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**Section 1: Significant Figures**

Chapter 2: Measurement and Problem Solving Section 1: Significant Figures

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Learning Objectives Determine which digits in a number are significant. Round numbers to the correct number of significant figures. Determine the correct number of significant figures in calculations.

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Uncertainties Measurements are only as reliable as the instrument used, and the care with which the measurement is made.

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**Uncertainties Accuracy: Precision:**

How close a measurement is to the correct or accepted value Precision: How close a set of measurements are to each other

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Uncertainties The percent error is how far off your own measurement or calculation is from the correct value.

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Question: What value should you record as the length of the beetle? Explain your answer. (The ruler is showing cm)

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Significant Figures Significant figures are the digits in a measurement that are known for certain, plus one uncertain digit. As scientists, we do not care about any digits that are not significant. When we take measurements in lab we only report significant figures.

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Significant Figures In the example, we are certain that the beetle is at least 1.5 cm in length, but the hundredths place is uncertain. We would therefore report this value as 1.55 The one and five are certain digits The second five is uncertain (depending on your opinion, you might report this as 1.54 or 1.56 – these would also be correct) Therefore, this measurement has three total significant figures (two certain digits and one uncertain digit)

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Significant Figures 1.55

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**Significant Figures How to determine significant figures:**

All nonzero digits are significant. Interior zeros (zeros between two nonzero numbers) are significant. Trailing zeros (zeros to the right of a nonzero number) that fall after a decimal point are significant.

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Significant Figures Trailing zeros that fall before a decimal point are significant. Leading zeros (zeros to the left of the first nonzero number) are NOT significant. They serve only to locate the decimal point. Trailing zeros at the end of a number with no decimal point are NOT significant.

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Significant Figures Exact numbers are not counted as significant figures. Exact counting of discrete objects Numbers that are part of an equation Defined quantities (e.g. a dozen, a mole, etc) Some conversion factors are defined quantities, while others are not. 1 in. = 2.54 cm exact

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**Significant Figures 0.0035 1 dozen = 12 1.080 100.00 2371 100,000**

How many significant figures are in each number? 0.0035 1 dozen = 12 1.080 100.00 2371 100,000 2.9 × 105

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**Significant Figures Rules for Rounding:**

When numbers are used in a calculation, the result is rounded to reflect the significant figures of the data. For calculations involving multiple steps, round only the final answer—do not round off between steps. This practice prevents small rounding errors from affecting the final answer.

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Significant Figures Round down if the last digit dropped is 4 or less; round up if the last digit dropped is 5 or more.

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**Significant Figures Round to 3 sig-figs: 4.287 4.29**

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**Significant Figures Multiplication and Division Rule:**

The result of multiplication or division carries the same number of significant figures as the factor with the fewest significant figures.

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Significant Figures The intermediate result (in blue) is rounded to two significant figures to reflect the least precisely known factor (0.10), which has two significant figures.

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Significant Figures The intermediate result (in blue) is rounded to three significant figures to reflect the least precisely known factor (6.10), which has three significant figures.

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**Significant Figures Addition and Subtraction Rule:**

In addition or subtraction calculations, the result carries the same number of decimal places as the quantity carrying the fewest decimal places.

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Significant Figures We round the intermediate answer (in blue) to two decimal places because the quantity with the fewest decimal places (5.74) has two decimal places.

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Significant Figures We round the intermediate answer (in blue) to one decimal place because the quantity with the fewest decimal places (4.8) has one decimal place.

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Significant Figures In calculations involving both multiplication /division and addition/subtraction: do the steps in parentheses first determine the correct number of significant figures in the intermediate answer without rounding then do the remaining steps.

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**Significant Figures 3.489 × (5.67 – 2.3) 3.489 × 3.37 = 11.758 = 12**

do the step in parentheses first – 2.3 = 3.37 Use the subtraction rule to determine that the intermediate answer has only one significant decimal place. To avoid small errors, it is best not to round at this point; instead, underline the least significant figure as a reminder. 3.489 × 3.37 = = 12 Use the multiplication rule to determine that the intermediate answer (11.758) rounds to two significant figures (12) because it is limited by the two significant figures in 3.37.

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**Significant Figures 56.55 × 0.920 ÷ 34.2585**

Perform each calculation to the correct number of significant figures. 56.55 × ÷

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**Significant Figures 1.10 × 0.512 × 1.301 × 0.005 ÷ 3.4**

Perform each calculation to the correct number of significant figures. 1.10 × × × ÷ 3.4

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Significant Figures Perform each calculation to the correct number of significant figures.

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Significant Figures Perform each calculation to the correct number of significant figures.

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**Significant Figures 6.78 × 5.903 × (5.489 – 5.01)**

Perform each calculation to the correct number of significant figures. 6.78 × × (5.489 – 5.01)

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**Significant Figures 3.897 × (782.3 – 451.88)**

Perform each calculation to the correct number of significant figures. 3.897 × (782.3 – )

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**Significant Figures 3 + (15/12) (3 + 15)/12**

Which calculation would have its result reported to the greater number of significant figures? 3 + (15/12) (3 + 15)/12

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