Question

Round off the following to three significant digits: (a) \(1.454 \times 10^{1}\) (b) \(1.455 \times 10^{2}\) (c) \(1.508 \times 10^{-3}\) (d) \(1.503 \times 10^{-4}\)

Short answer

(a) 1.45 × 10¹, (b) 1.46 × 10², (c) 1.51 × 10⁻³, (d) 1.50 × 10⁻⁴.

Step-by-step solution

Understand Significant Digits

Significant digits are the meaningful digits in a number, starting from the first non-zero digit. We need to round each given number to three significant digits.

Analyze Number (a)

For number (a) \(1.454 \times 10^{1}\): The number here effectively is 14.54. The first three significant digits are 1, 4, and 5. To round to three significant digits, look at the digits beyond the third (which is 4 in this case). Since 4 is less than 5, we don't add 1 to the third significant digit.

Round (a)

Thus, 14.54 becomes 14.5 when rounded to three significant digits. Therefore, \(1.454 \times 10^{1}\) becomes \(1.45 \times 10^{1}\).

Analyze Number (b)

For number (b) \(1.455 \times 10^{2}\): The number here effectively is 145.5. The first three significant digits are 1, 4, and 5. Look at the digit after the third significant digit, which is 5. Since it is 5 or more, we add 1 to the third significant digit.

Round (b)

Thus, 145.5 becomes 146 when rounded to three significant digits. Therefore, \(1.455 \times 10^{2}\) becomes \(1.46 \times 10^{2}\).

Analyze Number (c)

For number (c) \(1.508 \times 10^{-3}\): The number here effectively is 0.001508. The first three significant digits are 1, 5, and 0. Look at the digit after the third significant digit, which is 8. Since it is more than 5, we add 1 to the third significant digit.

Round (c)

Thus, 0.001508 becomes 0.00151 when rounded to three significant digits. Therefore, \(1.508 \times 10^{-3}\) remains \(1.51 \times 10^{-3}\).

Analyze Number (d)

For number (d) \(1.503 \times 10^{-4}\): The number here effectively is 0.0001503. The first three significant digits are 1, 5, and 0. Look at the digit after the third significant digit, which is 3. Since it is less than 5, we do not add 1 to the third significant digit.

Round (d)

Thus, 0.0001503 remains 0.000150 when rounded to three significant digits. Therefore, \(1.503 \times 10^{-4}\) becomes \(1.50 \times 10^{-4}\).

Key concepts

Rounding Numbers

Rounding numbers is a process of simplifying a number while maintaining its value as close as possible to the original. This is often done by reducing the number of significant digits in a number. Significant digits, also known as significant figures, are the digits in a number that contribute to its precision. When rounding to a specific number of significant digits, the rules are simple:

• Look at the number immediately after your desired last significant digit.
• If this number is 5 or more, round up by adding 1 to the last significant digit.
• If this number is less than 5, keep the last significant digit as it is and drop the remaining digits.

For example, when rounding 1.454 to three significant digits, you see that the digit after the third significant digit is 4. Since 4 is less than 5, you leave the third significant digit unaltered. Understanding these basic rules helps simplify and manage numerical data efficiently.

Scientific Notation

Scientific notation is a way of expressing very large or very small numbers in a compact form. It simplifies calculations and makes them easier to read and understand. A number is expressed in scientific notation as the product of a number between 1 and 10, and a power of 10. For example, the number 145.5 can be written as \(1.455 \times 10^{2}\).

When dealing with scientific notation, rounding numbers typically maintains the number in this exponential form. After rounding the number of significant digits as required, you simply adjust the power of ten accordingly if the number itself changes significantly in magnitude. Scientific notation facilitates clearer documentation of data, especially in scientific contexts where precision and ease of computation are crucial.

Measurement Precision

Measurement precision refers to the consistency and repeatability of measurements. It is distinct from accuracy, which is how well a measurement aligns with the true value. The precision of a measurement is often reflected in the number of significant digits recorded.

• High precision implies that a measuring tool can consistently repeat the same measurement with minimal variance.
• Rounding can adjust precision by reducing the number of significant digits, which might be necessary to reflect the actual accuracy of a measurement instrument.
• Choosing the proper level of precision is essential as it affects how measurements are interpreted and utilized in calculations.

For example, rounding 1.503 to three significant digits results in 1.50, which might better represent the precision of the measurement device used. Understanding the precision requirement in specific contexts can prevent misinterpretation of data and ensures clarity in scientific communication.