Scientific Notation and Significant Figures

Let’s take as an example: 5,700,000

Shift the decimal point to the left until we get a number between 1 and 10:

$$5.700000 \quad \text{(6 positions)}$$

The exponent is positive and equal to the number of positions moved:

$$5.7 \times 10^{6}$$

Let’s take as an example: 0.0065

Shift the decimal point to the right until we get a number between 1 and 10:

$$6.5 \quad \text{(3 positions)}$$

The exponent is negative and equal to the number of positions moved:

$$6.5 \times 10^{-3}$$

All digits different from zero are significant.

Examples:

• 2.4 → 2 S.F. (2 and 4 are significant)
• 893 → 3 S.F. (8, 9 and 3 are significant)

Zeros located between non-zero digits are significant.

Examples:

• 105 → 3 S.F. (1, 0 and 5 are significant)
• 2005 → 4 S.F. (2, 0, 0 and 5 are significant)

Zeros to the left of the first non-zero digit are NOT significant.

Examples:

• 0.45 → 2 S.F. (only 4 and 5 are significant)
• 0.039 → 2 S.F. (only 3 and 9 are significant)

Zeros to the right after the decimal point ARE significant.

Examples:

• 25.0 → 3 S.F. (2, 5 and 0 are significant)
• 3.00 → 3 S.F. (3, 0 and 0 are significant)

Zeros at the end of a whole number are ambiguous.
Solution: use scientific notation to clarify.

Example for 500:

• $5.00 \times 10^{2}$ → 3 S.F.
• $5.0 \times 10^{2}$ → 2 S.F.
• $5 \times 10^{2}$ → 1 S.F.