. Trailing zeros are significant only if a decimal point is present (e.g., $12.

00$has 4 SF, but$1200$ has 2 SF). When performing calculations, the result's precision is limited by the least precise input. For addition/subtraction, the answer is rounded to the fewest decimal places.

For multiplication/division, the answer is rounded to the fewest significant figures. Rounding rules involve checking the digit to be dropped: if less than 5, keep the preceding digit; if greater than 5, round up; if exactly 5, round to the nearest even number.

Exact numbers (counts or definitions) have infinite significant figures and don't limit precision. Understanding these rules ensures that scientific results accurately reflect the uncertainty of the measurements.

Important Differences

vs Decimal Places

Aspect	This Topic	Decimal Places
Definition	Significant figures are all the digits in a number that carry meaning contributing to its precision, including the last uncertain digit.	Decimal places refer to the number of digits present after the decimal point in a number.
Purpose	To indicate the precision of a measurement and the reliability of a value. Reflects the instrument's capability.	To specify the position of the decimal point and the magnitude of the fractional part of a number.
Rule for Zeros	Leading zeros are not significant. Trailing zeros are significant only if a decimal point is present.	All digits after the decimal point, including trailing zeros, are counted as decimal places.
Impact on Calculations (Addition/Subtraction)	Not directly used for rounding in addition/subtraction; decimal places rule applies.	The result is rounded to the least number of decimal places among the input values.
Impact on Calculations (Multiplication/Division)	The result is rounded to the least number of significant figures among the input values.	Decimal places are not the primary factor for rounding in multiplication/division; significant figures rule applies.
Example	$0.0025, ext{g}$ has 2 significant figures. $12.00, ext{cm}$ has 4 significant figures.	$0.0025, ext{g}$ has 4 decimal places. $12.00, ext{cm}$ has 2 decimal places.

While both significant figures and decimal places deal with digits in a number, they serve different purposes and follow distinct rules. Significant figures convey the overall precision of a measurement, considering all meaningful digits, and are crucial for multiplication and division. Decimal places, on the other hand, specifically count digits after the decimal point and are the primary determinant for rounding in addition and subtraction. A number can have many decimal places but few significant figures (e.g., $0.00012$) or vice-versa (e.g., $12345.0$). Understanding this distinction is vital for correctly handling experimental data and calculations in physics.