Chapter 3 Discussion/Participation

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Using & Understanding Mathematics: A Quantitative Reasoning Approach

Eighth Edition

Chapter 3

Numbers in the Real World

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Unit 3C Dealing with Uncertainty

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Significant Digits

Type of Digit	Significance
Nonzero digits	Always significant
Zeros that follow a nonzero digit and lie to the right of the decimal point (as in 4.20 or 3.00)	Always significant
Zeros between nonzero digits (as in 4002 or 3.06) or other significant zeros (such as the first zero in 30.0	Always significant
Zeros to the left of the first nonzero digit (as in 0.006 or 0.00052)	Never significant
Zeros to the right of the last nonzero digit but before the decimal point as in (40,000 or 210)	Not significant unless stated otherwise

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Example 1: Counting Significant Digits (1 of 4)

State the number of significant digits and the implied meaning of the following numbers.

a. a time of 11.90 seconds

b. a length of 0.000067 meter

c. a weight of 0.0030 gram

d. a population reported as 240,000

e. a population reported as

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Example 1: Counting Significant Digits (2 of 4)

Solution

a. a time of 11.90 seconds

4 significant digits and implies a measurement to the nearest 0.01 second

b. a length of 0.000067 meter

2 significant digits and implies a measurement to the nearest 0.000001 meter. Note that we can rewrite this number as 67 micrometers, showing clearly that it has only 2 significant digits

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Example 1: Counting Significant Digits (3 of 4)

c. a weight of 0.0030 gram

2 significant digits. The leading zeros are not significant because they serve only as placeholders, as we can see by rewriting the number as 3.0 milligrams. The final zero is significant because there is no reason to include it unless it was measured.

d. a population reported as 240,000

the number has 2 significant digits and implies a measurement to the nearest 10,000 people

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Example 1: Counting Significant Digits (4 of 4)

e. a population reported as

3 significant digits. Although this number means 240,000, the scientific notation shows that the first zero is significant, so it implies a measurement to the nearest 1000 people.

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Example 2: Rounding with Significant Digits (1 of 2)

For each of the following operations, give your answer with the specified number of significant digits.

give your answer with 2 significant

digits

give your answer with 4 significant

digits

Solution

Because we are asked

to give the answer with 2 significant digits, we round to

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Example 2: Rounding with Significant Digits (2 of 2)

give your answer with 4 significant

digits

Solution

Because we are asked

to give the answer with 4 significant digits, we round to

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Types of Measurement Error

Random errors occur because of random and inherently unpredictable events in the measurement process.

Systematic errors occur when there is a problem in the measurement system that affects all measurements in the same way, such as making them all too low or too high by the same amount.

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Provide students with a few concrete examples that occur in daily living in which one or both quantitative errors might be noticed.

Size of Errors

The absolute error describes how far a measured (or claimed) value lies from the true value.

absolute error = measured value − true value

The relative error compares the size of the error to the true value.

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Provide students with a few concrete examples that occur in daily living in which one or both quantitative errors might be noticed.

Example 3: Absolute and Relative Error

Find the absolute and relative error: A projected budget surplus of $17 billion turns out to be $25 billion at the end of the fiscal year.

absolute error = measured value − true value

= $25 billion − $17 billion = $8 billion

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Work through some simple calculations with students and verify that this skill is understood before preceding.

Describing Results

Accuracy describes how closely a measurement approximates a true value. An accurate measurement has a small relative error.

Precision describes the amount of detail in a measurement.

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Perhaps you might discuss a few examples from contemporary issues in the media.

Combining Measured Numbers

Rounding rule for addition or subtraction: Round the answer to the same precision as the least precise number in the problem.

Rounding rule for multiplication or division: Round the answer to the same number of significant digits as the measurement with the fewest significant digits.

To avoid errors, round only after completing all the operations, not during the intermediate steps.

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Perhaps you might discuss a few examples from contemporary issues in the media.

Example 4: Combining Measured Numbers (1 of 4)

a. A book written 30 years ago states that the oldest Mayan ruins are 2000 years old. How old are they now?

Solution

Because the book is 30 years old, we might be tempted to add 30 years to 2000 years to get 2030 years for the age of the ruins. However, 2000 years is the less precise of the two numbers: It is precise only to the nearest 1000 years, while 30 years is precise to the nearest 10 years. Therefore, the answer also should be precise only to the nearest 1000 years:

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Example 4: Combining Measured Numbers (2 of 4)

A book written 30 years ago states that the oldest Mayan ruins are 2000 years old. How old are they now?

Given the precision of the age of the ruins, they are still 2000 years old, despite the 30-year age of the book.

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