1.10.1 General Rules for Writing Significant Figures

Once we understand what significant figures are, the next step is to learn how to identify them correctly in any given number. The number of significant figures in a measurement tells us about its precision. To avoid confusion and to communicate measurements clearly and consistently, physicists follow a set of standard rules for counting significant figures.

Definition

Significant figures are all the accurately known digits in a measurement, plus the first doubtful digit. The rules for writing significant figures are a set of guidelines used to determine which digits in a reported number are significant and which are not.

Main Idea

The main idea is that not all digits in a number carry the same weight of information. Some digits are known with certainty, some are estimated, and some are merely placeholders. By following a consistent set of rules, we can clearly identify the significant figures, ensuring that the precision of a measurement is communicated accurately and without ambiguity. This is essential for scientific communication and for correctly performing calculations.

Core Concept

The text outlines four simple rules to determine how many significant figures are contained in a reported measurement:

Rule 1 All digits reported as a direct result of a measurement are significant. - This is the fundamental principle. If a digit is part of the measurement, it counts as significant.

Rule 2 The reported NONZERO digits (all digits from 1 to 9) are always significant. - Example from text: The number 23.457 has 5 significant figures (2, 3, 4, 5, 7).

• Example from text: In "29,000 spectators," the digits 2 and 9 are significant, but the three zeros are not significant. They only indicate the estimate of how many watched the match.
• Note: To avoid ambiguity, such numbers should be written in scientific notation with the exact number of significant figures: 2.90 × 10⁵ (3 sig figs), 2.900 × 10⁵ (4 sig figs), or 2.9000 × 10⁵ (5 sig figs).

• Example from text: 0.0029 has two significant figures (2 and 9). The zeros after a nonzero digit indicate a measurement, so these zeros are significant.
• Example from text: 0.00290 has three significant figures (2, 9, and the final 0).

Detailed Explanation

The rules for significant figures are designed to be straightforward and unambiguous. They help us distinguish between digits that represent measured values and zeros that are only there to position the decimal point.

Rule 1: All Direct Measurement Digits are Significant This rule means that every digit you record from an instrument is significant. For example, if a digital balance reads 25.43 g, all four digits are significant because they are a direct result of the measurement.

Rule 2: Non-Zero Digits are Always Significant This is the simplest rule. Any digit from 1 to 9 is always significant. For example, in 123.45, all five digits are significant.

• 2.9 × 10⁴ has 2 significant figures.
• 2.90 × 10⁴ has 3 significant figures.
• 2.900 × 10⁴ has 4 significant figures.

Rule 4: Leading Zeros are Not Significant Zeros that come before the first non-zero digit are never significant. They simply show the position of the decimal point. For example, in 0.0029, the zeros before the 2 are not significant. Only the 2 and 9 are significant.

Zeros in the Middle are Significant Although not explicitly listed as a separate rule in the text, the examples show that zeros between two significant figures are significant. For example, in 100.8, both zeros are significant because they are between the 1 and the 8.

Intermediate Level

The rules for significant figures are essential for maintaining the integrity of scientific data. The ambiguity of zeros is the main challenge. Rule 3 addresses the problem of trailing zeros in large numbers, recommending scientific notation as the unambiguous solution. Rule 4 clarifies that leading zeros are never significant. The examples in the text (such as 100.8 s, 0.00580 km, and 210.0 g) demonstrate how to apply these rules in different cases. The key takeaway is that the last significant digit is the first uncertain digit, and its position determines the precision of the measurement.

Important Points

• - Rule 1: All digits from a measurement are significant.
• - Rule 2: All non-zero digits (1-9) are significant.
• - Rule 3: Trailing zeros in large numbers (without a decimal point) are not significant. Use scientific notation to avoid ambiguity.
• - Rule 4: Leading zeros (before the first non-zero digit) are not significant.
• - Zeros between non-zero digits are significant.

Remember

• - Non-zero digits = Significant.
• - Leading zeros = Not significant.
• - Trailing zeros in large numbers = Ambiguous (use scientific notation).
• - Trailing zeros after a decimal = Significant.
• - Zeros between significant digits = Significant.

Common Mistakes

• - Counting leading zeros as significant: Forgetting that zeros before the first non-zero digit (like in 0.003) are not significant.
• - Counting trailing zeros in large numbers as significant: Assuming that the zeros in 29,000 are significant. They are not, unless indicated otherwise.
• - Forgetting zeros in the middle: Overlooking that zeros between non-zero digits are significant (e.g., in 100.8).

Quick Revision

• All digits from a direct measurement are significant.
• All non-zero digits (1-9) are always significant.
• In large numbers (without a decimal point), trailing zeros are not significant. Scientific notation should be used to remove ambiguity.
• In small numbers, leading zeros (before the first non-zero digit) are not significant.
• Zeros between non-zero digits are significant.

60-Second Revision

• All measured digits count.
• Non-zero digits (1-9) count.
• Trailing zeros in large numbers don't count (use sci. notation).
• Leading zeros don't count (0.0029 has 2 sig figs).
• Zeros between non-zero digits count (100.8 has 4 sig figs).

Frequently Asked Questions (FAQs)

• Q1. How many significant figures are in the number 100.8 s?
• Answer: All four digits (1, 0, 0, 8) are significant. The zeros are significant because they are between two non-zero digits.

Q2. How many significant figures are in the number 0.00580 km? Answer: There are three significant figures: 5, 8, and the final 0. The leading zeros are not significant. The final zero is significant because it comes after a non-zero digit in a decimal.

Q3. How many significant figures are in the number 210.0 g? Answer: There are four significant figures: 2, 1, 0, and the final 0. The zero between the 1 and the 0 is significant, and the trailing zero after the decimal point is also significant.

Q4. Why are the zeros in 29,000 not significant? Answer: The text explains that in figures reported as larger than the digit 1, the digit 0 is not significant when it follows a nonzero digit to indicate place. The zeros in 29,000 are placeholders to indicate the magnitude of the number, not the precision of the measurement.

Q5. How can you remove the ambiguity of trailing zeros in a large number? Answer: Use scientific notation. For example, 29,000 can be written as 2.9 × 10⁴ (2 significant figures), 2.90 × 10⁴ (3 significant figures), or 2.900 × 10⁴ (4 significant figures).

• Q6. What is the rule for zeros between two non-zero digits?
• Answer: Zeros between two non-zero digits are significant. For example, in 100.8, both zeros are significant because they are between the 1 and the 8.

Multiple Choice Questions (MCQs)

• Q1. How many significant figures are in the number 23.457?
• A. 3
• B. 4
• C. 5
• D. 6

• Correct Answer: C
• Explanation: All non-zero digits are significant, so there are 5 significant figures.

• Q2. How many significant figures are in the number 0.0029?
• A. 1
• B. 2
• C. 3
• D. 4

• Correct Answer: B
• Explanation: The leading zeros are not significant. The significant figures are 2 and 9.

• Q3. How many significant figures are in the number 0.00290?
• A. 1
• B. 2
• C. 3
• D. 4

• Correct Answer: C
• Explanation: The leading zeros are not significant. The digits 2, 9, and the trailing zero are significant.

• Q4. How many significant figures are in the number 100.8 s?
• A. 2
• B. 3
• C. 4
• D. 5

• Correct Answer: C
• Explanation: All four digits (1, 0, 0, 8) are significant. The zeros are between non-zero digits.

• Q5. In the number 29,000, which digits are significant?
• A. 2, 9, and all three zeros
• B. 2 and 9 only
• C. 2 only
• D. None of the digits

• Correct Answer: B
• Explanation: The text states that the digits 2 and 9 are significant, but the zeros are not.

• Q6. Zeros after a decimal point that come before non-zero digits are:
• A. Always significant
• B. Sometimes significant
• C. Not significant
• D. Always zero

• Correct Answer: C
• Explanation: The text states that zeros after a decimal point that come before nonzero digits are not significant and serve only as placeholders.

• Q7. How many significant figures are in the number 210.0 g?
• A. 2
• B. 3
• C. 4
• D. 5

Correct Answer: C Explanation: The final zero is significant since it comes after the decimal point. The zero between the last zero and 1 is also significant, giving 4 significant figures.

• Q8. How many significant figures are in the number 5,980,000,000,000,000,000,000,000 kg (mass of Earth) written as 5.98 × 10²⁴ kg?
• A. 2
• B. 3
• C. 4
• D. 5

• Correct Answer: B
• Explanation: The number 5.98 × 10²⁴ has three significant figures (5, 9, 8).

• Q9. What is the rule for non-zero digits?
• A. They are never significant.
• B. They are sometimes significant.
• C. They are always significant.
• D. They are only significant in decimals.

• Correct Answer: C
• Explanation: The text states that the reported non-zero digits (all digits from 1 to 9) are always significant.

• Q10. To avoid confusion about the number of significant figures in a large number like 29,000, it should be written as:
• A. 29000.
• B. 2.9 × 10⁴
• C. 290 × 10²
• D. 0.29 × 10⁵

Correct Answer: B Explanation: The text states that to avoid confusion, figures should be written in scientific notation with the exact number of significant figures, such as 2.90 × 10⁵.

Frequently Asked Questions (FAQs)

Q1. What is the difference between significant figures and decimal places? Answer: Significant figures count all the digits that carry meaning about the precision of a number, starting from the first non-zero digit. Decimal places refer specifically to the number of digits after the decimal point. For example, 0.0045 has two significant figures but four decimal places.

Q2. How do you determine the number of significant figures in a calculated result? Answer: For multiplication and division, the result should have the same number of significant figures as the measurement with the fewest significant figures. For addition and subtraction, the result should have the same number of decimal places as the measurement with the fewest decimal places.

Q3. Are zeros always significant? Answer: No. Zeros are significant if they are between non-zero digits or if they are trailing zeros after a decimal point. They are not significant if they are leading zeros (before the first non-zero digit).

Q4. Why is it important to use significant figures when reporting a measurement? Answer: Using significant figures is important because it communicates the precision of the measurement. Reporting too many digits suggests a level of accuracy that the instrument does not have, which is misleading.

Q5. What is the difference between the number 100 and 100.? Answer: The number "100" is ambiguous; it could have 1, 2, or 3 significant figures. The number "100." (with a decimal point) has three significant figures, as the decimal point indicates that the zeros are measured and significant.

Q6. How do you round off a number to the correct number of significant figures? Answer: Identify the last significant figure you want to keep. Look at the next digit to its right. If it is 5 or greater, increase the last significant figure by 1. If it is less than 5, leave the last significant figure as it is. Any digits further to the right are dropped.

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