Question

Round off each of the following measurements to three significant figures: a. \(1.854 \mathrm{~kg}\) b. \(88.2038 \mathrm{~L}\) c. \(0.004738265 \mathrm{~cm}\) d. \(8807 \mathrm{~m}\) e. \(1.832 \times 10^{5} \mathrm{~s}\)

Short answer

a. 1.85 kg, b. 88.2 L, c. 0.00474 cm, d. 8810 m, e. 1.83 × 10^5 s.

Step-by-step solution

Understanding Significant Figures

Significant figures are the digits in a number that contribute to its precision. Rounding to three significant figures involves keeping only the first three digits that are not zeros starting from the left.

Rounding 1.854 kg

The number 1.854 has four digits. To round to three significant figures, look at the fourth digit (4). Since it is less than 5, do not round up the third digit. Thus, the rounded value is 1.85 kg.

Rounding 88.2038 L

The number 88.2038 has six digits. To round to three significant figures, look at the fourth digit (2). Since it is less than 5, do not round up the third digit. Thus, the rounded value is 88.2 L.

Rounding 0.004738265 cm

The number 0.004738265 has nine digits. The first three significant figures are 473 (ignoring the leading zeros). Look at the fourth digit (8). Since it is 5 or greater, round up the third digit. Thus, the rounded value is 0.00474 cm.

Rounding 8807 m

The number 8807 has four digits. To round to three significant figures, look at the fourth digit (7). Since it is 5 or greater, round up the third digit. Thus, the rounded value is 8810 m.

Rounding 1.832 × 10^5 s

The number 1.832 × 10^5 has four significant figures. To round to the first three, look at the fourth digit (2). Since it is less than 5, do not round up the third digit. Thus, the rounded value is 1.83 × 10^5 s.

Key concepts

Rounding Numbers

Rounding numbers is a crucial skill in mathematics and science. It helps simplify calculations and makes numbers easier to work with. To round a number to a specific number of significant figures, follow these steps:
Identify the digit up to which you need to round.
Look at the next digit (right of the last significant figure you want to keep).
If this digit is 5 or greater, increase the last significant figure by 1. If it is less than 5, leave the last significant figure unchanged.
For example, if you need to round 88.2038 to three significant figures, you will end up with 88.2 because the fourth digit (2) is less than 5. Remember, leading zeros (e.g., 0.004738265) are not considered significant.

Precision in Measurements

Precision refers to how close repeated measurements of the same object are to each other. It is an important aspect of measurements in science and engineering. Precision is often reflected by the number of significant figures in a measurement. More significant figures typically indicate a more precise measurement.
A precise measurement means there is less variation and more consistency in the results. However, don't confuse precision with accuracy. Precision is all about consistency, while accuracy measures how close you are to the true value.
For example, if you measure the length of a table multiple times and get values like 2.563 cm, 2.565 cm, and 2.564 cm, your measurements are precise because they are close to each other.

Scientific Notation

Scientific notation is a method used to express very large or very small numbers in a compact form. It is written as the product of a number between 1 and 10 and a power of 10. For example, 1000 can be written as 1 × 10³ and 0.00086 as 8.6 × 10⁻⁴.
This notation makes it easy to handle and communicate large quantities or tiny measurements. When rounding numbers in scientific notation to a certain number of significant figures, the same rounding rules apply.
For example, rounding 1.832 × 10⁵ to three significant figures results in 1.83 × 10⁵ because the fourth digit (2) is less than 5. Always ensure the significant figures are properly maintained in scientific notation.

Measurement Accuracy

Accuracy in measurements tells how close a measurement is to the true value. Accuracy depends on both the systematic errors (bias) and the random errors in the measurement process.
In contrast to precision, which indicates the consistency of repeated measurements, accuracy indicates how close those measurements are to the actual value. High accuracy and low precision can occur, but both are preferable in scientific experiments.
Improving accuracy involves using better instruments, calibrating equipment correctly, and refining measurement techniques. For instance, if you repeatedly measure the mass of a block as 1.854 kg, but the true mass is 1.860 kg, your measurements are precise but not accurate. Adjusting and recalibrating your scale could improve accuracy.