Question

Write \(28.403951\) to (a) 6 s.f., (b) 5 s.f., (c) 4 s.f., (d) 3 s.f., (e) 2s.f., (f) 1 s.f.

Short answer

Question: Write the number 28.403951 to (a) 6 significant figures, (b) 5 significant figures, (c) 4 significant figures, (d) 3 significant figures, (e) 2 significant figures, and (f) 1 significant figure. Answer: (a) 28.403952, (b) 28.404, (c) 28.40, (d) 28.4, (e) 28.5, and (f) 30.

Step-by-step solution

(a) 6 Significant Figures

The number \(28.403951\) already has 8 significant figures, so we will keep the first 6 digits and consider the 7th digit, which is 5. Since it is equal to or greater than 5, we will round up. Thus, the number to 6 significant figures will be \(28.403\,95\,\underline{2}\).

(b) 5 Significant Figures

Keep the first 5 digits and consider the 6th digit, which is 9. Since it is equal to or greater than 5, we will round up. The number to 5 significant figures will be \(28.40\,3\,\underline{10}\,\). But we cannot have "10" as a digit, so carry over 1 to the previous digit (3). Therefore, the number to 5 significant figures will be \(28.404\).

(c) 4 Significant Figures

Keep the first 4 digits and consider the 5th digit, which is 3. Since it is less than 5, we don't need to round up. The number to 4 significant figures will be \(28.40\).

(d) 3 Significant Figures

Keep the first 3 digits and consider the 4th digit, which is 4. Since it is less than 5, we don't need to round up. The number to 3 significant figures will be \(28.4\).

(e) 2 Significant Figures

Keep the first 2 digits and consider the 3rd digit, which is 8. Since it is equal to or greater than 5, we will round up. The number to 2 significant figures will be \(28\,\underline{5}\).

(f) 1 Significant Figure

Keep the first digit and consider the 2nd digit, which is 8. Since it is equal to or greater than 5, we will round up. The number to 1 significant figure will be \(\underline{3}0\).

Key concepts

Rounding Numbers

Rounding numbers is a fundamental concept frequently used in mathematics to simplify figures for better readability and ease of calculation. It involves adjusting a number to reduce its digits while maintaining approximate accuracy.

The most common rule in rounding is to check the first digit to be removed. If this digit is 5 or higher, then the last retained digit is increased by one. However, if the digit is lower than 5, the last retained digit remains unchanged.

• For example, rounding the number 28.403951 based on different significant figures (s.f.) or decimal places is a practical skill.
• Consider rounding to 6 s.f.: The number 28.403951, when rounded to 6 s.f., becomes 28.40395 because the digit right after the sixth number is 5, prompting an increase in the sixth digit.
• Likewise, rounding to 5 s.f. yields 28.404. After checking the sixth digit, which is 9, rounding up turns the fifth digit from 3 to 4.

Understanding rounding helps ensure numbers are presented with the desired level of precision.

Precision in Measurement

Precision in measurement refers to how closely different measurements or calculations are to one another. It indicates the level of detail when representing a quantity or value.

In scientific computations and daily applications, this precision helps in avoiding overstating the accuracy of the values used. A number’s precision affects inaccuracies and direct calculations involving variables, especially when significant figures (s.f.) are considered.

• For example, when rounding 28.403951 to various significant figures, precision dictates the exactitude and correctness of the results intended for specific applications.
• When dealing with 4 s.f., we observe that 28.403951 becomes 28.40. The last digit here is the “4” and does not change because the next number is not high enough to require an adjustment.

Maintaining appropriate precision ensures the reliability and integrity of results in measurement and calculation.

Numerical Representation

Numerical representation involves expressing numbers in different formats, which include integer and decimal systems, among others. This representation plays a crucial role when working with significant figures.

Significant figures are used to convey the precision of a value clearly and concisely. They indicate the number of meaningful digits and help clarify the reliability of measurements and calculations.

• The number 28.403951 showcases numerical representation through various degrees of significant figures in our example.
• Reducing to 3 significant figures simplifies it to 28.4. Here, the number retains only three digits that contribute essentially to its precision.
• Representing to merely 1 significant figure, the number becomes 30. The drastic reduction here indicates a focus on the most critical digit while sacrificing detailed accuracy for simplicity.

This concept helps users comprehend and utilize numbers effectively across multiple contexts.