In which of the following pairs do both numbers contain the same number of significant figures?
a. 0.005 75 g and 5.75 × 10^-3 g
b. 405 K and 405.0 K
c. 150 000 s and 1.50 × 10⁴ s
d. 3.8 × 10^-2 L and 3.0 × 10⁵ L

Verified step by step guidance

1. Step 1: Recall the rules for determining significant figures. Significant figures include all non-zero digits, any zeros between significant digits, and trailing zeros in a decimal number. Leading zeros are not significant.
2. Step 2: Analyze option (a): 0.005 75 g and 5.75 x 10⁻³ g. For 0.005 75 g, the leading zeros are not significant, so it has 3 significant figures. For 5.75 x 10⁻³ g, all digits in the coefficient (5.75) are significant, so it also has 3 significant figures.
3. Step 3: Analyze option (b): 405 K and 405.0 K. For 405 K, the trailing zero is not significant because there is no decimal point, so it has 3 significant figures. For 405.0 K, the trailing zero is significant because of the decimal point, so it has 4 significant figures.
4. Step 4: Analyze option (c): 150 000 s and 1.50 x 10⁴ s. For 150 000 s, the number of significant figures depends on whether the trailing zeros are considered significant, which is not clear without additional notation (e.g., a decimal point or scientific notation). For 1.50 x 10⁴ s, the coefficient (1.50) has 3 significant figures.
5. Step 5: Analyze option (d): 3.8 x 10⁻² L and 3.0 x 10⁻⁵ L. For 3.8 x 10⁻² L, the coefficient (3.8) has 2 significant figures. For 3.0 x 10⁻⁵ L, the coefficient (3.0) has 2 significant figures because the trailing zero is significant due to the decimal point.

Key Concepts

Significant Figures

Significant figures are the digits in a number that contribute to its precision. This includes all non-zero digits, any zeros between significant digits, and trailing zeros in the decimal portion. Understanding significant figures is crucial for accurately reporting measurements and ensuring that calculations reflect the precision of the data.

Scientific Notation

Scientific notation is a way of expressing numbers that are too large or too small to be conveniently written in decimal form. It is typically formatted as a number between 1 and 10 multiplied by a power of ten. This notation helps in easily identifying significant figures, as the digits in the coefficient represent the significant figures of the number.

Measurement Precision

Measurement precision refers to the degree to which repeated measurements under unchanged conditions show the same results. It is closely related to significant figures, as a measurement's precision is indicated by the number of significant figures it contains. Understanding precision is essential for comparing measurements and ensuring consistency in scientific calculations.