Notes on Significant Figures, Zeros, Scientific Notation, and SI Units
Significant Figures, Measurements, and SI Units
Significance of a number in science
- A measured number has digits that carry meaning about how confident we are in the value.
- The collection of all digits that carry meaning (both certain and uncertain) is called the significant figures of that number.
- The rightmost digit is the uncertain digit; all digits to its left are the certain digits.
- Counting numbers have infinite certainty (infinite significant figures) because they are exact counts, not measurements.
- The term "significant figures" does not mean all digits are uncertain; it means the digits that carry meaningful precision, with the rightmost digit being uncertain when it is a measurement.
Zero types and their significance
- Leading zeros: zeros before the first nonzero digit. Never significant. Example: 0.0032 has two significant figures (3 and 2).
- Captive (trapped) zeros: zeros between nonzero digits. Always significant. Example: 101 has three sig figs; 1.03 has three sig figs.
- Trailing zeros: zeros at the end of a number.
- If there is a decimal point, trailing zeros are significant. Example: 12.340 has five sig figs.
- If there is no decimal point, trailing zeros are not considered significant (they may be placeholders). Example: 300 has one sig fig.
- Summary rule from the lesson:
- Significant if there is a decimal present; trailing zeros without a decimal are not significant.
Reading measurements from a graduated cylinder (meniscus)
- Read the liquid level from the bottom of the meniscus.
- Example concept from the lesson: read the bottom of the meniscus to determine the value.
Significant figures in a measured number (practical implications)
- For a given measurement, identify:
- The certain digits (all digits to the left of the rightmost digit that is considered certain).
- The uncertain digit (the rightmost digit that is uncertain).
- In a graduated cylinder example, if you have a reading near 6.6 mL and you estimate the tenths place, you might report something like 6.7 mL if you are confident about the tenths digit.
- When a device with finer marks is used, you may become confident in additional digits (e.g., hundredths).
Scientific notation: purpose and notation rules
- Purpose: to express numbers such that only significant digits are shown, avoiding ambiguity about precision.
- General format: write the first significant digit, place a decimal, then write the remaining significant digits.
- Example steps:
- 10^0 = 1, 10^1 = 10, 10^2 = 100, 10^3 = 1000. For any x ≠ 0, x^0 = 1.
- Negative exponents move the decimal to the left (in the denominator):
10^{-n} = \frac{1}{10^{n}}. Conversely, multiplying by 10^n moves the decimal to the right. - Example: To express 1500 with two significant figures using scientific notation:
- The correct form is 1.5 \times 10^{3}. (Because the digits 1 and 5 are the two significant figures and the exponent places the decimal correctly.)
- Example discussion from the lesson:
- If the first significant digit is 7 and there are four digits total (e.g., 7.03 with an implied decimal position), you might get something like 7.03 \times 10^{4}. The exact form depends on where the decimal is placed after ensuring only the significant digits are shown.
- Important reminder: when converting numbers to scientific notation, include only significant figures and place the decimal after the first significant digit.
Rules for combining significant figures in calculations
- Addition and subtraction depend on decimal places (the least precise decimal place governs).
- Multiplication and division depend on the number of significant figures (the smallest number of sig figs in any factor governs).
- When you must carry calculations across multiple steps and there will be further use of intermediate results, do not round off at intermediate steps; keep the full precision from the calculation, and only round at the final answer if it’s the last step.
- If continuing calculations, you may store extra digits (e.g., as a subscript of non-significant digits) to indicate the precision carried through, though this is typically avoided in routine calculator work.
Examples and scenario sketches from the lesson
- Example type: translating 2 sig figs in 1,500 to scientific notation, taking into account whether the zero is significant (depends on decimal presence).
- Example type: converting 6.613 (a number with decimal) to the appropriate sig figs and decimal places depending on the operation; if only two sig figs are required for a calculation, you round accordingly (e.g., 6.6, then later adjust if more precision is needed).
- Example: When reporting a product of measurements, the final result must reflect the smallest number of sig figs among the factors; if a product yields 91.66 but only three sig figs are allowed, report as 91.7.
Counting numbers vs measurements
- Counting numbers have infinite sig figs (e.g., you have 24 markers; the number 24 has infinite certainty for the count).
- In calculations, counting numbers are treated as having unlimited precision, so they do not limit the sig figs in the product or quotient.
