The Set of Integers
On a January morning in Chicago, the thermometer reads $-4\ ^\circ\text{F}$. You owe your friend $\$15$ for pizza. A research submarine is hovering $85$ feet below the surface. None of these can be written with just $0, 1, 2, 3, \ldots$ You need integers, numbers that can also say "below zero."
What Is an Integer?
A positive number is often written with a plus sign ($+5$), but $+5$ and $5$ are the same number.
Many situations come in opposite pairs. Zero is the dividing line between them.
| Positive ($+$) | Negative ($-$) |
|---|---|
| gain, earn, deposit | lose, owe, withdraw |
| above sea level, floor 3 | below sea level, basement 2 |
| above $0\ ^\circ\text{F}$ | below $0\ ^\circ\text{F}$ |
| go up, forward, right | go down, backward, left |
The numbers $1, 2, 3, \ldots$ are the natural numbers, written $\mathbb{N}$ (also called the counting numbers). Add $0$ and you get the whole numbers: $0, 1, 2, 3, \ldots$ Every natural number and every whole number is an integer, but not every integer is a whole number. The negative integers are not whole numbers and not natural numbers.
To turn a situation into an integer, decide which direction is positive. Then use the size of the change as the number and the direction as the sign. Losing $\$8$ is $-8$, and finding $\$8$ is $+8$.
Express the following real-life situations related to social media and the daily life of a teenager with an integer ($+$ or $-$): a) You have gained $150$ new followers…
Express the following real-life situations related to social media and the daily life of a teenager with an integer ($+$ or $-$):
- a) You have gained $150$ new followers on your TikTok account this week.
- b) A friend owes you $\$12$ for the birthday present you bought together.
- c) The street thermometer reads $4$ degrees below zero.
- d) You have gone down to the second basement of the shopping center to leave your bicycle.
Given the following list of numbers, classify them by indicating which belong to the set of natural numbers ($\mathbb{N}$) and which are integers ($\mathbb{Z}$) but not…
Given the following list of numbers, classify them by indicating which belong to the set of natural numbers ($\mathbb{N}$) and which are integers ($\mathbb{Z}$) but not natural numbers:
$$ -7, \quad +5, \quad -12, \quad 0, \quad +10, \quad -1, \quad 45 $$Write down the integer that represents the final state of a young person's bank account in these two independent situations: a) They had $\$25…
Write down the integer that represents the final state of a young person's bank account in these two independent situations:
- a) They had $\$25$ saved and bought a video game that cost $\$30$ (the bank allows them to go into negative balance).
- b) They owed their sister $\$10$ and their grandfather gave them a $\$10$ bill to settle the debt.
The Number Line
A number line puts integers in order. Positive numbers go to the right of $0$ and negative numbers go to the left. A vertical number line works the same way: up is positive and down is negative.
To read a point, start at $0$ and count the marks. Each mark here is $1$ unit.
Look at the following number line and identify which integer corresponds to each of the points marked with the letters $A$, $B$ and $C$.
Look at the following number line and identify which integer corresponds to each of the points marked with the letters $A$, $B$ and $C$.
Look at the following number line where four points have been marked: $A$, $B$, $C$ and $D$. Identify the integer that corresponds to each letter knowing that point $C$…
Look at the following number line where four points have been marked: $A$, $B$, $C$ and $D$. Identify the integer that corresponds to each letter knowing that point $C$ represents the number $-2$ and that the distance between each small mark is one unit.
Opposites and Absolute Value
Opposites always have the same absolute value. The opposite of $0$ is $0$.
Dates are integers too. We write years AD as positive and years BC as negative, so $300$ BC is $-300$ and AD $1776$ is $+1776$. In this section's problems, $0$ stands for the dividing point of the timeline.
A drone is flying at an altitude of $+35$ feet above sea level, while a diver is exploring a coral reef at a depth of $-15$ feet. a) Write down the absolute value of…
A drone is flying at an altitude of $+35$ feet above sea level, while a diver is exploring a coral reef at a depth of $-15$ feet.
