4. Integer Numbers
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Chapter 1
4.
Integer Numbers
This lesson provides a comprehensive guide to understanding integer numbers, focusing on key concepts like absolute value, additive inverse, and the number line. These integers are not just positive numbers; they include negatives and zero as well. The lesson explains how to use the number line to visualize and compare these numbers. Absolute value helps in understanding the 'distance' a number is from zero, while additive inverse is about finding a number that, when added to a given number, results in zero. These concepts have practical applications in various fields, from accounting to engineering, and are fundamental in developing strong mathematical skills.
Show less Show more expand_more| Student Learning Objectives: |
|---|
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| | 13 Theory slides |
| | 9 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Integer Numbers
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Players typically lose and win points based on their actions in video games. The points won can be represented by natural and whole numbers but not the points lost. Integers, however, can represent these losses. This lesson presents attributes and uses of integer numbers used in such situations.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Explore
Measuring Temperatures
Play with the thermometer below to measure different temperatures.
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Discussion
Integers
Integers are whole numbers and negative numbers without fractions or decimal parts. An integer can be positive, negative or zero. The set of all integers is denoted by Z.
Z={...- 2, - 1, 0,1,2...}
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Discussion
Graphing Integer Numbers
Similarly to natural and whole numbers, integer numbers can be displayed on a number line. This process helps to compare numbers and visualize which are greater and smaller.
Concept
Number Line
A number line can display integer numbers by extending the line to the left- and right-hand sides of 0. The positive integers, or natural numbers, are to the right of 0, and the negative integers are to the left of 0.
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Pop Quiz
Integer Numbers on a Number Line
Determine the integer represented by the point on the number line.
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Example
The Nature Scavenger Hunt Adventure
Maya is taking part in a nature scavenger hunt at summer camp. In this game, players are divided into teams; each has a map, a list of items to find, and tasks to complete. There are many ways for teams to either earn or lose points.
| Action or Task | Points |
|---|---|
| Find a hidden item (I) | 2 |
| Cheating (C) | -7 |
| Complete a physical challenge (P) | 4 |
| Solve a puzzle (S) | 7 |
| Take a wrong turn (W) | -3 |
| Run out of time before finishing a task (T) | -5 |
The team that gets the most points will be the winner. Maya wants a better idea of the meaning of these points to help her team win. She thinks that plotting them on a number line should help. Which of the following graphs matches Maya's?
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Hint
Check whether the points given by each action or task are negative or positive. When the number is positive, count that number of units going to the right of 0 on the number line. If the number of points is negative, count that number of units to the left of 0.
Solution
The points assigned to each task or action are integers. Let's mark them on a number line. For example, finding a hidden item (I) earns a team 2 points. We will mark it on the number line by moving two units to the right of 0 to display the points earned.
Next, -7 points are assigned to cheating (C), so the team loses 7 points if they cheat. We should move seven units to the left of 0 to show the points given by this action.
Let's follow a similar method to graph the remaining points.
The final graph matches option B. Now that Maya has a better understanding of the game, she can help her team win this wonderful adventure.
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Example
Recording Temperatures
Maya measures the temperatures of various things around her summer camp. She created a table with the temperatures she recorded.
| Temperatures Measured |
|---|
| -1^(∘) C |
| 9 ^(∘) C |
| -4^(∘) C |
| 3 ^(∘) C |
| 12^(∘) C |
She now wants to know the coldest and warmest temperature she recorded. Help Maya order these temperatures from the coldest to the warmest on a number line.
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Solution
We will use a number line to compare the temperatures. The first recorded temperature is -1^(∘) C. Let's move one unit to the left of zero and draw a point to plot this temperature.
The second temperature recorded is 9^(∘) C. We plot this by moving 9 units to the right of 0.
Now, let's plot the remaining temperatures.
On the final number line, the farthest point to the left is the coldest temperature and the farthest point to the right is the warmest.
