Scientific Notation and Metric Prefixes - Vocabulary Review

Scientific Notation: Purpose and Scope

  • Scientific notation helps scientists manage very large and very small numbers (e.g., astronomy distances, subatomic scales).
  • Large numbers (in meters, miles) have many zeros; small numbers (atoms, nuclei, interatomic distances) have zeros before the meaningful digits.
  • The exam will test: converting a regular number to scientific notation and converting back; identifying if a number is large or small based on the exponent; understanding zeros and decimal placement.

How to Convert Between Standard Numbers and Scientific Notation

  • Rule of thumb: Move the decimal point to create a coefficient between 1 and 9.999…
  • If the original number is bigger than 10 (large number), move the decimal point to the left until the first nonzero digit is before the decimal point.
  • The number of moves to the left determines the exponent: positive exponent indicates a large number.
  • If the original number is smaller than 1 (very small), move the decimal point to the right until you have a digit between 1 and 9 before the decimal.
  • The number of moves to the right determines the exponent: negative exponent indicates a small number.
  • Example intuition: a bouncing decimal point imagery helps students visualize moving the decimal left (large numbers) or right (small numbers).
  • Positive exponent means moved left; negative exponent means moved right.
  • If a reader sees a number in blue and a red exponent, the red exponent shows how many times the decimal was moved to the left (positive) or right (negative).

Examples from the Transcript

  • Large number example: 12 g of carbon atoms (carbon’s atomic mass ~12 amu; molar mass ~12 g/mol)
    • A mole contains N_A = 6.022 imes 10^{23} atoms.
    • Therefore, 12 g of carbon corresponds to roughly N_A = 6.022 imes 10^{23} atoms (a huge count of atoms).
    • If you had a single carbon atom, its mass is extremely small (a tiny fraction of a gram), illustrating a negative exponent potential.
    • A single carbon atom’s mass is ~1.99 imes 10^{-23} ext{ g} (illustrative scale; how many zeros and where the decimal sits matters).
  • Converting back from scientific notation to a standard number:
    • Given 6.022 imes 10^{23}, move the decimal point 23 places to the right to get the standard number (a very large number).
    • If the exponent is negative, move the decimal point to the left (divide by powers of 10).
  • Quick practice example (noted in class):
    • Convert 4.31 × 10^4 and 3.9 × 10^3 for addition:
    • 3.9 × 10^3 = 0.39 × 10^4
    • Sum = (4.31 + 0.39) × 10^4 = 4.70 × 10^4 = 4.7 × 10^4 (keep significant figures as measured).
    • For multiplication: (4.0 × 10^4) × (7.0 × 10^3) = (4.0 × 7.0) × 10^(4+3) = 28.0 × 10^7 = 2.8 × 10^8 (in proper scientific notation).
    • For division: (8.5 × 10^4) ÷ (5.0 × 10^9) = (8.5 ÷ 5.0) × 10^(4−9) = 1.7 × 10^(−5).
  • Calculator tip: many calculators use an “ee” function for entering scientific notation (e.g., 5.0e3). Practice by hand as well to avoid overflow or mis-entry.
  • Common pitfalls:
    • Mixing up the sign of the exponent (positive vs negative).
    • Forgetting to rewrite the result in formal scientific notation if the coefficient is not between 1 and 9.
    • Not counting the exact number of decimal moves when aligning exponents for addition/subtraction.

Quick Discussion Prompt from Class

  • A present-day data point (e.g., a debt clock value) can be expressed in scientific notation; students discussed proper formatting.
  • Correct formal notation example discussed: ideally 3.73 imes 10^{13} (notably, 37.3 imes 10^{12} is informal; proper form requires a coefficient between 1 and 9).

