1.10 Significant Figures

In physics, when we make a measurement, we need to communicate how precise that measurement is. Writing down every digit from a calculator display can be misleading because it suggests a level of accuracy that the instrument does not have. Significant figures are a way of expressing the precision of a measured value by indicating which digits are reliably known and which are uncertain.

Definition

Significant figures are all the accurately known digits in a measurement, plus the first doubtful (or uncertain) digit. They represent the greatest accuracy possible with the instrument being used.

Main Idea

The main idea is that a measurement is only as precise as the instrument used. Significant figures provide a standardized way to report measurements, showing the reader exactly how much confidence we have in each digit. A measurement with more significant figures is more precise than one with fewer. When performing calculations, the result should not have more significant figures than the least precise measurement used in the calculation.

Core Concept

• Exact Values: These are values that are counted clearly. For example, if you count 3 pencils, the number "3" is exact. There is no uncertainty.
• Measured Values: These are values obtained from a measurement. They always contain some uncertainty. For example, the length of a pencil measured with a ruler is a measured value.

• The blue pencil's length seems to be greater than 17 cm but shorter than 17.1 cm.
• The red pencil's length seems to be greater than 8 cm but shorter than 8.1 cm.
• The best estimate for the blue pencil is 17.05 cm and for the red pencil is 8.05 cm.
• The first numbers (17.0 and 8.0) are certain. The last digit (0.05) has been estimated and is not certain.
• The two certain numbers, together with one uncertain number, represent the greatest accuracy possible with the ruler. These are the significant figures.

Detailed Explanation

General Rules for Writing Significant Figures The text provides a set of rules to determine how many significant figures are in a reported measurement:

• Example: The number 23.457 has 5 significant figures.
• Example: In "29,000 spectators," the digits 2 and 9 are significant, but the zeros are not. To avoid confusion, this should be written in scientific notation with the exact number of significant figures: 2.90 × 10⁴ (3 sig figs), 2.900 × 10⁴ (4 sig figs), or 2.9000 × 10⁴ (5 sig figs).
• Example: 0.0029 has two significant figures (2 and 9).
• Zeros after a non-zero digit in a decimal indicate a measurement, so they are significant. For example, 0.00290 has three significant figures.
• Zeros between two significant figures are also significant. For example, 100.8 has four significant figures.

• Choose the smaller place value for an accurate final result.
• If the digit in the smallest place is less than 5, leave it as it is (rounding down).
• If the digit is greater than or equal to 5, add 1 to that digit (rounding up).
• Find the digit you want to round and look at the digit to its right.
• If the digit to the right is less than 5, treat it as zero.
• If the digit to the right is 5 or greater, add 1 to that digit and treat all other digits as zero.

Intermediate Level

Significant figures are essential for maintaining the integrity of scientific data. When you report a measurement, the last digit is always an estimate and is the uncertain one. The number of significant figures tells you the precision of the instrument. A measurement of 2.5 cm (2 sig figs) is less precise than 2.50 cm (3 sig figs) because the latter implies the instrument can measure to the hundredths place, while the former only to the tenths place. When performing calculations, the result should not be reported with more significant figures than the least precise measurement used. This ensures that you are not overstating the precision of your result. For example, if you multiply a length (2 sig figs) by a width (3 sig figs), the result should only have 2 significant figures.

Important Points

• - Significant figures indicate the precision of a measurement.
• - All non-zero digits are significant.
• - Zeros between non-zero digits are significant.
• - Leading zeros (before non-zero digits) are not significant.
• - Trailing zeros after a decimal point are significant.
• - The last digit of a measurement is the estimated (uncertain) digit.
• - Rounding rules must be followed to maintain the correct number of significant figures.

Remember

• - Significant Figures = Certain Digits + First Uncertain Digit.
• - Non-zero digits are always significant.
• - Zeros between significant digits are significant.
• - Leading zeros are not significant (0.0029 has 2 sig figs).
• - Trailing zeros after a decimal point are significant (0.00290 has 3 sig figs).