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Discussion
Opposite Numbers
Consider the lucky number -7. Or is it lucky number 7? In any case, let's see what happens when 7 is added to -7. -7+7=0 Notice that 7 cancels out -7. This is because 7 is the additive inverse of -7. This property of integer numbers is worth exploring deeper!
Concept
Additive Inverse
The additive inverse of a number is another number such that their sum equals 0. If y is the additive inverse of x, then the following equation holds true.
x+y=0
Given a number, its additive inverse — also called its opposite number — can be found by changing the sign. Some examples of additive inverses are listed in the table.
| Number | Additive Inverse | Sum |
|---|---|---|
| 5 | - 5 | 5+(- 5)=0 |
| - 21 | 21 | - 21+21=0 |
| a | - a | a+(- a)=0 |
| - b | b | - b+b=0 |
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Pop Quiz
Finding the Additive Inverse of Integer Numbers
Find the additive inverse of the indicated number.
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Discussion
Comparing Distances Written as Integer Numbers
We often compare quantities that express distances. Knowing which number represents a farther distance is simple if the numbers are positive. What if they are negative? This is where the absolute value comes to the rescue.
Concept
Absolute Value
The absolute value of a number a is the distance between a and 0 on the number line. It is denoted as |a| and is always a non-negative value.
For any integer number a, these two properties hold true.
| Property | Algebraic Representation | Example |
|---|---|---|
| Non-negativity | |a| ≥ 0 | |7| = 7 and 7≥0 |
| Symmetry | | - a| = |a| | |-7| = 7 and |7| = 7 |-7| = |7| |
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Pop Quiz
Comparison of Numbers
Determine whether the given statement is true or false. Some statements involve absolute values.
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Example
Measuring Elevation
At a particular animal rehabilitation center, the animals have space to roam. Maya takes photos of the animals and notes their elevation compared to her standing position.
| Animal | Elevation (ft) |
|---|---|
| Kingfisher | 10 |
| River otter | -8 |
| Porcupine | 7 |
| Sturgeon | -12 |
| Painted turtle | -3 |
Use Maya's notes to identify which animal is farthest from her standing position in elevation.
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Hint
Use the absolute values to order the elevations of the animals from Maya's position.
Solution
The elevations of the animals contain negative and positive integers. We will use the absolute value of the elevations to get the distances the animals are from Maya.
| Animal | Elevation (ft) |
|---|---|
| Kingfisher | 10 |
| River otter | -8 |
| Porcupine | 7 |
| Sturgeon | -12 |
| Painted turtle | -3 |
The absolute value of a negative number is its additive inverse, while the absolute value of a positive number is equal to itself. Let's move 10 units up from 0 on a vertical number line to display the distance of the kingfisher.
Next, the river otter's elevation is -8 feet. Its absolute value is |-8|=8. That means the river otter's distance is 8 units from 0 in the positive direction.
Use a similar reasoning to identify the remaining distances.
The graph shows that the sturgeon is the farthest from Maya, even though it has a negative elevation! On the other side of the spectrum, the painted turtle is the closest to her.
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Closure
The Bear Sculpture
Maya has $20 in her camp store account. She wants to buy a custom bear sculpture that costs $30. After the purchase, she checks her balance.
Maya's balance is -$10. What does this amount mean? A negative balance in an account usually means that money is owed. The amount owed is the absolute value of the negative balance. Maya's Debt: |-10|=$10 Maya owes $10.
| Balance | Meaning |
|---|---|
| Negative | Debt or money owed |
| 0 | No debt and no credit |
| Positive | Credit |
After Maya adds $10 to her account, the balance is 0. This means the sculpture is fully paid for and no money is owed.
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Integer Numbers
Exercises
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Let's consider the given situation.
A student loses 4 points for not doing her homework.
A loss indicates a negative number. This means we must use a negative integer to represent these 4 points the student lost for not doing her homework. The negative integer for this situation is -4.
Let's look at the given situation.