- a) Write down the absolute value of both numbers: $|+35|$ and $|-15|$.
- b) Which of the two is at a greater distance from sea level ($0$ feet)?
- c) Write down the opposite of the diver's position.
In a large shopping mall, the elevator buttons represent the floors. The ground floor is $0$, the upper floors are positive ($+1, +2...$) and the parking garage…
In a large shopping mall, the elevator buttons represent the floors. The ground floor is $0$, the upper floors are positive ($+1, +2...$) and the parking garage basements are negative ($-1, -2...$).
- a) If you are on floor $-3$ and go up to the opposite floor, which floor do you arrive at?
- b) What is the absolute value of the number of floor $-2$?
- c) Which floor is further from the street (floor $0$): $+4$ or $-5$?
An archaeologist finds a coin at an excavation site. Upon investigation, they discover that it was minted in the year $45$ BC. If the archaeologist lives in the year…
An archaeologist finds a coin at an excavation site. Upon investigation, they discover that it was minted in the year $45$ BC. If the archaeologist lives in the year $2024$ AD, answer:
- a) What integer would you assign to each date?
- b) What is the opposite of the year the coin was minted?
- c) Represent both years approximately on a timeline.
Comparing and Ordering Integers
On a horizontal number line, numbers get larger as you move right. So the number farther to the right is always greater.
The symbol $<$ means "is less than" and $>$ means "is greater than." The small end points at the smaller number.
At a ski resort, the following minimum temperatures have been recorded over four days. Order them from lowest to highest using the $<$ symbol: Monday: $-6$ $^\circ C$…
At a ski resort, the following minimum temperatures have been recorded over four days. Order them from lowest to highest using the $<$ symbol:
-
Monday: $-6$ $^\circ C$
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Tuesday: $+2$ $^\circ C$
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Wednesday: $-9$ $^\circ C$
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Thursday: $0$ $^\circ C$
In the physics laboratory, the freezing temperatures of various liquids are being analyzed. The results are: Liquid P: $-8$ ºF Liquid Q: $-15$ ºF Liquid R: $-3$ ºF…
In the physics laboratory, the freezing temperatures of various liquids are being analyzed. The results are:
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Liquid P: $-8$ ºF
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Liquid Q: $-15$ ºF
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Liquid R: $-3$ ºF
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Liquid S: $-12$ ºF
Order the liquids from lowest to highest temperature using the $<$ symbol and explain which of them has the greatest absolute value.
Compare the following pairs of integers using the $>$ or $<$ signs and justify your answer based on their position on the number line: a) $-15$ \\\\\_ $-20$ b) $+3$…
Compare the following pairs of integers using the $>$ or $<$ signs and justify your answer based on their position on the number line:
- a) $-15$ _____ $-20$
- b) $+3$ _____ $-100$
- c) $0$ _____ $-5$
- d) $op(+7)$ _____ $-6$
Solving Number Clues
Some problems give clues about a number instead of naming it. Turn each clue into a picture on the number line.
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"$|x| = 7$" means $x$ is $7$ units from $0$, so $x = 7$ or $x = -7$.
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"$|x| < 4$" means $x$ is less than $4$ units from $0$, so $x$ is $-3, -2, -1, 0, 1, 2$, or $3$.
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"The opposite of $x$ is negative" means $x$ itself is positive.
Open circles show numbers that are not included (the clue says "less than," not "less than or equal to"). A strategy that works well:
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List every integer that fits the first clue.
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Cross out the ones that break the next clue.
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Test the remaining candidates in any clue that is hard to picture, such as one that adds $x$ and $|x|$.
Write down all the integers $x$ that simultaneously satisfy these two conditions: 1. Their absolute value is less than $6$ ($|x| < 6$). 2. It is a number strictly less…
Write down all the integers $x$ that simultaneously satisfy these two conditions:
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Their absolute value is less than $6$ ($|x| < 6$).
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It is a number strictly less than $-2$ ($x < -2$).
Represent the solution on a number line.
Find all the integers $x$ that simultaneously satisfy the following three conditions: 1. Their absolute value is greater than $4$ but less than $10$. 2. The opposite of…
Find all the integers $x$ that simultaneously satisfy the following three conditions:
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Their absolute value is greater than $4$ but less than $10$.
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The opposite of $x$ is a negative number.
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If you add its absolute value to $x$, the result is less than $16$.
Moving Along the Number Line
Many problems tell a story of ups and downs. Keep a running position: start where the story starts and update after each move.
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Up, forward, right, gain: add a positive amount.
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Down, backward, left, loss: subtract, which moves you toward smaller numbers.
You can also group the moves. Add all the "up" amounts, add all the "down" amounts, and subtract. This gives the net change.
Some stories use words that change the next move:
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"Twice as long, in the opposite direction" means multiply the size by $2$ and switch the sign.
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"The opposite of the square you are on" means jump to the number on the other side of $0$.
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"Double the absolute value of $-3$" means $2 \times 3 = 6$ units, in whatever direction the story says.
An underwater exploration drone performs the following maneuvers starting from sea level (which we represent as the value $0$): It ascends $12$ feet to calibrate its…
An underwater exploration drone performs the following maneuvers starting from sea level (which we represent as the value $0$):
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It ascends $12$ feet to calibrate its signal.
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It descends $25$ feet to take coral samples.
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It goes up $8$ feet to avoid a strong current.
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It goes down another $10$ feet to enter an underwater cave.
Indicate the final position of the drone with respect to sea level using integers and draw a vertical diagram in your notebook to represent these positions.
In a board game, your counter is on the square that represents the integer $-4$. You roll a special die and get the following consecutive moves: 1. You move forward $6$…
In a board game, your counter is on the square that represents the integer $-4$. You roll a special die and get the following consecutive moves:
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You move forward $6$ spaces.
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You move backward double what the absolute value of $-3$ indicates.
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You move to the opposite of the square you are on at that moment.
Which square do you finish the game on? Indicate all the intermediate steps.
We are designing a video game where a character must jump on platforms numbered with integers. The character starts on platform $0$. The first jump is $+5$ units. The…
We are designing a video game where a character must jump on platforms numbered with integers. The character starts on platform $0$.
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The first jump is $+5$ units.
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The second jump is twice as long as the first, but in the opposite direction.
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The third jump takes them to a platform whose number is the opposite of the number of the platform they were on after the second jump.
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The fourth jump is $-12$ units.
Which platform are they on at the end? Draw the path on the number line indicating each step.
Distance and the Middle Point
The distance between two integers is the number of units between them on the number line.
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On opposite sides of $0$, add the absolute values: the distance between $-12$ and $20$ is $12 + 20 = 32$.
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On the same side of $0$, subtract the smaller absolute value from the greater: the distance between $-18$ and $-30$ is $30 - 18 = 12$.
A clue like "$b$ is $3$ units from $a$" gives two possible numbers: $3$ to the right or $3$ to the left. A second clue such as "$b < a$" picks the left one.
A number that is equally far from two others is the middle point. Find the distance between them, take half, and move that far from either end.
Two divers, Emma and Daniel, are in the ocean. Emma is at a depth of $18$ feet below sea level. Daniel is $12$ feet from Emma, but he is in a deeper position. a) At what…
Two divers, Emma and Daniel, are in the ocean. Emma is at a depth of $18$ feet below sea level. Daniel is $12$ feet from Emma, but he is in a deeper position.
- a) At what depth is Daniel? Express this as an integer.
- b) If Emma goes up $5$ feet, what will her new position be?
- c) What is the absolute distance between Emma's new position and Daniel's position?
Three cities have very curious extreme temperatures on a winter's day: City A has a temperature that is the opposite of $6$. City B has a temperature such that its…
Three cities have very curious extreme temperatures on a winter's day:
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City A has a temperature that is the opposite of $6$.
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City B has a temperature such that its absolute value is $4$, and it is warmer than city A.
