Fundamental Concepts in Measurement and Scientific Notation

Measurement and Scientific Notation

Scientific Notation

Scientific notation is a method of expressing very large or very small numbers in the form a × 10n, where a is a number between 1 and 10, and n is an integer. This notation simplifies calculations and clearly indicates the precision of measurements.

• Converting to Scientific Notation: Move the decimal point so that only one nonzero digit remains to the left. Count the number of places moved to determine the exponent.

• Examples:

  • 144,000 → $1.44 \times 10^5$

  • 0.000000034 → $3.4 \times 10^{-8}$

  • 0.066 → $6.6 \times 10^{-2}$

  • 4,300 → $4.3 \times 10^3$

Significant Figures

Significant figures (SFs) are the digits in a measurement that are known with certainty plus one estimated digit. They reflect the precision of a measured quantity.

• Rules for Counting Significant Figures:

  • All nonzero digits are significant.

  • Zeros between nonzero digits are significant.

  • Leading zeros are not significant.

  • Trailing zeros are significant only if there is a decimal point.

• Examples:

  • 2.3055 cm (5 SFs, measured)

  • 3.40 × 104 km (3 SFs, measured)

  • 2.54 cm = 1 in. (exact, infinite SFs)

  • 82 mg (2 SFs, measured)

Exact vs. Measured Numbers

• Exact Numbers: Values known with complete certainty, often from definitions or counting (e.g., 1 in = 2.54 cm).

• Measured Numbers: Values obtained by measurement, subject to uncertainty (e.g., 2.3055 cm).

Mathematical Operations with Significant Figures

Addition and Subtraction

When adding or subtracting, the result should have the same number of decimal places as the measurement with the fewest decimal places.

• Example: $3.25 + 14.8 + 7.06 = 25.1$ (rounded to 1 decimal place)

Multiplication and Division

When multiplying or dividing, the result should have the same number of significant figures as the measurement with the fewest significant figures.

• Example: $46.7 \times 3.2 = 150$ (rounded to 2 SFs)

Unit Conversions and Dimensional Analysis

Metric Prefixes and Units

The metric system uses prefixes to indicate multiples or fractions of base units. Understanding these is essential for converting between units.

Prefix	Symbol	Factor
kilo-	k	103
centi-	c	10-2
milli-	m	10-3
micro-	μ	10-6
nano-	n	10-9
pico-	p	10-12
femto-	f	10-15
giga-	G	109
mega-	M	106

Common Unit Abbreviations

• milligram: mg

• deciliter: dL

• kilometer: km

• picogram: pg

• gigagram: Gg

• megameter: Mm

• microliter: μL

• femtosecond: fs

Complete Names for Abbreviated Units

• kg: kilogram

• cL: centiliter

• ms: millisecond

• Pm: petameter

• dL: deciliter

• Ts: terasecond

• mcg: microgram

• pm: picometer

Using Prefixes to Name Quantities

• 1000 g = 1 kilogram (kg)

• 106 g = 1 megagram (Mg)

• 0.1 g = 1 decigram (dg)

• 0.001 g = 1 milligram (mg)

Unit Conversion Examples

• Liters to kiloliters: $873 \text{ L} = 0.873 \text{ kL}$

• Meters to millimeters: $1 \text{ m} = 1000 \text{ mm}$

• Centimeters to millimeters: $1 \text{ cm} = 10 \text{ mm}$

• Grams to milligrams: $3.7 \text{ g} = 3700 \text{ mg}$

• Nanoseconds to seconds: $2.05 \times 10^5 \text{ ns} = 2.05 \times 10^{-4} \text{ s}$

Applied Problems in Chemistry

Density Calculations

Density is defined as mass per unit volume. It is a physical property used to characterize substances.

• Formula: $\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

• Example: A bone sample has a mass of 2.15 g and a volume of 1.40 cm3. Its density is: $\text{Density} = \frac{2.15\ \text{g}}{1.40\ \text{cm}^3} = 1.54\ \text{g/cm}^3$

Medical Dosage Calculations

• Example: If a patient is given an order for 0.625 g of calcium carbonate and the solution contains 125 mg/mL, the volume given is: Convert 0.625 g to mg: $0.625\ \text{g} \times \frac{1000\ \text{mg}}{1\ \text{g}} = 625\ \text{mg}$ Volume: $\frac{625\ \text{mg}}{125\ \text{mg/mL}} = 5.0\ \text{mL}$

• Example: A doctor orders 5.0 mL of phenobarbital elixir (30 mg/7.5 mL). How many mg should be given? Set up a proportion: $\frac{30\ \text{mg}}{7.5\ \text{mL}} = \frac{x\ \text{mg}}{5.0\ \text{mL}}$ $x = \frac{30 \times 5.0}{7.5} = 20\ \text{mg}$

Dimensional Analysis in Everyday Contexts

• Example: If you walk 7500 m at a pace of 55.0 m/min, the time required is: $\text{Time} = \frac{7500\ \text{m}}{55.0\ \text{m/min}} = 136\ \text{min}$

Summary Table: Metric Prefixes and Units

Prefix	Symbol	Factor	Example Unit
kilo-	k	103	kilogram (kg)
centi-	c	10-2	centimeter (cm)
milli-	m	10-3	milligram (mg)
micro-	μ	10-6	microliter (μL)
nano-	n	10-9	nanometer (nm)
pico-	p	10-12	picogram (pg)
femto-	f	10-15	femtosecond (fs)
giga-	G	109	gigagram (Gg)
mega-	M	106	megameter (Mm)

Additional info: Some unit names and abbreviations were inferred based on standard SI conventions. Medical dosage and density calculation examples were expanded for clarity.