A football team gains 3 yards.
A gain indicates a number greater than 0. We then need a positive integer to represent this situation. This integer is 3.
Let's now analyze the third situation.
The temperature outside is 6 degrees below zero.
This situation describes a temperature below zero. Below zero means a number less than 0 on a number line. We must represent this situation with a negative integer. This integer is -6.
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Let's begin by looking at the given situation.
Tiffaniqua has a debt of $50.
Debt means that we owe money. This represents a negative situation. Therefore, we represent this situation with a negative number. This integer is -50.
Consider the given situation.
A diver is 20 meters below sea level.
In this case, the sea level represents the starting point of 0. The diver is below sea level, a number less than 0. This means that we need to represent this situation with the integer -20.
Finally, let's analyze the third situation.
A hiker climbs 300 meters up a mountain.
For this situation, the bottom or base of the mountain represents the starting point of 0. Because the hiker is above 0, it represents a positive number. This integer is 300.
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Solution
Write the following numbers from the smallest to the greatest. |-6|, 0, -10, 10, -8, |13|
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We can use a number line to plot the numbers. The smaller numbers will be more left and the greater righter. With this in mind, let's look at the first number in the list. |-6| Recall that the absolute value of a number is the distance between that number and 0. In addition, the absolute value of a negative number is its additive inverse. Because the additive inverse of -6 is 6, let's plot |-6| over the number 6.
The following number in the list is 0. Since we do not move any unit to the right or left of 0, we plot a point over the number 0 to represent this number.
We can now follow a similar process to graph the remaining points on the number line.
The numbers are now ordered on the number line. The more left is -10, meaning it is the smallest number in the list. In contrast, |13| is the rightest, meaning it is the greatest. Let's write all the given numbers from the smallest to the greatest. The Numbers From the Smallest to the Greatest -10, -8, 0, |-6|, 10, |13|
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Solution
The table contains the average winter temperature of some states of the US.
| State | Average Winter Temperature |
|---|---|
| Pennsylvania, PA | -2^(∘) C |
| Georgia, GA | 9^(∘) C |
| Tennessee, TN | 4^(∘) C |
| New York, NY | -5^(∘) C |
| Maine, ME | -8^(∘) C |
Which option contains the states ordered from the lowest average winter temperature to the highest?
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Let's begin by graphing the temperatures on a number line. Consider that the smallest temperatures will be more left and the higher will be further right. The first temperature is for Pennsylvania, PA. This place has an average winter temperature of -2^(∘) C. Let's draw a point on -2 on the number line.
Georgia, GA, has an average winter temperature of 9^(∘) C. Let's plot a point on the number 9 to represent this temperature.
We can now plot the remaining temperatures following a similar fashion.
Now that we graphed the temperatures, we can see that Maine has the lowest average winter temperature. At the same time, Georgia is the warmest place. Let's write the places according to their temperature in ascending order. Maine, New York, Pennsylvania, Tennessee, Georgia
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Solution
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Let's look at the given expression. |35| We are asked to find the absolute value of 35, which is a positive integer. The absolute value of a positive integer is equal to itself. This means that the absolute value of 35 is 35. |35|=35
Consider the given expression.
|-13|
This expression asks for the absolute value of -13. The absolute value of a negative integer is its additive inverse. Change the sign of a number to find its additive inverse. The additive inverse of -13 is then 13. We can now write the absolute value of -13.
|-13|=13
This expression asks for the absolute value of -7 and then applies to it a negative sign.
-|-7|
We first calculate the absolute value. The opposite of -7 is 7. This means that |-7|=7. Next, this result is transformed to negative, representing the value that simplifies the given expression.
-|-7|=-7
Finally, let's look at the last expression.
-|15|
Again, we first calculate the absolute value. 15 is a positive number, which means its absolute value is itself. The negative sign outside the absolute value transforms it into a negative. We can then write the value to this expression.
-|15|=-15
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Integer Numbers
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