Scientific notation and significant figures

Chemistry deals with very large and very small numbers, so we use scientific notation: N × 10ⁿ, where N is between 1 and 9.999… and n is a whole number. Count how far the decimal point moves: 6,500 = 6.5 × 10³, and 0.00048 = 4.8 × 10⁻⁴.

Every measurement has some uncertainty. Significant figures are the digits we are certain about plus one doubtful digit. In 25.6 cm, the 2 and 5 are certain and the 6 is the doubtful last digit, so there are three significant figures. The rules:

• All non-zero digits count.
• Zeros between non-zero digits count: 2005 has four.
• Leading zeros (to the left of the first non-zero digit) never count: 0.034 has two.
• Trailing zeros count only if there is a decimal point: 2.500 has four, but 200 has just one.
• Exact numbers (2 balls, 20 students) have infinite significant figures.

In calculations, the answer cannot be more certain than the least certain measurement:

• Adding or subtracting: keep the fewest decimal places. 4.25 + 3.1 = 7.35, reported as 7.4.
• Multiplying or dividing: keep the fewest significant figures. 3.0 × 2.25 = 6.75, reported as 6.8.

Rounding off: if the digit to be dropped is less than 5, leave the last digit as it is; if more than 5, raise it by one. If it is exactly 5, raise the last digit only if it is odd: 4.75 becomes 4.8, but 4.65 becomes 4.6.

Precision is how close repeated readings are to each other; accuracy is how close they are to the true value. Imagine four students measuring something whose true value is 3.00. A student whose readings are 2.50, 2.51, 2.49 is precise but not accurate; one whose readings are 3.01, 2.99, 3.00 is both.

To change units, use the unit factor method: multiply by a fraction that equals 1, such as 1 m / 100 cm, arranged so that the unwanted unit cancels. 250 cm × (1 m / 100 cm) = 2.5 m.