Ordering Rational Numbers and Finding Distance

On the number lines in this lesson, numbers get greater as you move to the right. So less than means to the left of, and greater than means to the right of. This holds for every pair of rational numbers, whatever their signs.

The line above runs from $-2$ to $2$ with a tick every $0.25$. The marker at $-0.75$ is the place of $-\frac{3}{4}$. Read the four markers from left to right and you have them in order from least to greatest:

$-1.5 < -0.75 < 0.5 < 1.25$

Now look at how far each marker is from $0$. The absolute value of a number $a$, written $|a|$, is its distance from $0$ on the number line. It is also called the magnitude of $a$. A distance is never negative, so $|a|$ is never negative: $\left|-1.5\right| = 1.5$ and $|1.25| = 1.25$.

Notice that $-1.5$ is farther from $0$ than any other marker, yet it is the least of the four numbers. Order follows position on the line, not magnitude.

To compare or order rational numbers, you only need comparisons of positive fractions and decimals, which you already know.

1. Sort by side. Negative numbers are left of $0$ and positive numbers are right of $0$. So every negative number is less than $0$, and $0$ is less than every positive number.
2. Use one form. Write the absolute values in the same form: fractions with a common denominator, or decimals. If a decimal repeats, keep the repeat marked with its bar.
3. Compare within each side. Of two positive numbers, the one with the larger absolute value is farther right, so it is greater. Of two negative numbers, the one with the larger absolute value is farther left, so it is less.

Now the second question: how far apart are two numbers? The distance between two numbers is the length of the piece of number line between them. It is how far you travel to get from one to the other, so it is the same in both directions and it is never negative.

Absolute values do the measuring, because each one is a distance from $0$. The line below marks $-1.5$, $-0.5$ and $0.75$, with a tick every $0.25$.

• Opposite sides of zero. Going from $-1.5$ to $0.75$, you pass through $0$. You travel $1.5$ to reach $0$, then $0.75$ more. Add: $1.5 + 0.75 = 2.25$.
• Same side of zero. Going from $-1.5$ to $-0.5$, you stop before $0$. The whole trip from $-1.5$ to $0$ would be $1.5$ long, and you skip its last $0.5$. Subtract: $1.5 - 0.5 = 1$.

Check by counting the gaps between ticks: $9$ gaps of $0.25$ make $2.25$, and $4$ gaps of $0.25$ make $1$.

The distance between two rational numbers $p$ and $q$ is written $|p-q|$, the absolute value of their difference. Here is why that fits.

A difference tells you what to add: $7-3=4$ because $3+4=7$. In the same way, $p-q$ is the number you add to $q$ to land on $p$. On the number line it describes the move from $q$ to $p$: how far, and in which direction (right is positive, left is negative). The absolute value keeps the how far and drops the direction. So $|p-q|$ is the length of the move, which is the distance between the two points.

Try it with $q=-1.5$ and $p=0.75$ on the line above. The move from $q$ to $p$ is $2.25$ to the right, and the move back is $2.25$ to the left. The length is $2.25$ either way, so $|p-q|$ and $|q-p|$ are both $2.25$, the same number you found by adding absolute values. So adding absolute values (opposite sides of $0$) or subtracting them (same side of $0$) gives $|p-q|$ using only arithmetic with positive numbers.

Takeaway. Position on the number line answers both questions. To order rational numbers, ask which is farther left: every negative number is less than every positive number, and of two negative numbers, the one with the larger absolute value is less. The distance between two rational numbers $p$ and $q$ is $|p-q|$. Find it with absolute values: add them when the numbers are on opposite sides of $0$, and subtract the smaller from the larger when they are on the same side. Next, you will look for errors in work that converts between fractions and decimals.