Significant Figures

What are Significant Figures?

Significant figures are the reliably known digits in a measurement plus the first doubtful digit. They indicate the precision of the measurement.

Example: If a measurement is $1.62\ \text{s}$, then:

• 1 and 6 are certain
• 2 is doubtful So, it has 3 significant figures.

Rules for Counting Significant Figures:

1. All non-zero digits are significant.

• Example: $2.71$ has 3 significant figures.

1. Zeroes between non-zero digits are significant.

• Example: $205$ has 3 significant figures.

1. Leading zeroes (before non-zero digit) are not significant.

• Example: $0.0056$ has 2 significant figures.

1. Trailing zeroes after decimal are significant.

• Example: $2.500$ has 4 significant figures.

1. Trailing zeroes without a decimal are not significant.

• Example: $2500$ has 2 significant figures, but $2.500 \times 10^3$ has 4.

Scientific Notation

Scientific notation avoids ambiguity in zeroes.

• $2.500 \times 10^3$ → 4 significant figures
• $3.1 \times 10^5$ → 2 significant figures

In scientific notation, all digits in the coefficient are significant.

Arithmetic with Significant Figures

• Multiplication/Division: Final answer should have the same number of significant figures as the input with the least significant figures.
• Addition/Subtraction: Final result should have the same number of decimal places as the number with the least decimal places.

Rounding Off Rules

1. If the digit to be dropped is < 5, retain the previous digit.
2. If > 5, increase the previous digit by 1.
3. If = 5, round to the nearest even digit.

• Example: $2.745$ → $2.74$ (since 4 is even)
• $2.735$ → $2.74$ (since 3 is odd)

Order of Magnitude

Used for rough estimates. Express a quantity as $a \times 10^b$ and round $a$ to either 1 or 10.

• Earth’s diameter: $1.28 \times 10^7\ \text{m}$ → order = $10^7$
• Hydrogen atom: $1.06 \times 10^{-10}\ \text{m}$ → order = $10^{-10}$

Example:

Each side of a cube is measured as $7.203\ \text{m}$. Find surface area and volume.

Solution:

• Significant figures = 4
• Surface Area = $6 \times (7.203)^2 = 311.3\ \text{m}^2$
• Volume = $(7.203)^3 = 373.7\ \text{m}^3$

Example:

$5.74\ g$ of a substance occupies $1.2\ cm^3$. Find its density.

• Mass: 3 significant figures
• Volume: 2 significant figures
• Density = $\frac{5.74}{1.2} = 4.8\ g/cm^3$ → final result has 2 significant figures