Lesson 2
Comparing Positive and Negative Numbers
2.1: Which One Doesn’t Belong: Inequalities
Which inequality doesn’t belong?
 \(\frac{5}{4} < 2\)
 \(8.5 > 0.95\)
 \(8.5 < 7\)
 \(10.00 < 100\)
2.2: Comparing Temperatures
Here are the low temperatures, in degrees Celsius, for a week in Anchorage, Alaska.
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.
 Which is warmer, the coldest temperature recorded in the USA, or the average temperature on Mars? Explain how you know.
 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.
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 \(\text263.15^\circ\text{C}\).
 Water boils at \(100^\circ \text{C}\). What is this temperature in \(\text{K}\)?
 Ammonia boils at \(\text35.5^\circ \text{C}\). What is the boiling point of ammonia in \(\text{K}\)?
 Explain why only positive numbers (and 0) are needed to record temperature in \(\text{K}\).
2.3: 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.
 \(\text2 < 4\)
 \(\text2 < \text7\)
 \(4 > \text7\)
 \(\text7 > 10\)
Drag each point to its proper place on the number line. Use your observations to help answer the questions that follow.

Andre says that \(\frac14\) is less than \(\text{}\frac {3}{4}\) because, of the two numbers, \(\frac14\) is closer to 0. Do you agree? Explain your reasoning.
 Answer each question. Be prepared to explain how you know.

Which number is greater: \(\frac14\) or \(\frac54\)?

Which number is farther from 0: \(\frac14\) or \(\frac54\)?

Which number is greater: \(\text{}\frac {3}{4}\) or \(\frac58\)?

Which number is farther from 0: \(\text{}\frac {3}{4}\) or \(\frac58\)?

Is the number that is farther from 0 always the greater number? Explain your reasoning.

Summary
Here is a number line labeled with positive and negative numbers. The number 4 is positive, so its location is 4 units to the right of 0 on the number line. The number 1.1 is negative, so its location is 1.1 units to the left of 0 on the number line.
We say that the opposite of 8.3 is 8.3, and that the opposite of \(\frac {\text{}3}{2}\) is \(\frac32\). Any pair of numbers that are equally far from 0 are called opposites.
Points \(A\) and \(B\) are opposites because they are both 2.5 units away from 0, even though \(A\) is to the left of 0 and \(B\) is to the right of 0.
A positive number has a negative number for its opposite. A negative number has a positive number for its opposite. The opposite of 0 is itself.
You have worked with positive numbers for many years. All of the positive numbers you have seen—whole and nonwhole numbers—can be thought of as fractions and can be located on a the number line.
To locate a nonwhole number on a number line, we can divide the distance between two whole numbers into fractional parts and then count the number of parts. For example, 2.7 can be written as \(2 \frac {7}{10}\). The segment between 2 and 3 can be partitioned into 10 equal parts or 10 tenths. From 2, we can count 7 of the tenths to locate 2.7 on the number line.
All of the fractions and their opposites are what we call rational numbers. For example, 4, 1.1, 8.3, 8.3, \(\frac {\text{}3}{2}\), and \(\frac32\) are all rational numbers.
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:
\(\displaystyle \text2.7 <\text 1.3\)
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:
\(\displaystyle \text1.3 >\text 2.7\)
In general, any number that is to the right of a number \(n\) is greater than \(n\)
We can also see that \(0.8 > \text1.3\) and \(0.8 > \text2.7\). In general, any positive number is greater than any negative number.
Glossary Entries
 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.
 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.
 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.
 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 \(\frac81\) and \(\text\frac81\).
Also, 0.75 and 0.75 are rational numbers because they can be written as \(\frac{75}{100}\) and \(\text\frac{75}{100}\).
 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.