Introduction and Mathematical Concepts in Physics

Quantitative Nature of Physics

• Physics uses numbers to describe quantities.
• In most cases, numbers in physics are approximate, not exact.
• Example: Saying a table is 2 meters long usually means it's closer to 2 meters than to 3 meters or 1 meter.
• Typical approximation: ±0.5 (e.g., 2 meters could mean 1.5 to 2.5 meters).

Precision and Measurement

• Precision in measurement is essential.
• Example of precision: 2000 meters vs. 2100 meters vs. 2130 meters.
• More digits imply higher precision.
• Discusses errors and precision in terms of significant figures (sig figs).

Significant Figures

• Definition: Digits that carry meaning contributing to a number's precision.
• Counting Sig Figs:
  • Numbers without zeros: simply count digits (e.g., 4367 has 4 sig figs).
  • Numbers with zeros: specific rules apply.

Rules for Counting Zeros

1. Leading Zeros: Not significant.
   • Example: 0.0023 has 2 sig figs.
2. Sandwiched Zeros: Always significant.
   • Example: 104 has 3 sig figs.
3. Trailing Zeros: Only significant if there's a decimal point.
   • Example: 1300 has 2 sig figs, but 1300.0 has 5 sig figs.

Examples of Significant Figures with Zeros

• 1.45 (3 sig figs)
• 1.005 (4 sig figs)
• 0.000450 (3 sig figs)
• 50.0060 (6 sig figs)

Scientific Notation

• Useful for large and small numbers.
• Format: First non-zero digit, followed by decimal point and remaining significant digits, multiplied by a power of 10.
• Example: 1,670,000 = 1.67 × 10^6
• Small number example: 0.00053 = 5.3 × 10^-4
• Positive exponents represent large numbers; negative exponents represent small numbers.

Using Scientific Notation for Sig Figs

• Ensures correct number of significant figures are represented.
• Example: 1,670,000 with 5 sig figs = 1.6700 × 10^6

Importance of Significant Figures

• Allows for proper representation of precision in calculations.
• Scientific notation helps in maintaining and clarifying the number of significant figures.