Units and the SI base units (five foundational concepts)
- Time: symbol is s (seconds)
- Temperature: symbol is K (Kelvin). Note: temperature in chemistry often uses Kelvin as the base unit.
- Distance (length): symbol is m (meter)
- Mass: symbol is kg (kilogram)
- Amount of substance: symbol is n (lowercase script n), with unit mole (mol)
- Other base quantities mentioned:
- Current: symbol A (ampere)
- Luminous intensity: symbol cd (candela)
- The system of units used in science is the International System of Units (SI, or Systeme International d’Unites in French). The statue of the base units is commonly listed as: seconds, Kelvin, meter, kilogram, and mole. The United States does not universally use the metric system in everyday life, though SI is the standard in scientific work.
Relationship between base and derived units
- Derived quantities are built from base units. Examples:
- Area = length × width ⇒ units: \text{m}^2
- Volume = length × width × height ⇒ units: \text{m}^3
- Velocity = distance / time ⇒ units: \frac{\text{m}}{\text{s}}
- Acceleration = distance / time^2 ⇒ units: \frac{\text{m}}{\text{s}^2}
- Some measurements can be expressed in non-SI units (e.g., miles, pounds), but conversions to SI are common in science courses.
Periodic table masses and simple addition for molecular mass
- Masses listed under element symbols on the periodic table typically have two decimals (e.g., Carbon: 12.01; Oxygen: 16.00).
- The mass of a compound like CO is the sum of the masses of its constituent atoms: m(\text{CO}) = m(\text{C}) + m(\text{O}) = 12.01 + 16.00 = 28.01. Each term has two decimals, so the sum has two decimals.
- Note: when reporting masses, you should keep the same decimal precision as the least precise input value in the sum (here, both inputs have two decimals).
Practical lab-oriented notes from the lesson
- When performing measurements (e.g., reading a graduated cylinder), know how precise your instrument is and report only as many digits as the instrument justifies (and as the rules of sig figs require).
- In lab practice, if you have to carry a calculation into subsequent steps, keep as many significant digits as possible until the final result, then apply the appropriate rounding rule.
- Practice problems and supplementary notes are available in the instructor’s module (referred to as the doctor Toomey supplementary notes module). These include problems with and without answers to reinforce the sig figs concepts, zeros, and scientific notation, and to match the homework you will do.
Quick reference reminders (from the lesson)
- Most math operations in science rely on these rules:
- Addition/subtraction: depends on the smallest number of decimals (decimal places).
- Multiplication/division: depends on the smallest number of significant figures.
- When uncertain, always write numbers with the appropriate significant figures and, if needed, convert to scientific notation so only significant digits are shown.
- In practice, use scientific notation to express numbers where appropriate, making explicit the significant digits and the exponent.
- Always include units when reporting numbers; a unit-less number is usually of limited value in scientific communication.
Final note on practice and study resources
- The instructor emphasized practicing with the posted problems in the supplementary notes module (first page with questions, second page with questions and answers).
- The goal is to become fluent with identifying significant figures, determining the role of each zero, applying the decimal-place rule for addition/subtraction, applying sig-fig rules for multiplication/division, and using scientific notation correctly.
Quick recap of key formulas and concepts used in the lesson
- Power and exponent basics:
- a^0 = 1, \quad a \neq 0
- a^{-n} = \frac{1}{a^{n}}
- Scientific notation form:
- Write the first significant digit, decimal point, then remaining significant digits, and multiply by 10^{k} to place the decimal appropriately.
- Kelvin conversion (temperature):
- T(\text{K}) = T(\circ\text{C}) + 273.15
- Basic SI base units (symbol):
- Time: \text{s}
- Temperature: \text{K}
- Length: \text{m}
- Mass: \text{kg}
- Amount: \text{mol} (symbol for amount of substance: n, script style)
- Extra SI bases sometimes used in teaching (not fundamental in this course): current \text{A} (ampere), luminous intensity \text{cd} (candela)
- Common derived quantities: area \text{m}^2, volume \text{m}^3, velocity \frac{\text{m}}{\text{s}}, acceleration \frac{\text{m}}{\text{s}^2}
Encouragement to practice
- Practice problems posted in the supplementary notes module are intended to solidify understanding of significant figures, zeros, decimal places, scientific notation, and unit conversions. Use them to reinforce today’s material before the next class.