The Metric System: History, Base Units, and Prefixes

  • History and motivation:
    • France (1790): Meetings to standardize measurement units (the original system was named the System International).
    • 1875: A signatory agreement in Paris among 17 nations to adopt standardized units.
  • US context: The US has historically used both the English (customary) system and the metric system; scientists often use metric units in lab work.
  • Base units highlighted for this course:
    • Length: meter, symbolized as ext{m}. Base unit for length; prefixes modify the size.
    • Mass: kilogram, symbolized as ext{kg}. In labs, grams are commonly used; 1 kg = 1000 g.
    • Volume: liter, symbolized as ext{L} (capital L). Milliliters are commonly used; 1 L = 1000 mL.
    • Time: second, symbolized as ext{s}.
    • Temperature: Kelvin, symbolized as ext{K}. Base unit for absolute temperature; degrees Celsius are also used in labs for practical temperature readings.
    • Amount of substance: mole, symbolized as ext{mol}. Avogadro’s number relates moles to entities: N_A = 6.022 imes 10^{23} entities per mole.
    • Luminous intensity: candela, symbolized as ext{cd} (not a focus here).
  • Notable points:
    • Celsius and Kelvin are commonly used; Fahrenheit is rarely used in the scientific curriculum except for conversion practice.
    • Prefixes represent powers of 10; base units are unchanged, prefixes change magnitude.

Metric Prefixes: Magnitude and Notation

  • Prefixes and their effects (base unit is the thing you’re measuring; e.g., meter, gram, second):
    • Decreasing prefixes (smaller than base):
    • deci (d) = 10^(-1)
    • centi (c) = 10^(-2)
    • milli (m) = 10^(-3)
    • micro (μ) = 10^(-6) (represented by the Greek letter μ; sometimes written as mu)
    • nano (n) = 10^(-9)
    • pico (p) = 10^(-12)
    • femto (f) = 10^(-15)
    • Increasing prefixes (larger than base):
    • deka/da (Da or DA) = 10^1
    • hecto (h) = 10^2
    • kilo (k) = 10^3
    • mega (M) = 10^6 (capital M)
    • giga (G) = 10^9 (capital G)
    • tera (T) = 10^12 (capital T)
  • Capitalization matters:
    • Lowercase m = milli (10^-3)
    • Uppercase M = mega (10^6)
    • Similar distinctions apply to other prefixes (Da, d, c, etc.).
  • Important examples:
    • 1 meter = 1 m base unit; 1 kilometer = 1 × 10^3 m = 1000 m;
    • 1 decimeter = 0.1 m = 1 × 10^(-1) m;
    • 1 centimeter = 0.01 m = 1 × 10^(-2) m;
    • 1 millimeter = 0.001 m = 1 × 10^(-3) m;
    • 1 micrometer = 1 × 10^(-6) m; 1 nanometer = 1 × 10^(-9) m; 1 picometer = 1 × 10^(-12) m; 1 femtometer = 1 × 10^(-15) m.
  • Example: prefix-to-base-unit conversion
    • 1 kilometer = 1 × 10^3 meters
    • 1 hectometer = 1 × 10^2 meters
    • 1 gigameter = 1 × 10^9 meters
    • 1 decimeter = 1 × 10^(-1) meters
    • 1 centimeter = 1 × 10^(-2) meters
  • Angstroms:
    • Angstrom symbol: Å; 1 Å = 1 × 10^(-10) meters
    • Angstroms, nanometers, and similar prefixes are used to describe molecular scales and atomic spacing; useful for bond lengths, molecular sizes, etc.

Volume and Related Conversions

  • Volume units: liter (L) and milliliter (mL);
    • 1 L = 1000 mL.
    • 1 mL = 1 cm^3 (cubic centimeter).
    • 1 cm = 1 × 10^(-2) meters; hence 1 cm^3 = (1 × 10^(-2) m)^3 = 1 × 10^(-6) m^3.
    • Therefore, 1 dm^3 = 1 L (since 1 dm = 0.1 m, so 1 dm^3 = 0.001 m^3 = 1 L).
  • Practical visualization:
    • A decimeter cube (1 dm on each side) has a volume of 1 dm^3 = 1 L = 1000 cm^3 = 1000 mL.
    • A common lab fact: 1 mL = 1 cm^3.
  • Practical note for coursework:
    • Do not cube liters or milliliters when dealing with volume units; the cube applies to length units, not the volume units themselves.
    • In medical contexts, one cubic centimeter (cc) often denotes 1 mL.
  • Preview for labs:
    • Visualization activity: building a decimeter cube box to confirm 1 L capacity.

