The Set of Integers

In everyday life we use integers to express opposite situations. Associate an integer (positive or negative) with each of the following situations:

1. a) The outdoor thermometer reads $4$ degrees below zero.
2. b) You owe $900$ pesos to a friend.
3. c) A diver is at a depth of $20$ meters.
4. d) An airplane flies at an altitude of $8,500$ meters.
5. e) You have won $50$ points in a level of your favorite video game.
6. f) The shopping center parking lot is on the third basement floor.

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:

$$ -7, \quad +5, \quad -12, \quad 0, \quad +10, \quad -1, \quad 45 $$

An archaeologist finds a coin at an excavation site. While researching it, he discovers that it was minted in the year $45$ BC. If the archaeologist lives in the year $2024$ AD, answer:

1. a) What integer would you assign to each date?

1. b) What is the opposite of the year the coin was minted?

1. c) Represent approximately, on a timeline that you will draw in your notebook, the year the coin was minted and the year in which the archaeologist lives. Mark the year $0$ and indicate which side corresponds to the years BC and which to the years AD.

The concept of modulus represents the distance of a number from zero. Calculate the following absolute values:

1. a) $|-7| =$
2. b) $|+12| =$
3. c) $|-25| =$
4. d) $|0| =$
5. e) $|-1| =$
6. f) $|+100| =$

Compare the following pairs of integers using the symbols greater than ($>$) or less than ($<$):

1. a) $+5$ ____ $-8$
2. b) $-3$ ____ $-1$
3. c) $0$ ____ $-5$
4. d) $-12$ ____ $-20$
5. e) $+10$ ____ $+2$
6. f) $-1$ ____ $0$

The opposite of an integer is the number that has the same modulus but a different sign. Write down the opposite of the following numbers:

1. a) $op(+6) =$
2. b) $op(-9) =$
3. c) $op(-15) =$
4. d) $op(+21) =$
5. e) $op(0) =$
6. f) $op(-1) =$

Reason whether the following statements are true or false, providing an example or counterexample in each case:

1. a) If $a$ and $b$ are two negative integers and $|a| < |b|$, then $a > b$.
2. b) The modulus of an integer is always greater than the number itself.
3. c) The distance between any integer and its opposite is always an even number.
4. d) If we add the modulus of a number to its opposite, the result is always zero.

Order the following integers from smallest to largest, using the symbol $<$ between them:

$$-5, \quad +3, \quad -1, \quad 0, \quad +7, \quad -8, \quad -2$$

At a ski resort, the following minimum temperatures were recorded over four days. Order them from lowest to highest using the symbol $<$:

• Monday: $-6$ $^\circ C$

• Tuesday: $+2$ $^\circ C$

• Wednesday: $-9$ $^\circ C$

• Thursday: $0$ $^\circ C$

In the physics laboratory, the freezing temperatures of several liquids are being analysed. The results are:

• Liquid P: $-8$ ºC

• Liquid Q: $-15$ ºC

• Liquid R: $-3$ ºC

• Liquid S: $-12$ ºC

Order the liquids from lowest to highest temperature using the symbol $<$ and explain which of them has the largest modulus.

Order the following mathematical elements from smallest to largest, having first calculated their numerical value:

$$-| -12 |, \quad op(+5), \quad | -8 |, \quad op(op(-3)), \quad -(-10), \quad 0, \quad -| -1 |$$

A temperature sensor in a cryogenics laboratory initially reads $12\,^{\circ}\text{C}$ (look at the scale). Because of an experiment, the temperature drops $18\,^{\circ}\text{C}$, then rises $5\,^{\circ}\text{C}$ and finally drops again $3\,^{\circ}\text{C}$.

1. a) Represent each of the temperature changes using integers.

1. b) What is the final temperature recorded by the sensor?

1. c) Is the final temperature above or below $0\,^{\circ}\text{C}$?

