36.3: Comparing Positive and Negative Numbers

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Lesson

Let's compare numbers on the number line.

Exercise \(\PageIndex{1}\): Which One Doesn't Belong: Inequalities

Which inequality doesn't belong?

\(\frac{5}{4}<2\)
\(8.5>0.95\)
\(8.5<7\)
\(10.00<100\)

Exercise \(\PageIndex{2}\): Comparing Temperatures

Here are the low temperatures, in degrees Celsius, for a week in Anchorage, Alaska.

Table \(\PageIndex{1}\)
day	Mon	Tues	Weds	Thurs	Fri	Sat	Sun
temperature	\(5\)	\(-1\)	\(-5.5\)	\(-2\)	\(3\)	\(4\)	\(0\)

Plot the temperatures on a number line. Which day of the week had the lowest low temperature?
The lowest temperature ever recorded in the United States was -62 degrees Celsius, in Prospect Creek Camp, Alaska. The average temperature on Mars is about -55 degrees Celsius.
1. Which is warmer, the coldest temperature recorded in the USA, or the average temperature on Mars? Explain how you know.
2. Write an inequality to show your answer.
On a winter day the low temperature in Anchorage, Alaska, was -21 degrees Celsius and the low temperature in Minneapolis, Minnesota, was -14 degrees Celsius.
Jada said, “I know that 14 is less than 21, so -14 is also less than -21. This means that it was colder in Minneapolis than in Anchorage.”
Do you agree? Explain your reasoning.

Are you ready for more?

Another temperature scale frequently used in science is the Kelvin scale. In this scale, 0 is the lowest possible temperature of anything in the universe, and it is -273.15 degrees in the Celsius scale. Each \(1\text{K}\) is the same as \(1^{\circ}\text{C}\), so \(10\text{K}\) is the same as \(-263.15^{\circ}\text{C}\).

Water boils at \(100^{\circ}\text{C}\). What is this temperature in \(\text{K}\)?
Ammonia boils at \(-35.5^{\circ}\text{C}\). What is the boiling point of ammonia in \(\text{K}\)?
Explain why only positive numbers (and 0) are needed to record temperature in \(\text{K}\).

Exercise \(\PageIndex{3}\): Rational Numbers on a Number Line

Plot the numbers -2, 4, -7, and 10 on the number line. Label each point with its numeric value.

Decide whether each inequality statement is true or false. Be prepared to explain your reasoning.
1. \(-2<4\)
2. \(-2<-7\)
3. \(4>-7\)
4. \(-7>10\)

Drag each point to its proper place on the number line. Use your observations to help answer the questions that follow.

Andre says that \(\frac{1}{4}\) is less than \(-\frac{3}{4}\) because, of the two numbers, \(\frac{1}{4}\) is closer to \(0\). Do you agree? Explain your reasoning.
Answer each question. Be prepared to explain how you know.
1. Which number is greater: \(\frac{1}{4}\) or \(\frac{5}{4}\)?
2. Which number is farther from 0: \(\frac{1}{4}\) or \(\frac{5}{4}\)?
3. Which number is greater: \(-\frac{3}{4}\) or \(\frac{5}{8}\)?
4. Which number is farther from 0: \(-\frac{3}{4}\) or \(\frac{5}{8}\)?
5. Is the number that is farther from 0 always the greater number? Explain your reasoning.

Summary

We use the words greater than and less than to compare numbers on the number line. For example, the numbers -2.7, 0.8, and -1.3, are shown on the number line.

Because -2.7 is to the left of -1.3, we say that -2.7 is less than -1.3. We write:

\(-2.7<-1.3\)

In general, any number that is to the left of a number \(n\) is less than \(n\).

We can see that -1.3 is greater than -2.7 because -1.3 is to the right of -2.7. We write:

\(-1.3>-2.7\)

In general, any number that is to the right of a number is greater than

We can also see that \(0.8>-1.3\) and \(0.8>-2.7\). In general, any positive number is greater than any negative number.

Glossary Entries

Definition: Negative Number

A negative number is a number that is less than zero. On a horizontal number line, negative numbers are usually shown to the left of 0.

Definition: Opposite

Two numbers are opposites if they are the same distance from 0 and on different sides of the number line.

For example, 4 is the opposite of -4, and -4 is the opposite of 4. They are both the same distance from 0. One is negative, and the other is positive.

Definition: Positive Number

A positive number is a number that is greater than zero. On a horizontal number line, positive numbers are usually shown to the right of 0.

Definition: Rational Number

A rational number is a fraction or the opposite of a fraction.

For example, 8 and -8 are rational numbers because they can be written as \(\frac{8}{1}\) and \(-\frac{8}{1}\).

Also, 0.75 and -0.75 are rational numbers because they can be written as \(\frac{75}{100}\) and \(-\frac{75}{100}\).

Definition: Sign

The sign of any number other than 0 is either positive or negative.

For example, the sign of 6 is positive. The sign of -6 is negative. Zero does not have a sign, because it is not positive or negative.

Practice

Exercise \(\PageIndex{4}\)

Decide whether each inequality statement is true or false. Explain your reasoning.

\(-5>2\)
\(3>-8\)
\(-12>-15\)
\(-12.5>-12\)

Exercise \(\PageIndex{5}\)

Here is a true statement: \(-8.7<-8.4\). Select all of the statements that are equivalent to \(-8.7<-8.4\).

-8.7 is further to the right on the number line than -8.4.
-8.7 is further to the left on the number line than -8.4.
-8.7 is less than -8.4.
-8.7 is greater than -8.4.
-8.4 is less than -8.7.
-8.4 is greater than -8.7.

Exercise \(\PageIndex{6}\)

Plot each of the following numbers on the number line. Label each point with its numeric value. \(0.4, -1.5, -1\frac{7}{10}, -\frac{11}{10}\)

(From Unit 7.1.2)

Exercise \(\PageIndex{7}\)

The table shows five states and the lowest point in each state.

Table \(\PageIndex{2}\)
state	lowest elevation (feet)
California	\(-282\)
Colorado	\(3350\)
Louisiana	\(-8\)
New Mexico	\(2842\)
Wyoming	\(3099\)

Put the states in order by their lowest elevation, from least to greatest.

(From Unit 7.1.4)

Exercise \(\PageIndex{8}\)

Each lap around the track is 400 meters.

How many meters does someone run if they run:
2 laps?
5 laps?
\(x\) laps?
If Noah ran 14 laps, how many meters did he run?
If Noah ran 7,600 meters, how many laps did he run?

(From Unit 6.1.6)

Exercise \(\PageIndex{9}\)

A stadium can seat 16,000 people at full capacity.

If there are 13,920 people in the stadium, what percentage of the capacity is filled? Explain or show your reasoning.
What percentage of the capacity is not filled?

(From Unit 3.4.7)