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Locating Numbers on the Number Line

Locating Positive and Negative Numbers on the Number Line

Do you live in a place that has very cold winters? Have you ever experienced a temperature below zero? If so, you are already familiar with negative numbers. A negative number is a number that is less than $$0.$$ Very cold temperatures are measured in degrees below zero and can be described by negative numbers.

For example, $$-1\text{°F}$$ (read as “negative one degree Fahrenheit”) is $$1\phantom{\rule{0.2em}{0ex}}\text{degree}$$ below $$0.$$ A minus sign is shown before a number to indicate that it is negative. The figure below shows $$-20\text{°F},$$ which is $$20\phantom{\rule{0.2em}{0ex}}\text{degrees}$$ below $$0.$$

Temperatures below zero are described by negative numbers.

Temperatures are not the only negative numbers. A bank overdraft is another example of a negative number. If a person writes a check for more than he has in his account, his balance will be negative.

Elevations can also be represented by negative numbers. The elevation at sea level is $$\text{0 feet}.$$ Elevations above sea level are positive and elevations below sea level are negative. The elevation of the Dead Sea, which borders Israel and Jordan, is about $$1,302\phantom{\rule{0.2em}{0ex}}\text{feet}$$ below sea level, so the elevation of the Dead Sea can be represented as $$-1,302\phantom{\rule{0.2em}{0ex}}\text{feet}.$$ See the figure below.

The surface of the Mediterranean Sea has an elevation of $$0\phantom{\rule{0.2em}{0ex}}\text{ft}.$$ The diagram shows that nearby mountains have higher (positive) elevations whereas the Dead Sea has a lower (negative) elevation.

Depths below the ocean surface are also described by negative numbers. A submarine, for example, might descend to a depth of $$500\phantom{\rule{0.2em}{0ex}}\text{feet}.$$ Its position would then be $$-500\phantom{\rule{0.2em}{0ex}}\text{feet}$$ as labeled in the figure below.

Depths below sea level are described by negative numbers. A submarine $$500\phantom{\rule{0.2em}{0ex}}\text{ft}$$ below sea level is at $$-500\phantom{\rule{0.2em}{0ex}}\text{ft}.$$

Both positive and negative numbers can be represented on a number line. Recall that the number line created in Mathematics 101 started at $$0$$ and showed the counting numbers increasing to the right as shown in the figure below. The counting numbers $$\text{(1, 2, 3, …)}$$ on the number line are all positive.

We could write a plus sign, $$+,$$ before a positive number such as $$+2$$ or $$+3,$$ but it is customary to omit the plus sign and write only the number. If there is no sign, the number is assumed to be positive.

Now we need to extend the number line to include negative numbers. We mark several units to the left of zero, keeping the intervals the same width as those on the positive side. We label the marks with negative numbers, starting with $$-1$$ at the first mark to the left of $$0,-2$$ at the next mark, and so on. See the figure below.

On a number line, positive numbers are to the right of zero. Negative numbers are to the left of zero. What about zero? Zero is neither positive nor negative.

The arrows at either end of the line indicate that the number line extends forever in each direction. There is no greatest positive number and there is no smallest negative number.

Example

Plot the numbers on a number line:

1. $$\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{1em}{0ex}}$$
2. $$\phantom{\rule{0.2em}{0ex}}-3\phantom{\rule{1em}{0ex}}$$
3. $$\phantom{\rule{0.2em}{0ex}}-2$$

Solution

Draw a number line. Mark $$0$$ in the center and label several units to the left and right.

1. To plot $$3,$$ start at $$0$$ and count three units to the right. Place a point as shown in the figure below.

2. To plot $$-3,$$ start at $$0$$ and count three units to the left. Place a point as shown in the figure below.

3. To plot $$-2,$$ start at $$0$$ and count two units to the left. Place a point as shown in the figure below.