Question

In which of the following pairs do both numbers contain the same number of significant figures? a. \(11.0 \mathrm{~m}\) and \(11.00 \mathrm{~m}\) b. \(405 \mathrm{~K}\) and \(504.0 \mathrm{~K}\) c. \(0.00012 \mathrm{~s}\) and \(12000 \mathrm{~s}\) d. \(250.0 \mathrm{~L}\) and \(2.5 \times 10^{-2} \mathrm{~L}\)

Short answer

(c)

Step-by-step solution

- Understand Significant Figures

Significant figures represent the number of meaningful digits in a number. These include all non-zero digits, any zeroes between significant figures, and trailing zeros in a decimal number.

- Analyze Pair a

For the pair (11.0 m and 11.00 m): The first number has 3 significant figures (1, 1, and the trailing zero). The second number has 4 significant figures (1, 1, and two trailing zeros).

- Analyze Pair b

For the pair (405 K and 504.0 K): The first number has 3 significant figures (4, 0, and 5). The second number has 4 significant figures (5, 0, 4, and the trailing zero).

- Analyze Pair c

For the pair (0.00012 s and 12000 s): The first number has 2 significant figures (1 and 2). The second number has only 2 significant figures as well if we consider all trailing zeros are placeholders (1 and 2).

- Analyze Pair d

For the pair (250.0 L and 2.5 x 10^{-2} L): The first number has 4 significant figures (2, 5, 0, and the trailing zero). The second number has 2 significant figures (2 and 5).

- Determine Matching Pair

From the analysis: Pair (c) has both numbers with the same number of significant figures (2 each).

Key concepts

meaningful digits

When we talk about significant figures, we refer to meaningful digits in measurements. This means every digit that adds to the accuracy and precision of a number is considered a meaningful digit.

Here are some rules to identify meaningful digits:

• All non-zero digits are always significant.
• Any zeros between significant digits are significant.
• Trailing zeros in a decimal number are significant.

The term ‘meaningful’ refers to the information the digit provides about the certainty of the measurement. For example, in the number 150.0, all four digits (1, 5, 0, and 0) are significant. This tells us the measurement is precise to the nearest tenth.

trailing zeros

Trailing zeros can sometimes be tricky. They are the zeros that appear at the end of a number after all the non-zero digits.

When are trailing zeros significant?

• They are significant when they come after a decimal point. For example, in 56.00, both zeros are significant.
• If there is no decimal point, trailing zeros are not usually considered significant. For instance, in 2000, we only consider the first digit as significant unless specified otherwise.

Understanding the role of trailing zeros is crucial, especially in scientific measurements where precision matters.

science education

Understanding significant figures is a fundamental part of science education. It helps students develop a clear understanding of how precise and accurate a measurement is.

Significant figures are used in various scientific calculations:

• They ensure consistency and accuracy in data reporting.
• They help scientists and researchers communicate their findings effectively.
• They prevent the misinterpretation of data by providing a clear indication of uncertainty.

By learning about significant figures, students gain a valuable tool for interpreting experimental data and refining their analytical skills. This knowledge extends beyond the classroom, applying to real-world scientific problems and research.