Significant Figures

Section 1.2: Significant Figures from Chapter 1: Units and Measurement (Class 11 CBSE Physics):

📘 Comprehensive Note: Significant Figures (Section 1.2 – Units and Measurement)

🔍 What Are Significant Figures?

In any measurement, absolute precision is impossible because every measurement contains some degree of uncertainty or error. To express the reliability and precision of such measurements, we use significant figures.

Definition:
Significant figures (or significant digits) are the digits in a measurement that are known reliably, plus the first uncertain digit.

• Example:
  If a pendulum’s period is measured as 1.62 s,

  • ‘1’ and ‘6’ are reliable digits.

  • ‘2’ is uncertain.

  • So, the number has three significant figures.

📏 Why Are Significant Figures Important?

• They indicate the precision of the measurement instrument.

• Prevent misleading representation of measurement data.

• Help maintain consistency in reporting scientific data.

🧮 Rules for Determining Significant Figures

1. All non-zero digits are significant.

   • Example: 123 has three significant figures.

2. Zeros between non-zero digits are significant.

   • Example: 2.308 has four significant figures.

3. Leading zeros (zeros to the left of the first non-zero digit in numbers < 1) are not significant.

   • Example: 0.002308 → only 2, 3, 0, 8 are significant → 4 significant figures.

4. Trailing zeros without a decimal point are not significant.

   • Example:

     • 123000 has three significant figures.

     • 123000. has six significant figures (due to the decimal point).

5. Trailing zeros in numbers with a decimal point are significant.

   • Example:

     • 3.500 has four significant figures.

     • 0.06900 has four significant figures.

6. Zeros to the left of the decimal point in numbers less than 1 (like 0.1250) are not significant; however, the trailing zeros are.

   • 0.1250 has four significant figures.

🔁 Effect of Changing Units

Changing units does not change the number of significant figures.

• Example:

  • 2.308 cm = 0.02308 m = 23.08 mm = 23080 µm

  • All forms have four significant figures.

🧪 Ambiguity in Trailing Zeros

• If a number is written without a decimal, trailing zeros may not be interpreted as significant.

• Example:

  • 4.700 m = 4700 mm → This may seem to have 2 significant figures.

  • But the original measurement (4.700 m) has four.

🔬 Solution: Use Scientific Notation

To avoid ambiguity, scientific notation is ideal.

Format: a × 10ᵇ
Where:

• ‘a’ has all significant digits

• ‘b’ is the exponent or order of magnitude

• Example:

  • 4.700 m = 4.700 × 10² cm = 4.700 × 10³ mm = 4.700 × 10⁻³ km

  • In all cases, the number has four significant figures

🔁 Changing the power of 10 does not affect the number of significant figures.

🔢 Order of Magnitude

When an estimate is enough, round ‘a’ to 1 (if ≤5) or 10 (if >5), and write the number as 10ᵇ.

• Diameter of Earth = 1.28 × 10⁷ m → Order of magnitude: 10⁷

• Diameter of Hydrogen atom = 1.06 × 10⁻¹⁰ m → Order of magnitude: 10⁻¹⁰

So, the Earth is 17 orders of magnitude larger than a hydrogen atom.

📐 Exact Numbers in Formulas

• Constants like 2, π, or n used in formulas are considered exact and have infinite significant figures.

  • Example:

    • r = d/2 → ‘2’ is exact

    • T = t/n → ‘n’ is exact

  • These numbers do not limit the significant figures of the result.

✅ Summary: Key Guidelines

Case	Significant?
Non-zero digits	Yes
Zeros between non-zero digits	Yes
Leading zeros (e.g. 0.0023)	No
Trailing zeros (no decimal, e.g. 2300)	No
Trailing zeros (with decimal, e.g. 2.300)	Yes
Scientific notation zeros in ‘a’ (e.g. 4.700)	Yes
Powers of 10 in scientific notation	No (do not count)
Exact numbers (like π, 2, or 100 in ratio)	Infinite sig. figs