Common Mistakes

• - Counting leading zeros: Forgetting that zeros before a non-zero digit are just placeholders and not significant.
• - Misinterpreting trailing zeros: Not recognizing that a zero after a decimal point can be significant, indicating the precision of the measurement.
• - Not rounding correctly: Failing to apply the rounding rules correctly, especially when the digit to be dropped is 5 or greater.
• - Over-reporting precision: Reporting a calculated result with more significant figures than the measurements it was derived from.

Quick Revision

Significant figures are all the accurately known digits in a measurement plus the first doubtful digit. They indicate the precision of the measurement. The rules for determining significant figures include: all non-zero digits are significant; zeros between non-zero digits are significant; leading zeros are not significant; and trailing zeros after a decimal point are significant. When rounding, if the digit to be dropped is 5 or greater, round up the last significant digit.

60-Second Revision

• Rules: Non-zero digits count. Zeros between count. Leading zeros don't count. Trailing zeros after a decimal count.
• Example: 0.00580 km has 3 sig figs (5, 8, 0).
• Purpose: Communicate the precision of a measurement.

Frequently Asked Questions (FAQs)

Q1. What are significant figures? Answer: Significant figures are all the accurately known digits in a measurement, plus the first uncertain (doubtful) digit. They represent the precision of the measurement.

Q2. Why are significant figures important in physics? Answer: They are important because they communicate the precision and uncertainty of a measurement. When performing calculations, significant figures ensure that the result is not reported with more precision than the original measurements justify.

• The leading zeros are not significant.
• The 5 and 8 are significant.
• The trailing zero after the 8 is significant.

• The 2 and 1 are significant.
• The zero between 1 and 0 is significant.
• The final zero after the decimal point is significant.

Q5. What is the rule for rounding off numbers? Answer: When rounding off a number, look at the digit to the right of the last significant figure you want to keep. If that digit is less than 5, leave the last significant figure as it is. If it is 5 or greater, increase the last significant figure by 1.

Q6. What is the difference between an exact value and a measured value? Answer: An exact value is a counted number (e.g., 3 pencils), which has no uncertainty. A measured value is obtained from a measurement (e.g., 17.05 cm), which always has some uncertainty in its last digit.

Multiple Choice Questions (MCQs)

• Q1. Significant figures include:
• A. Only the accurately known digits
• B. Only the doubtful digits
• C. All accurately known digits plus the first doubtful digit
• D. All digits in a number

• Correct Answer: C
• Explanation: The text defines significant figures as "All the accurately known figures and the first doubtful figure."

• Q2. 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 (2, 3, 4, 5, 7).

• Q3. 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, so there are 2.

• Q4. 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 (after a decimal point) are significant, so there are 3.

• Q5. How many significant figures are in the number 100.8 s?
• A. 2
• B. 3
• C. 4
• D. 5

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

• Q6. 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 in figures larger than 1, zeros that follow a non-zero digit to indicate place value are not significant.

• Q7. 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 place holders."

• Q8. What is the rule for rounding off a number when the digit to the right is less than 5?
• A. Round up
• B. Round down (leave as it is)
• C. Double the digit
• D. Change the digit to zero

• Correct Answer: B
• Explanation: The text states that if the digit in the smallest place is less than 5, it should be left as it is (rounding down).

• Q9. What is the rule for rounding off a number when the digit to the right is 5 or greater?
• A. Round down
• B. Leave it as it is
• C. Round up (add 1)
• D. Change the digit to zero

• Correct Answer: C
• Explanation: The text states that if the digit is greater than or equal to 5, add 1 to that digit (rounding up).

• Q10. The number 5,980,000,000,000,000,000,000,000 kg (mass of Earth) written in scientific notation with three significant figures is:
• A. 5.98 × 10²⁴ kg
• B. 5.980 × 10²⁴ kg
• C. 5.9 × 10²⁴ kg
• D. 6.0 × 10²⁴ kg

• Correct Answer: A
• Explanation: The text states that the mass of Earth is 5.98 × 10²⁴ kg, which has three significant figures (5, 9, 8).

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.

You: