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Players typically lose and win points based on their actions in video games. The wins can be represented by natural and whole numbers but not losses. Integers, however, can represent 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.

Natural Numbers
Whole Numbers
Number Line

Explore

Measuring Temperatures

Look at the thermometer and play with it to measure different temperatures.

How can temperatures below zero be expressed?

Discussion

Integers

Integers are whole numbers and negative numbers without fractions or decimal parts. The following are some examples of integers and non-integers.

Integers	Non-integers
$17,$ $0,$ $- 1,$ $121,$ $- 67$	$π,$ $- \frac{3}{4},$ $1.5,$ $- 12.9,$ $\frac{1}{2}$

An integer can be positive, negative or zero. The set of all integers is denoted by $Z .$

$Z = {\dots - 2, - 1, 0, 1, 2 \dots}$

The dots indicate that the set continues to increase or decrease by one unit at a time indefinitely. An example of the use of integers is to represent temperatures below zero. One of the lowest temperatures recorded on earth was about

- 8 9^{\circ} C .

Discussion

Graphing Integer Numbers

Similar to natural and whole numbers, integer numbers can be displayed on a number line. This helps to compare numbers and visualize which are greater and smaller.

Concept

Number Line

A number line can display integer numbers by extending it on the left- and right-hand side of zero. The positive integers or natural numbers will be on the right-hand side, and the negative integers will be on the left-hand side of zero.

Recall the thermometer shown previously. A thermometer is also a number line. It can be extended down to measure positive and negative temperatures.

Pop Quiz

Integer Numbers on a Number Line

Determine the integer represented by the point on the number line.

Random Generator of Integer Numbers Numbers

Example

The Nature Scavenger Hunt Adventure

Maya is attending a teen-based summer camp. She is now taking part in a nature scavenger hunt. In this game, players are divided into teams; each has a map, a list of items to find, and tasks to complete. Each action, not only completed task, is worth 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 more 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?

Hint

Identify if 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. On the other hand, if the number of points is negative, count that number of units to the left of $0 .$

Solution

The points for each task or action are integers. Display each on a number line and label it. Consider that finding a hidden item (I) gives

2

points. That requires moving two units to the right of

0

to display the points earned.

Next, cheating (C) is represented by

- 7 .

This means that the team losses

7

points if they cheat. Move seven units to the left of zero to represent the points given by this action.

Follow a similar method to graph the remaining points.

This graph matches option B. Now that Maya has a better understanding of the game, she can help her team to win this wonderful adventure.

Example

Recording Temperatures Just Like Grandma

Maya is loving the summer camp. The staff even gave her what they call an explorer's box. One of the items in it is a thermometer. Her grandma used one just like it! Maya is excited to record temperatures all week wherever they go.

She created a table with the temperatures she recorded.

Temperatures Measured
$- 1^{\circ} C$
$9^{\circ} C$
$- 4^{\circ} C$
$3^{\circ} C$
$1 2^{\circ} C$

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.

Hint

On a number line, the smaller numbers are further left and the greater ones are further right. Plot one temperature at a time as a point on a number line. A negative temperature is on the left-hand side of $0 .$ A positive temperature is on the right-hand side of $0 .$

Solution

A number line helps to compare numbers. The smaller numbers are further left and the greater ones are further right. Now, because the list contains positive and negative integers, the number line must contain $0 .$

The first recorded temperature is $- 1^{\circ} C .$ Move one unit to the left of zero and draw a point over the number line to draw this temperature.

The second recorded temperature is $9^{\circ} C .$ Plot this temperature by following a similar method. In this case, move $9$ units to the right-hand side of $0 .$

The graph shows that the first temperature Maya recorded is less than the second. Now, plot the remaining temperatures following a similar reasoning.

The number line shows that the coldest recorded temperature was

- 4^{\circ} C

and the warmest was

1 2^{\circ} C .

The temperatures can now be written from the coldest to the warmest.

- 4^{\circ} C, - 1^{\circ} C, 3^{\circ} C, 9^{\circ} C, 1 2^{\circ} C

Discussion

Opposite Numbers

Consider the lucky number

- 7 .

Well, perhaps the sign needs to change to get

7

for it to be lucky. Moving on, what happens when

7

is added to

- 7 .

- 7 + 7 = 0

Notice that

7

cancels

- 7 .

