# The Properties of Real Numbers

## Systems of Measurement

In this section we will see how to convert among different types of units, such as feet to miles or kilograms to pounds. The basic idea in all of the unit conversions will be to use a form of  the multiplicative identity, to change the units but not the value of a quantity.

### Make Unit Conversions in the U.S. System

There are two systems of measurement commonly used around the world. Most countries use the metric system. The United States uses a different system of measurement, usually called the U.S. system. We will look at the U.S. system first.
The U.S. system of measurement uses units of inch, foot, yard, and mile to measure length and pound and ton to measure weight. For capacity, the units used are cup, pint, quart and gallons. Both the U.S. system and the metric system measure time in seconds, minutes, or hours.
The equivalencies among the basic units of the U.S. system of measurement are listed in . The table also shows, in parentheses, the common abbreviations for each measurement.
In many real-life applications, we need to convert between units of measurement. We will use the identity property of multiplication to do these conversions. We’ll restate the Identity Property of Multiplication here for easy reference.
To use the identity property of multiplication, we write  in a form that will help us convert the units. For example, suppose we want to convert inches to feet. We know that  foot is equal to  inches, so we can write  as the fraction  When we multiply by this fraction, we do not change the value but just change the units.
But  also equals  How do we decide whether to multiply by  or  We choose the fraction that will make the units we want to convert from divide out. For example, suppose we wanted to convert  inches to feet. If we choose the fraction that has inches in the denominator, we can eliminate the inches.
On the other hand, if we wanted to convert  feet to inches, we would choose the fraction that has feet in the denominator.
We treat the unit words like factors and ‘divide out’ common units like we do common factors.
When we use the Identity Property of Multiplication to convert units, we need to make sure the units we want to change from will divide out. Usually this means we want the conversion fraction to have those units in the denominator.
Sometimes to convert from one unit to another, we may need to use several other units in between, so we will need to multiply several fractions.

### Use Mixed Units of Measurement in the U.S. System

Performing arithmetic operations on measurements with mixed units of measures requires care. Be sure to add or subtract like units.

### Make Unit Conversions in the Metric System

In the metric system, units are related by powers of  The root words of their names reflect this relation. For example, the basic unit for measuring length is a meter. One kilometer is  meters; the prefix kilo- means thousand. One centimeter is  of a meter, because the prefix centi- means one one-hundredth (just like one cent is  of one dollar).
The equivalencies of measurements in the metric system are shown in . The common abbreviations for each measurement are given in parentheses.
To make conversions in the metric system, we will use the same technique we did in the U.S. system. Using the identity property of multiplication, we will multiply by a conversion factor of one to get to the correct units.
Have you ever run a  or  race? The lengths of those races are measured in kilometers. The metric system is commonly used in the United States when talking about the length of a race.
Since the metric system is based on multiples of ten, conversions involve multiplying by multiples of ten. In Decimal Operations, we learned how to simplify these calculations by just moving the decimal.
To multiply by  we move the decimal to the right  places, respectively. To multiply by  we move the decimal to the left  places respectively.
We can apply this pattern when we make measurement conversions in the metric system.
In , we changed  grams to kilograms by multiplying by  This is the same as moving the decimal  places to the left.

### Use Mixed Units of Measurement in the Metric System

Performing arithmetic operations on measurements with mixed units of measures in the metric system requires the same care we used in the U.S. system. But it may be easier because of the relation of the units to the powers of  We still must make sure to add or subtract like units.

### Convert Between U.S. and Metric Systems of Measurement

Many measurements in the United States are made in metric units. A drink may come in  bottles, calcium may come in  capsules, and we may run a  race. To work easily in both systems, we need to be able to convert between the two systems.
 shows some of the most common conversions.
We make conversions between the systems just as we do within the systems—by multiplying by unit conversion factors.
The conversion factors in  are not exact, but the approximations they give are close enough for everyday purposes. In , we rounded the number of fluid ounces to the nearest tenth.

### Convert Between Fahrenheit and Celsius Temperatures

Have you ever been in a foreign country and heard the weather forecast? If the forecast is for  What does that mean?
The U.S. and metric systems use different scales to measure temperature. The U.S. system uses degrees Fahrenheit, written  The metric system uses degrees Celsius, written   shows the relationship between the two systems.
If we know the temperature in one system, we can use a formula to convert it to the other system.

### Section Exercises


### Practice Makes Perfect

Make Unit Conversions in the U.S. System 
In the following exercises, convert the units.
Use Mixed Units of Measurement in the U.S. System 
In the following exercises, solve and write your answer in mixed units.
Make Unit Conversions in the Metric System 
In the following exercises, convert the units.
Use Mixed Units of Measurement in the Metric System 
In the following exercises, solve and write your answer in mixed units.
Convert Between U.S. and Metric Systems 
In the following exercises, make the unit conversions. Round to the nearest tenth.
Convert between Fahrenheit and Celsius 
In the following exercises, convert the Fahrenheit temperature to degrees Celsius. Round to the nearest tenth.
In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.



ⓑ Overall, after looking at the checklist, do you think you are well-prepared for the next chapter? Why or why not?


### Chapter Review Exercises


### Rational and Irrational Numbers

In the following exercises, write as the ratio of two integers.
In the following exercises, determine which of the numbers is rational.
In the following exercises, identify whether each given number is rational or irrational.
In the following exercises, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers for each set of numbers.

### Commutative and Associative Properties

In the following exercises, use the commutative property to rewrite the given expression.
In the following exercises, use the associative property to rewrite the given expression.
In the following exercises, evaluate each expression for the given value.
In the following exercises, simplify using the commutative and associative properties.

### Distributive Property

In the following exercises, simplify using the distributive property.
In the following exercises, evaluate using the distributive property.

### Properties of Identities, Inverses, and Zero

In the following exercises, identify whether each example is using the identity property of addition or multiplication.
In the following exercises, find the additive inverse.
In the following exercises, find the multiplicative inverse.
In the following exercises, simplify.

### Systems of Measurement

In the following exercises, convert between U.S. units. Round to the nearest tenth.
In the following exercises, solve, and state your answer in mixed units.
In the following exercises, convert between metric units.
In the following exercises, solve.
In the following exercises, convert between U.S. and metric units. Round to the nearest tenth.
In the following exercises, convert the Fahrenheit temperatures to degrees Celsius. Round to the nearest tenth.
In the following exercises, convert the Celsius temperatures to degrees Fahrenheit. Round to the nearest tenth.

### Chapter Practice Test

In the following exercises, simplify the given expression.
In the following exercises, solve using the appropriate unit conversions.