Two divers, Elena and Carlos, are in the sea. Elena is at a depth of $18$ meters below sea level. Carlos is $12$ meters from Elena, but he is in a deeper position.

1. a) At what depth is Carlos? Express it as an integer.
2. b) If Elena goes up $5$ meters, what will her new position be?
3. c) What is the absolute distance between Elena's new position and Carlos's position?

A delivery driver gets into the elevator of a shopping center on the ground floor ($0$). First, he goes up to the storeroom on floor $+5$, then goes down $8$ floors to collect goods in the parking lot, and finally goes up $2$ floors to make a delivery.

1. a) Which floor is the parking lot on? Express the result as an integer.
2. b) On which floor does he make the final delivery?
3. c) How many floors in total has the delivery driver traveled throughout his route? (Hint: use the concept of absolute value).

In a board game, your counter is on the square representing the integer $-4$. You roll a special die and get the following consecutive moves:

1. Move forward $6$ positions.

2. Move back twice what the modulus of $-3$ indicates.

3. Move to the opposite of the square you are currently on.

Which square do you end up on at the end of the game? Show all intermediate steps.

Two points, $A$ and $B$, have been marked on a number line. Knowing that the distance between point $A$ and point $B$ is $15$ units and that point $A$ corresponds to the integer $-6$:

1. a) Determine which integer point $B$ represents if it is to the right of $A$.
2. b) Calculate the value of $|A| + |B|$.
3. c) If we place a new point $C$ right in the middle of the segment joining $A$ and $B$, what number would $C$ represent?

An archaeologist is studying an ancient dynasty. He knows that the first king of the dynasty was born in $42$ BC and died at $65$ years of age. His son, the heir, was born when the father was $28$ years old and reigned for $35$ years after his father's death.

1. a) Use integers to represent the years of birth and death of both figures.
2. b) In which year did the son's reign end? Express the result in chronological format (BC or AD).

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 second jump is twice as long as the first, but in the opposite direction.

• The third jump takes it to a platform whose number is the opposite of the number of the platform it was on after the second jump.

• The fourth jump is $-12$ units.

Which platform is it on at the end? Draw the path on the number line showing each step.

You can do it on paper and send a photo with your answer.

Write down the integers that simultaneously satisfy the following three conditions:

1. Their modulus is greater than $2$ and less than $6$.

2. It is a negative number.

3. When you add its opposite to it, the result is $0$.

Represent the numbers found on a number line.

Imagine two integers, $a$ and $b$. We know that:

1. The distance between $a$ and $0$ is $7$ units.

2. The distance between $a$ and $b$ is $3$ units.

3. $b$ is less than $a$.

Determine all possible values for $a$ and for $b$. For each case, write down the modulus of the sum $|a + b|$.

Identify the Cartesian coordinates $(x, y)$ of the points $P$ and $Q$ represented on the following grid. Remember that the first number is for the horizontal axis (x-axis) and the second for the vertical axis (y-axis).

In a strategy game, a pirate ship moves around a map with coordinates. Its starting position is the point $P(-3, 2)$.

1. a) If the ship moves $5$ units to the right, what is its new abscissa coordinate ($x$)?

1. b) If from that new position it moves down $4$ units, what is its ordinate coordinate ($y$)?

1. c) In your notebook, plot the starting point and the final point (the ship's position after the two movements) on a coordinate axis system.

In the following Cartesian plane, a robot starts at point $P(-2, 3)$. It makes three consecutive moves:

1. It moves $5$ units to the right.

2. It moves $7$ units down.

3. It moves $4$ units to the left.

In which quadrant does the robot end up? Do this in your notebook. State the final coordinates of the point and calculate the modulus of the sum of its new x-coordinate ($x$) and its new y-coordinate ($y$).

On a Cartesian plane, a square has two of its opposite vertices at points $A(-2, 3)$ and $C(4, -3)$.

1. Draw the axes and plot points $A$ and $C$.

2. 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).

3. What are the coordinates of the center point where the diagonals of the square intersect?