### Properties of Rational Numbers

One day, Jason leaves his house and starts walking to school. After three blocks, he stops to tie his shoe and leaves his lunch bag sitting on the curb. Two blocks farther on, he realizes his lunch is missing and goes back to get it. After picking up his lunch, he walks six more blocks to arrive at school. How far is the school from Jason’s house? And how far did Jason actually walk to get there?

**Graph and Compare Integers**

**Integers** are the counting numbers (1, 2, 3...), the negative opposites of the counting numbers (-1, -2, -3...), and zero. There are an infinite number of integers and examples are 0, 3, 76, -2, -11, and 995.

#### Comparing Numbers

Compare the numbers 2 and -5.

When we plot numbers on a number line, the **greatest** number is farthest to the right, and the **least** is farthest to the left.

In the diagram above, we can see that 2 is farther to the right on the number line than -5, so we say that 2 is greater than -5. We use the symbol “

**Classifying Rational Numbers**

When we divide an integer **rational number**. It’s called this because it is the **ratio** of one number to another, and we can write it in fraction form as **numerator** and the bottom number is called the **denominator**.)

You can think of a rational number as a fraction of a cake. If you cut the cake into

For example, when we see the rational number

With the rational number

The rational number

**Proper fractions** are rational numbers where the numerator is less than the denominator. A proper fraction represents a number less than one.

**Improper fractions** are rational numbers where the numerator is greater than or equal to the denominator. An improper fraction can be rewritten as a mixed number – an integer plus a proper fraction. For example,

**Equivalent fractions** are two fractions that represent the same amount. For example, look at a visual representation of the rational number

You can see that the shaded regions are the same size, so the two fractions are equivalent. We can convert one fraction into the other by **reducing** the fraction, or writing it in lowest terms. To do this, we write out the prime factors of both the numerator and the denominator and cancel matching factors that appear in both the numerator **and** denominator.

Reducing a fraction doesn’t change the value of the fraction—it just simplifies the way we write it. Once we’ve canceled all common factors, the fraction is in its **simplest form**.

#### Classifying and Simplifying Numbers

Classify and simplify the following rational numbers

a)

3 and 7 are both prime, so we can't factor them. That means

b)

**Order Rational Numbers**

Ordering rational numbers is simply a matter of arranging them by increasing value—least first and greatest last.

#### Ordering Fractions

1. Put the following fractions in order from least to greatest*:*

Simple fractions are easy to order—we just know, for example, that one-half is greater than one quarter, and that two thirds is bigger than one-half. But how do we compare more complex fractions?

*2.* Which is greater, *or* *?*

In order to determine this, we need to rewrite the fractions so we can compare them more easily. If we rewrite each as an equivalent fraction so that both have the same denominators, then we can compare them directly. To do this, we need to find the **lowest common denominator** (LCD), or the least common multiple of the two denominators.

The lowest common multiple of 7 and 9 is 63. Our fraction will be represented by a shape divided into 63 sections. This time we will use a rectangle cut into 9 by

7 divides into 63 nine times, so

We can multiply the numerator and the denominator both by 9 because that’s really just the opposite of reducing the fraction. To get back from

The fractions

Therefore,

9 divides into 63 seven times, so

By writing the fractions with a **common denominator** of 63, we can easily compare them. If we take the 28 shaded boxes out of 63 (from our image of

Since

**Graph and Order Rational Numbers**

To plot non-integer rational numbers (fractions) on the number line, we can convert them to mixed numbers (graphing is one of the few occasions in algebra when it’s better to use mixed numbers than improper fractions), or we can convert them to decimal form.

#### Plotting Numbers on a Number Line

Plot the following rational numbers on the number line.

a)

If we divide the number line into sub-intervals based on the denominator of the fraction, we can look at the fraction’s numerator to determine how many of these sub-intervals we need to include.

b)

\begin{align*}-\frac{3}{7}\end{align*} falls between 0 and -1. We divide the interval into seven units, and move left from zero by three of those units.

### Examples

#### Example 1

Classify and simplify the rational number \begin{align*}\frac{50}{60}\end{align*}.

\begin{align*}\frac{50}{60}\end{align*} is a proper fraction, and we can simplify it as follows: \begin{align*}\frac{50}{60} = \frac{5 \cdot 5 \cdot 2}{5 \cdot 3 \cdot 2 \cdot 2} = \frac{5}{3 \cdot 2} = \frac{5}{6}\end{align*}.

#### Example 2

Plot the rational number \begin{align*}\frac{17}{5}\end{align*} on the number line.

\begin{align*}\frac{17}{5}\end{align*} as a mixed number is \begin{align*}3\frac{2}{5}\end{align*} and falls between 3 and 4. We divide the interval into five units, and move over two units.

Another way to graph this fraction would be as a decimal. \begin{align*}3\frac{2}{5}\end{align*} is equal to 3.4, so instead of dividing the interval between 3 and 4 into 5 units, we could divide it into 10 units (each representing a distance of 0.1) and then count over 4 units. We would end up at the same place on the number line either way.

### Review

- Order the numbers 2, \begin{align*}-\frac{5}{2}\end{align*}, and \begin{align*}\frac{5}{2}\end{align*}from least to greatest (the question from the beginning of the lesson).
- The tick-marks on the number line represent evenly spaced integers. Find the values of \begin{align*}a, b, c, d, \text{ and } e\end{align*}.

In 3-5, determine what fraction of the whole each shaded region represents.

For 6-10, place the following sets of rational numbers in order, from least to greatest.

- \begin{align*}\frac{1}{2}, \frac{1}{3}, \frac{1}{4}\end{align*}
- \begin{align*}\frac{1}{10}, \frac{1}{2}, \frac{2}{5}, \frac{1}{4}, \frac{7}{20}\end{align*}
- \begin{align*}\frac{39}{60}, \frac{49}{80}, \frac{59}{100}\end{align*}
- \begin{align*}\frac{7}{11}, \frac{8}{13}, \frac{12}{19}\end{align*}
- \begin{align*}\frac{9}{5}, \frac{22}{15}, \frac{4}{3}\end{align*}

For 11-15, find the simplest form of the following rational numbers.

- \begin{align*}\frac{22}{44}\end{align*}
- \begin{align*}\frac{9}{27}\end{align*}
- \begin{align*}\frac{12}{18}\end{align*}
- \begin{align*}\frac{315}{420}\end{align*}
- \begin{align*}\frac{244}{168}\end{align*}

### Review (Answers)

To view the Review answers, open this PDF file and look for section 2.1.