Chem Chapter 1

Overview: There are two general types of numbers used in making quantitative measurements:
- Exact numbers have no uncertainty associated with them.
- Inexact numbers do have a degree of uncertainty.
- Concept: Exact vs Inexact Numbers.

Exact numbers include the following:
- Numbers obtained by counting
- Examples: $5$ students in the lab.
- Examples: There are $2$ hydrogen atoms in every molecule of $ext{H}_2 ext{O}$ .
- Definitions (Conversions between units within a system):
- Metric: $1\ ext{m} = 100\ \text{cm}$
- English: $1\ \text{ft} = 12\ \text{in}$
- Mathematical constants, unless rounded:
- Example: $\pi$ is the ratio of a circle’s circumference to its diameter and is an exact number.
Key idea: Exact numbers do not introduce uncertainty and often come from counting, definitions, or well-established constants.

Inexact numbers: Numbers obtained by a measurement are inexact because there is always uncertainty associated with the measuring device used to obtain the value.

1) Any non-zero digit is significant
- Examples: $352$ has 3 sig figs, $3.514$ has 4 sig figs.
- Rule cue: count all non-zero digits as significant.
2) Zeros that occur between non-zero digits are significant
- Examples: $402$ has 3 sig figs, $1.202$ has 4 sig figs.
3) Zeros to the left of the first non-zero digit are not significant
- Examples: $0.0258$ has 3 sig figs, $0.000659$ has 3 sig figs.
4) Zeros to the right of the last non-zero digit after a decimal point are significant
- Examples: $0.120$ has 3 sig figs, $0.2500$ has 4 sig figs.
5) Zeros to the right of the last non-zero digit in a number that does not contain a decimal point are ambiguous; use scientific notation for clarity
- Examples: $100$ (ambiguous), $1.0 \times 10^{2}$ (2 sig figs), $1.00 \times 10^{2}$ (3 sig figs).
- Rule takeaway: use scientific notation when the number could be interpreted ambiguously.

Guiding principle: The final answer cannot have more decimal places than the original number with the least number of decimal places.
Example (illustrative):
- Numbers: $42.5, 5.1, 25.12, 450.22, 450.2$
- Sum: $42.5 + 5.1 + 25.12 + 450.22 + 450.2 = 573.14$
- Least decimal places among addends: 1 (from 42.5, 5.1, 450.2).
- Final rounded result: $573.1$ (to 1 decimal place).

Guiding principle: The final answer must have the same number of significant figures as the original number with the fewest sig figs.
Example (illustrative):
- Consider numbers with varying sig figs: $42.5$ (3 sig figs), $2.0$ (2 sig figs), $850.2$ (4 sig figs),
  $8508.5$ (5 sig figs), and $102$ (3 sig figs).
- The fewest sig figs among these is 2 (from $2.0$ ).
- A representative combined expression: $\frac{42.5 \times 2.0 \times 850.2}{8508.5 \times 102}$
- Result should be reported with 2 sig figs. Approximate final value: $0.083$ (two sig figs; i.e., $8.3\times 10^{-2}$ ).
Note: The exact numbers (like conversion factors) do not limit the sig figs of the result.

Exact numbers do not limit the number of significant figures in a calculation.
Example 1 (unit conversion with a measured value):
- $7.55\ \text{cm} \times \left(\dfrac{10\ \text{mm}}{1\ \text{cm}}\right) = 75.5\ \text{mm}$
- Explanation: The factor $\dfrac{10\ \text{mm}}{1\ \text{cm}}$ is exact, so the result retains the sig figs from the measured value $7.55\ \text{cm}$ (which has 3 sig figs). The final result $75.5\ \text{mm}$ has 3 sig figs.
Example 2 (exact constants like $\pi$ ):
- $\pi(3.5\ \text{cm})^2 = 38\ \text{cm}^2$
- Here, $3.5\ \text{cm}$ has 2 sig figs; the result is reported with 2 sig figs (i.e., $38\ \text{cm}^2$ ).

Guiding principle: In multistep calculations, round only at the last step to avoid accumulation of rounding errors.
Important practice: Keep track of the number of significant figures that would be obtained after each individual step.
Example (illustrative workflow):
- Step 1 (addition): compute the sum with the proper decimal places before applying multiplication/division.
- Step 2 (multiplication): apply significant-figures rule to the product, limiting to the fewest sig figs from the factors used in that step.
- Final step: round to the correct number of sig figs only once, after all steps are complete.
Note: The transcript contains multiple multistep examples (one showing a sum then a product; another showing a combination of products and differences). The key takeaway is to avoid early rounding and to respect decimal-place and sig-fig rules at each operation.

Exact numbers have no uncertainty (counting, unit conversions, mathematical constants in exact form).
Inexact numbers carry measurement uncertainty.
Significant figures rules determine how many meaningful digits to report, with specifics for zeros and for scientific notation.
Addition/subtraction is governed by decimal places (fewer decimal places controls the precision).
Multiplication/division is governed by sig figs (fewest sig figs among inputs controls the precision).
Exact numbers, including unit conversions, do not limit the sig figs in a calculation.
In multistep calculations, delay rounding until the final result while tracking the theoretical sig figs at each step.

Metric conversion example: $1\ \text{m} = 100\ \text{cm}$
English unit conversion example: $1\ \text{ft} = 12\ \text{in}$
Pi as an exact value: $\pi$
Significance examples (sig figs): $352$ , $3.514$ , $402$ , $1.202$ , $0.0258$ , $0.000659$ , $0.120$ , $0.2500$ , $100$ , $1.0 \times 10^{2}$ , $1.00 \times 10^{2}$
Multistep and rounding concepts: examples include sums/products like $5.19 + 0.012 = 5.20$ and $5.20 \times 2.1 = 10.9$
Exact-number-conversion example: $7.55\ \text{cm} \times \left(\dfrac{10\ \text{mm}}{1\ \text{cm}}\right) = 75.5\ \text{mm}$
Circular-area example with pi: $\pi(3.5\ \text{cm})^2 = 38\ \text{cm}^2$
(Note: Some lines in the transcript were garbled or incomplete. The notes above preserve the core concepts, rules, and representative examples.)