Quick Reference: Key Formulas and Conversions

  • Scientific notation general form: a imes 10^{n} with 1 \le a < 10 and integer n.
  • For a positive exponent (large number): exponent n is positive; decimal moved left n times.
  • For a negative exponent (small number): exponent n is negative; decimal moved right |n| times.
  • Addition/Subtraction in sci notation:
    • Align the exponents by adjusting the coefficient of the term with the smaller exponent until both terms share the same exponent, then add coefficients. If the sum of coefficients causes the result to exceed 10, move decimal and increase the exponent by 1.
    • Example: 4.31\times 10^{4} + 3.9\times 10^{3} = (4.31 + 0.39) \times 10^{4} = 4.70 \times 10^{4} = 4.7\times 10^{4}.n- Multiplication in sci notation:
    • Multiply the coefficients and add exponents: (a\times 10^{m}) \times (b\times 10^{n}) = (ab) \times 10^{m+n}.
    • Example: (4.0\times 10^{4}) \times (7.0\times 10^{3}) = 28.0 \times 10^{7} = 2.8 \times 10^{8}. If the coefficient exceeds 10, adjust to formal notation by moving decimal to the left and increasing the exponent accordingly.
  • Division in sci notation:
    • Divide the coefficients and subtract exponents: (a\times 10^{m}) / (b\times 10^{n}) = (a/b) \times 10^{m-n}.
    • Example: (8.5\times 10^{4}) / (5.0\times 10^{9}) = (8.5/5.0) \times 10^{4-9} = 1.7 \times 10^{-5}.$n
  • Calculator tip: many calculators use an exponent notation (ee); e.g., inputting 5.0e3 yields 5.0 \times 10^{3}.
  • Dimensional consistency reminder:
    • If you move the decimal to the left to obtain a proper scientific notation for a number > 10, you increase the exponent by 1.
    • If you move the decimal to the right to obtain a number < 1, you decrease the exponent by 1.

Notable Concepts Introduced (Contextual Points)

  • Abbreviations and capitalization matter in prefixes:
    • Example: kilometer is written as ext{k m} (lowercase k for kilo), while mega is ext{M} (uppercase M).
    • Micro is represented by the Greek letter mu (μ); sometimes written as MCg in certain contexts (e.g., micrograms in consumer labeling), but scientific texts use μ as the prefix for 10^(-6).
  • Angstroms and molecular dimensions:
    • Angstrom (Å) = 1.0\times 10^{-10}\,\text{m}; useful for describing bonds and molecular sizes.
  • Avogadro’s number and the mole concept:
    • N_A = 6.022\times 10^{23}$$ entities per mole.
    • A mole is a fixed number of entities (atoms, ions, molecules) analogous to a dozen, which helps relate mass to amount of substance.
  • Key takeaway about significant figures (brief preview):
    • Significant figures reflect the precision of the measuring instrument; excess digits beyond the measurement precision should be rounded according to rules discussed later in the course.
  • Significance of the “notational discipline”:
    • Formal scientific notation (coefficients between 1 and 9) is the standard in scientific communication; informal notations may be used in quick approximate comparisons but should be converted to formal form for publication or grading.

Next Steps in the Course (What to Expect)

  • Friday: Significant figures in more detail (to be discussed and practiced).
  • Continued practice with conversions, including more complex unit conversions across prefixes and base units.
  • Deeper exploration of logarithms and their relationship to significant figures (brief preview here; covered in later modules).
  • Hands-on lab activities to visualize volume and prefix concepts (e.g., decimeter cube proving 1 L).

Summary Notes

  • Scientific notation is essential for writing and calculating with very large or small numbers compactly and accurately.
  • Mastery involves converting to/from standard notation, correctly adjusting exponents for addition/subtraction, multiplication, and division.
  • The metric system provides a coherent framework for units and prefixes that scale by powers of 10, with careful attention to capitalization and the distinction between base units and prefixes.
  • Volume and length relationships (e.g., 1 L = 1 dm^3 = 1000 cm^3; 1 mL = 1 cm^3) are foundational for practical lab work and conversions.
  • Angstroms and prefixes up to tera (and beyond) equip scientists to discuss sizes from atomic to cosmic scales with clarity.
  • Avogadro’s number links the macroscopic world (grams) to the microscopic world (atoms and molecules) via the mole concept.