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City C has a temperature that is exactly the same distance from the temperature of A as it is from the temperature of B.
Calculate the temperature of the three cities and order them from lowest to highest using the $<$ symbol.
On a scientific expedition, an exploration drone and a mini-submarine set off from the deck of a ship that is exactly at sea level (altitude $0$ feet). 1. The drone…
On a scientific expedition, an exploration drone and a mini-submarine set off from the deck of a ship that is exactly at sea level (altitude $0$ feet).
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The drone ascends $145$ feet, then descends $60$ feet and finally goes back up $25$ feet.
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The submarine descends $85$ feet, goes up $30$ feet and goes back down until it reaches a depth that doubles its previous distance from sea level.
If at that moment the drone drops a probe that travels through the air and water until it reaches the submarine, what total distance does the probe travel? Represent the situation on a vertical axis.
<line x1="0" y1="310" x2="400" y2="310"/>
<line x1="0" y1="370" x2="400" y2="370"/>
Working Backwards
Sometimes you know where the story ends and need to find where it began. Undo each step in reverse order: add instead of subtract, divide instead of multiply.
Timelines are chains of moves too. Add the years someone lived to their birth year to get the year they died.
A historian is analyzing the life of an ancient philosopher. The data they have is as follows: The philosopher was born in the year $42$ BC (which we represent as…
A historian is analyzing the life of an ancient philosopher. The data they have is as follows:
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The philosopher was born in the year $42$ BC (which we represent as $-42$).
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His son was born when the philosopher was $31$ years old.
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The philosopher lived $12$ years longer than his son.
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The son died in the year $28$ AD (which we represent as $+28$).
How many years did the philosopher live? In what year did he die? Is it possible that the philosopher knew his grandson if he was born the year the son died? Justify your answer using integers.
A bank account has had the following movements over the last five days: Monday: Deposit of an unknown amount of money $X$. Tuesday: Payment of a bill of $\$120…
A bank account has had the following movements over the last five days:
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Monday: Deposit of an unknown amount of money $X$.
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Tuesday: Payment of a bill of $\$120$.
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Wednesday: Collection of an invoice of $\$200$.
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Thursday: The balance tripled compared to the previous day (Wednesday).
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Friday: $\$150$ was withdrawn and the final balance was exactly $\$240$.
What was the account balance on Monday before the movements started? Set up the situation using a chain of operations with integers.
Integers in the Coordinate Plane
Two number lines, one horizontal (the x-axis) and one vertical (the y-axis), cross at $0$. Together they make the coordinate plane. Every point has an ordered pair $(x, y)$.
Moving a point changes only one coordinate at a time. Moving right or left changes $x$. Moving up or down changes $y$.
To find the other corners of a rectangle with sides parallel to the axes, mix the coordinates of the two given opposite corners. To find its center, find the halfway point on each axis.
Identify the Cartesian coordinates $(x, y)$ of the points $P$ and $Q$ represented on the following grid. Remember that the first number is the horizontal axis (x-axis)…
Identify the Cartesian coordinates $(x, y)$ of the points $P$ and $Q$ represented on the following grid. Remember that the first number is the horizontal axis (x-axis) and the second is the vertical axis (y-axis).
On a Cartesian plane, a square has two of its opposite vertices at the points $A(-2, 3)$ and $C(4, -3)$. 1. Draw the axes and plot the points $A$ and $C$. 2. Determine…
On a Cartesian plane, a square has two of its opposite vertices at the points $A(-2, 3)$ and $C(4, -3)$.
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Draw the axes and plot the points $A$ and $C$.
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Determine the coordinates of the other two vertices, $B$ and $D$, knowing that the square is not rotated (its sides are parallel to the axes).
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What are the coordinates of the central point where the diagonals of the square intersect?
Back to the start: A thermometer at $-4\ ^\circ\text{F}$, a debt of $-\$15$, and a submarine at $-85$ feet are all negative integers. Each one measures a value below a reference point: $0\ ^\circ\text{F}$, a balance of $\$0$, and sea level.