Understanding Significant Figures

When we measure something, there’s always a tiny bit of uncertainty. Significant figures help us show how precise a measurement is by including all the reliable digits plus the first uncertain one. For example, if you measure a pendulum’s period as 1.62 seconds, “1” and “6” are certain, while “2” is uncertain—so there are 3 significant figures.

Rules for Counting Significant Figures

  • Non-zero digits are always significant. (e.g., 2.308 has 4 significant figures).
  • Zeros between non-zero digits are significant. (e.g., 202 has 3 significant figures).
  • Leading zeros (before the first non-zero digit) are NOT significant. (e.g., 0.0023 has 2 significant figures).
  • Trailing zeros (after a decimal point) ARE significant. (e.g., 3.500 has 4 significant figures).
  • Trailing zeros without a decimal are NOT significant. (e.g., 1500 has 2 significant figures).

Pro Tip: Use scientific notation (like \(4.700 \times 10^3\)) to avoid confusion with trailing zeros. Here, all digits in the base number (4.700) are significant!

Arithmetic with Significant Figures

Calculations must reflect the precision of the original measurements:

  • Multiplication/Division: Round the result to match the least number of significant figures in the original data.
    Example: \( \frac{4.237 \, \text{g}}{2.51 \, \text{cm}^3} = 1.69 \, \text{g/cm}^3 \) (3 significant figures).
  • Addition/Subtraction: Round to match the least number of decimal places.
    Example: \(436.32 \, \text{g} + 227.2 \, \text{g} = 663.5 \, \text{g}\) (1 decimal place).

Rounding Off

Follow these rules to drop uncertain digits:

  • If the digit to drop is > 5, round up the preceding digit (e.g., 2.746 → 2.75).
  • If it’s < 5, leave the preceding digit unchanged (e.g., 2.743 → 2.74).
  • If it’s 5, round up if the preceding digit is odd, else leave it (e.g., 2.735 → 2.74; 2.745 → 2.74).

Uncertainty in Calculations

Combining measurements adds their uncertainties. For example:

  • A rectangle’s sides: \(16.2 \pm 0.1 \, \text{cm}\) and \(10.1 \pm 0.1 \, \text{cm}\).
  • Area: \(164 \pm 3 \, \text{cm}^2\) (uncertainty combines errors).

Key NEET Concepts

  1. Significant Figures in Measurements: Rules for counting and applying them in calculations.
  2. Scientific Notation: Essential for expressing precision (e.g., \(3.00 \times 10^8 \, \text{m/s}\)).
  3. Rounding Off: Critical for lab data and multi-step problems.
  4. Error Propagation: How uncertainties affect final results (e.g., density calculations).
  5. Unit Conversions: Significant figures remain unchanged across units (e.g., 2.308 cm = 23.08 mm).

Example Problems

1. Cube Measurements:
Side = 7.203 m (4 significant figures).
Surface area = \(6 \times (7.203)^2 = 311.3 \, \text{m}^2\).
Volume = \((7.203)^3 = 373.7 \, \text{m}^3\).

2. Density Calculation:
Mass = 5.74 g (3 sig figs), Volume = 1.2 cm³ (2 sig figs).
Density = \(\frac{5.74}{1.2} = 4.8 \, \text{g/cm}^3\) (rounded to 2 sig figs).

Remember: Exact numbers (like “2” in \(2\pi r\)) have infinite significant figures!