The reason for that is

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 opposite number — can be found by changing the sign. Some examples of additive inverses are listed in the table.

Number	Additive Inverse	Sum
$5$	$- 5$	$5 + (- 5) = 0$
$- 21$	$21$	$- 21 + 21 = 0$
$0$	$- 0$	$0 + (- 0) = 0$
$a$	$- a$	$a + (- a) = 0$
$- b$	$b$	$- b + b = 0$

Pop Quiz

Finding the Additive Inverse of Integer Numbers

The additive inverse of a number is another number such that the sum of these two numbers equals $0 .$ Use this information to find the additive inverse of the indicated number.

An applet asking for the additive inverse of random integer numbers.

Discussion

Comparing Distances Written as Integer Numbers

It is common to compare quantities that express distances. They help identify which quantity is further than another. Knowing which number represents a further distance is simple if the numbers are positive. What if they are negative? No fears — 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 it is always a non-negative value.

Interactive number line illustrating the concept of absolute value

The absolute value of a negative number is its opposite value, while the absolute value of a positive number is equal to itself.

∣ a ∣ = {a - a if a \geq 0 if a < 0

For any integer number $a,$ these two properties hold true.

Property	Algebraic Representation	Example
Non-negativity	$∣ a ∣ \geq 0$	$∣ 7 ∣ = 7$ and $7 \geq 0$
Symmetry	$∣ - a ∣ = ∣ a ∣$	$∣ - 7 ∣ = 7$ and $∣ 7 ∣ = 7 ∣ - 7 ∣ = ∣ 7 ∣$

The non-negative property says that the absolute value of any integer must always be greater than or equal than 0. In addition, the symmetry property means that the absolute value of a number and the absolute value of its opposite are the same.

Pop Quiz

Comparison of Numbers

Time to identify and compare numbers. See the given statement and evaluate if it is true or false. Some statements involve absolute values.

Example

Measuring Animals' Elevation

Near the camp is an animal rehabilitation center. There, the animals have space to roam. Maya gets the chance to visit! She takes photos of the animals and notes their elevation compared to her standing position.

Animal	Elevation (In Feet)
Kingfisher	$10$ feet
River otter	$- 8$ feet
Porcupine	$7$ feet
Sturgeon	$- 12$ feet
Painted turtle	$- 3$ feet

Use Maya's notes to identify which animal is furthest from her standing position according to elevation.

What is the animal closest to Maya's standing position?

Hint

Use the absolute values to order the elevations of the animals from Maya's position.

Solution

The elevations contain negative and positive integers. Apply the absolute value of the elevations to get the distances the animals are from Maya's position.

Animal	Elevation (In Feet)
Kingfisher	$10$ feet
River otter	$- 8$ feet
Porcupine	$7$ feet
Sturgeon	$- 12$ feet
Painted turtle	$- 3$ feet

The absolute value of a negative number is its additive inverse. In contrast, the absolute value of a positive number is equal to itself. Now, move $10$ units up from $0$ in 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 distance of the sturgeon is the furthest. That is even with a negative elevation! On the other side of the spectrum, the painted turtle is the closest.

Closure

The Bear Sculpture

Many campers trust Maya's math skills and entrust their money to her. Vincenzo, has already given her $$ 20 .$ He then goes to the gift shop and orders a custom bear sculpture. Before paying, he runs to Maya to retrieve his $$ 20 .$ Maya sees the ticket and notices that the bear sculpture costs $$ 30 .$ She shows his total balance to Vincenzo.

Vicenzo's balance after buying the present

What does this amount represent? The negative amount in Vincenzo's balance means he has a debt or money owed. The amount he is in debt is given by the absolute value of the total.

Total Vincenzo ’ s Debt : ∣ - 10 ∣ = $ 10

What if Vincenzo's balance has a positive amount? That means he would still have cash and credit to buy more items. What if the balance is at

0 ?

That means Vincenzo has no debt and no money.

Balance	Meaning
Negative	Debt or money owed
$0$	No debt and no credit
Positive	Cash and credit

Vincenzo needs to pay that $$ 10$ because the bear sculpture costs more than he actually has. Scouring for loose change, he finds $$ 10$ in a secret pocket in his backpack. That bear sculpture is all his!

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Catch-Up and Review

Hint

Solution

Hint

Solution

Hint

Solution