Numbers in Science

Exact and Uncertain Numbers

In scientific measurements and calculations, it is crucial to distinguish between exact numbers and those with uncertainty. This distinction affects the precision and reliability of results in physics.

• Exact Numbers: Values known with complete certainty, often arising from definitions or counting.

  • Counting numbers: 1, 2, 3, ...

  • Mathematical constants: π, e, etc.

  • Mathematical functions: For example,

  • Reference values: e.g., (speed of light)

• Numbers with Uncertainty: Any value obtained from a measurement, which inherently includes some degree of error or uncertainty.

• Uncertainty in Measurement: The uncertainty is often estimated as half the smallest division on the measuring device, representing the limit of precision.

Significant Figures

Definition and Importance

Significant figures ("sig figs") indicate the number of reliably known digits in a measured or calculated quantity. They are essential for expressing the precision of measurements and ensuring consistency in scientific communication.

• The number of significant figures is typically assumed to be accurate to ±1 in the last digit.

• To determine significant figures:

  • Start with the first non-zero digit on the left and count all digits to the right.

  • Examples: 0.0006307 and 5.900 both have 4 significant digits.

  • Numbers like 100 may have 1, 2, or 3 significant digits depending on context; a trailing decimal point (e.g., 100.) indicates 3 significant figures.

• Scientific Notation: Expressing numbers in scientific notation (e.g., ) helps clarify the number of significant figures.

Rules for Calculations

• Addition and Subtraction: The result should have the same number of decimal places as the measurement with the least precise decimal place.

  • Example: (rounded to the tenths digit)

• Multiplication and Division: The result should have the same number of significant figures as the value with the fewest significant figures.

  • Example: (rounded to 3 significant figures)

• Constants: Treat mathematical and physical constants as having infinite significant figures unless otherwise specified.

• Rounding: Only round the final result of a calculation, not intermediate steps. If intermediate rounding is unavoidable, keep at least one extra digit.

Examples

• Adding: (rounded to the nearest whole number if 864 has no decimal places)

• Scientific notation: (coefficient is 1.23, power is 4)

System of Units

International System of Units (SI)

The International System of Units (SI) is the standard system used in science, based on meters (m), kilograms (kg), and seconds (s). Understanding the scale of physical quantities is essential for checking the plausibility of answers.

• SI is abbreviated from the French "Système International d'Unités".

• Common base units: meter (m), kilogram (kg), second (s).

Table: Approximate Lengths, Masses, and Time Intervals

Prefixes

Metric Prefixes

Metric prefixes are used to express multiples or fractions of units, making it easier to handle very large or small quantities. Memorizing common prefixes is essential for scientific literacy.

• Prefixes range from (yocto) to (yotta).

• Common prefixes: kilo (k, ), mega (M, ), giga (G, ), milli (m, ), micro (μ, ), nano (n, ), pico (p, ).

• Examples:

  • Modern SSDs have terabytes (TB) of storage ( bytes).

  • Blood glucose levels: 70–100 mg/dL; cholesterol: <200 mg/dL.

  • Radiation dosage: 0.1 mSv (millisieverts).

  • Molecular bond lengths: typically in picometers (pm).

• In medicine, micrograms (μg) are sometimes written as "mcg" to avoid confusion.

Table: Common Metric Prefixes

Conversions

Unit Conversions

Unit conversions are a fundamental skill in physics, allowing you to translate measurements into different units as needed for calculations or comparisons.

• Conversions are often performed using conversion factors, which are ratios equal to 1 (e.g., ).

• To convert units, multiply by the appropriate conversion factor(s) to cancel out the original units and introduce the desired units.

• Example: To convert 75 mg/dL to μg/cm³, use the relationships and .

• It is often necessary to convert all quantities to standard metric units before performing calculations, especially in complex formulas.

Generalized Conversion Factors

Many scientific relationships are linear and can be treated as conversion factors. Trigonometric functions, for example, relate the sides of a right triangle and can be used as ratios in calculations.

• Example: can be used to convert between side lengths in a triangle.

• Physical constants and relationships (e.g., ) also act as conversion factors between units.

Example Problem

• Wafer Slicing: If 400 chips are etched from a silicon wafer of known thickness, and the wafer is cut from a cylindrical crystal of length 25 cm, the maximum number of chips is determined by dividing the total length by the thickness of each chip.

• Additional info: This type of problem demonstrates the practical use of unit conversions and significant figures in real-world physics applications.