Notes on Significant Figures, Zeros, Scientific Notation, and SI Units

Significant Figures

A collection of all digits in a measured number that carry meaning about its confidence and precision, including both certain and uncertain digits.

Leading Zeros

Zeros that appear before the first nonzero digit in a number. They are never significant.

Captive (Trapped) Zeros

Zeros located between nonzero digits in a number. They are always significant.

Trailing Zeros

Zeros at the end of a number. They are significant if there is a decimal point present; otherwise, they are generally not considered significant (may be placeholders).

Uncertain Digit

The rightmost digit in a measured number that is estimated and thus carries some uncertainty.

Scientific Notation

A method to express numbers by showing only significant digits, in the format of a first significant digit, a decimal, remaining significant digits, multiplied by a power of 10.

SI Base Units (Five Foundational)

The standard units in science: second (s) for time, Kelvin (K) for temperature, meter (m) for distance/length, kilogram (kg) for mass, and mole (mol) for the amount of substance (symbol "n").

Derived Units

Units created by combining base units through multiplication or division, such as \text{m}^2 for area, \text{m}^3 for volume, or \text{m/s} for velocity.

Rule for Addition/Subtraction with Sig Figs

The result of an addition or subtraction operation must have the same number of decimal places as the measurement with the fewest decimal places.

Rule for Multiplication/Division with Sig Figs

The result of a multiplication or division operation must have the same number of significant figures as the measurement with the fewest significant figures.

Counting Numbers

Numbers that represent exact counts rather than measurements. They

Significant Figures

The collection of all digits in a measured number that carry meaning about its precision, including both certain and the single uncertain digit.

Leading Zeros

Zeros before the first nonzero digit; they are never significant (e.g., 0.0032 has two significant figures).

Captive (Trapped) Zeros

Zeros located between nonzero digits; they are always significant (e.g., 101 has three significant figures).

Trailing Zeros (with Decimal)

Zeros at the end of a number when a decimal point is present; these are significant (e.g., 12.340 has five significant figures).

Trailing Zeros (without Decimal)

Zeros at the end of a number when no decimal point is present; these are generally not considered significant as they may just be placeholders (e.g., 300 has one significant figure).

Counting Numbers

Exact counts, not measurements; they have infinite certainty and infinite significant figures, thus not limiting precision in calculations.

Scientific Notation

A method to express numbers to clearly show only significant digits and avoid ambiguity about precision, typically in the form a imes 10^{k} where 1
j ext{a} ext{<} 10.

SI Unit for Time

Seconds (s).

SI Unit for Temperature

Kelvin (K).

SI Unit for Length

Meter (m).

SI Unit for Mass

Kilogram (kg).

SI Unit for Amount of Substance

Mole (mol), with symbol for amount of substance as n.

Significant Figures Rule: Addition/Subtraction

The result of addition or subtraction must have the same number of decimal places as the measurement with the fewest decimal places.

Significant Figures Rule: Multiplication/Division

The result of multiplication or division must have the same number of significant figures as the measurement with the fewest significant figures.

Meniscus

The curve in the upper surface of a liquid in a container; measurements from a graduated cylinder are read from the bottom of the meniscus.

Kelvin Conversion

To convert temperature from Celsius to Kelvin, use the formula T(\text{K}) = T(\degree \text{C}) + 273.15.