Essential Question
- Why are scientific units and significant figures important, and how do we use them correctly in scientific calculations?
Overview
This lecture discusses the arbitrary nature and importance of scientific units, how to convert between them, and the rules for using significant figures in calculations. It also shows practical unit conversions and highlights the consequences of unit errors.
Scientific Units: Definitions & Importance
- Units like kilograms, seconds, and volts are arbitrary standards created by agreement.
- The kilogram is defined by the mass of the International Prototype Kilogram (IPK); the IPK is the kilogram itself, and all other kilograms are defined by comparison to it.
- The International System of Units (SI) has seven base units; all other units are derived from these (e.g., speed = length/time, force = mass × acceleration).
- Some units are named after scientists (e.g., hertz, watt), but as units, they are written in lowercase.
- The second is based on the Earth's rotation, but since the Earth's rotation is slowing, seconds are becoming less tied to physical reality.
- Using correct units is crucial in science and engineering; mistakes can have serious consequences, such as the Mars Climate Orbiter crash caused by a unit conversion error.
Unit Conversion
- Units cancel out in calculations; always ensure only the desired units remain at the end.
- Convert values by multiplying by fractions that represent equivalent values (e.g., 1 hour/60 minutes or 1 minute/60 seconds).
- Example: Converting 60 miles per hour to light-years per second:
- 60 miles/hour × (1 hour/60 minutes) × (1 minute/60 seconds) × (1 lightyear/5.9 × 10¹² miles)
- Units cancel hours, minutes, and miles, leaving light-years per second.
- Result: 60 mph = 9.3 × 10⁻¹² lightyears/second.
- Use unit analysis to check for errors and confirm that the results make sense.
- All units, like miles or kilometers, are arbitrary; more universal units (like light-years per second) can be used for comparison.
Significant Figures (Sig Figs)
- There are two types of numbers:
- Exact numbers: defined values with no uncertainty (e.g., 60 seconds in a minute, 12 eggs in a dozen).
- Measured numbers: obtained by measurement, always include some uncertainty.
- Measured numbers show both the value and the precision of the measurement. For example, a speedometer reading of 60 mph could mean anything from about 59.87 to 60.49 mph.
- Significant figures are the digits in a measurement that are known plus the first uncertain digit.
- Zeros can be placeholders or significant; scientific notation clarifies which zeros are significant.
Scientific Notation
- Scientific notation expresses numbers as a coefficient times a power of ten (e.g., 6.0 × 10¹).
- Only the digits shown in scientific notation are significant. For example, 6.0 × 10¹ has two significant figures, while 6 × 10¹ has one.
- Scientific notation helps preserve and clearly show significant digits, especially with very large or small numbers.
- Example: 2.4590 × 10⁻⁴ = 0.00024590 (five significant figures).
Rules for Significant Figures in Calculations
- For addition and subtraction: the answer should have the same number of decimal places as the measurement with the fewest decimal places.
- Example: 1,495.2 + 1.9903 = 1,497.1903 → rounded to 1,497.2 (one decimal place).
- For multiplication and division: the answer should have the same number of significant figures as the measurement with the fewest significant figures.
- Example: 60 × 5.0839 = 305.034 → rounded to 3.0 × 10² (two significant figures).
- Extra digits beyond the correct number of significant figures are misleading and should not be included, as they imply a false level of precision.
Key Terms & Definitions
- Unit — Standard measurement quantity (e.g., kilogram, second).
- SI Base Units — Seven fundamental units from which all others are derived (e.g., meter, kilogram, second).
- Significant Figures (Sig Figs) — Digits in a measured or calculated value that show its precision.
- Scientific Notation — A way to express large or small numbers as a coefficient times a power of ten.
- Exact Number — Number with no uncertainty, defined by definition.
- Measured Number — Number obtained by measurement, includes uncertainty.
Configurations / Action Items
- Practice converting between different units, making sure to cancel units as needed. Try converting everyday speeds or distances into more unusual units (e.g., lightyears per second).
- Use scientific notation to clearly show significant digits and avoid confusion about which digits are significant.
- Apply significant figure rules in all calculations, in chemistry and other sciences, to ensure results are accurate and not misleading.
- Always check that your final units make sense and that your answer has the correct number of significant figures.