Question

State the number of signiticant digits in each of the following: (a) \(1.050 \times 10^{2} \mathrm{~cm}\) (b) \(2 \times 10^{3} \mathrm{~cm}\) (c) \(3.00 \times 10^{-4} \mathrm{~cm}\) (d) \(5.0 \times 10^{-5} \mathrm{~cm}\)

Short answer

(a) 4 significant digits, (b) 1 significant digit, (c) 3 significant digits, (d) 2 significant digits.

Step-by-step solution

Understanding Significant Digits

Significant digits (or figures) are the digits in a number that contribute to its accuracy. This includes all non-zero digits, zeroes between significant digits, and trailing zeroes after a decimal point.

Counting Significant Digits - Part (a)

For the number \(1.050 \times 10^{2} \mathrm{~cm}\), identify and count the significant figures. All digits (1, 0, 5, 0) are significant because 1 and 5 are non-zero, and the zero to the right of a decimal and after a non-zero digit is also significant. This number has 4 significant digits.

Counting Significant Digits - Part (b)

Examining \(2 \times 10^{3} \mathrm{~cm}\), only the digit 2 is given with certainty. Because it is presented without any zeroes after a decimal or within the number, those are not counted as significant in this context. Therefore, it has 1 significant digit.

Counting Significant Digits - Part (c)

For \(3.00 \times 10^{-4} \mathrm{~cm}\), the digits 3, 0, and 0 are significant. The first zero is significant as it is between a non-zero digit and the decimal, while the second is significant as it is a trailing zero in a decimal number. This has 3 significant digits.

Counting Significant Digits - Part (d)

In \(5.0 \times 10^{-5} \mathrm{~cm}\), the digit 5 is significant, and the zero is also significant because it is a trailing zero in a decimal number. Thus, this number contains 2 significant digits.

Key concepts

Significant Digits Counting

Understanding significant digits is crucial in maintaining the precision of a measurement. These are the numbers that carry meaningful contributions to its precision. Consider digits that are always significant, like:

• Non-zero digits. For example, in the number 256, each digit is significant.
• Zeroes that are between significant digits, like in 1.05, where the zero is significant.
• Trailing zeroes after a decimal point, as seen in 4.500, include all digits as significant.

When counting significant digits in scientific notation, only consider the numbers before the multiplication sign (×10). For example, in the expression "3.05 × 10⁴," there are 3 significant digits: 3, 0, and 5. It's also important to recognize when zeroes are not significant, such as leading zeroes in a decimal number like 0.0025, where none of the leading zeros are significant.

Scientific Notation

Scientific notation is a way of writing numbers that accommodates values that are extremely large or small, making them more readable. It is expressed as the product of a number, usually between 1 and 10, and a power of ten. For example:

• The number 3800 can be written as "3.8 × 10³", where "3.8" is the significant part, and "10³" tells us the decimal placement.
• In "4.56 × 10⁻²", the negative power indicates that the decimal point moves two places to the left, resulting in 0.0456.

In scientific notation, it's easier to identify significant digits. The numbers in front of the multiplication with "10" show the digits that reflect the precision of the measurement. This notation is particularly useful in disciplines like chemistry and physics, where you frequently encounter numbers like the speed of light or the scale of atoms.

Accuracy in Measurements

Accuracy reflects how close a measured value is to the true or accepted value. It’s a different concept from precision, which considers how closely repeated measurements agree with each other. When describing accuracy:

• A highly accurate measurement is one that has little error in comparison with the true value.
• The number of significant digits usually conveys the measurement's accuracy. More significant digits imply higher accuracy.

In practice, understanding and counting significant figures helps ensure that measurements remain precise and reliable. Accurate reporting through significant figures means that every measurement related to an experiment or calculation is meaningful and reproducible, providing a solid foundation for further analysis and conclusions.