Exercises: The set of integers

In a survival video game session, your character's actions affect their experience points. Express the following situations using an integer:

1. a) You find a chest and gain $450$ points.
2. b) You are defeated by an enemy and lose $125$ points.
3. c) You fall into a trap and your score drops by $50$ points.
4. d) You complete a side quest and receive $80$ points.

Calculate the absolute value and the opposite of the following integers:

1. a) $-14$
2. b) $+21$
3. c) $-5$
4. d) $+9$

Look at the following number line and state which integers correspond to the points marked with the letters $A$, $B$, $C$, and $D$.

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

1. a) $-8 \quad \dots \quad -2$
2. b) $+5 \quad \dots \quad -10$
3. c) $0 \quad \dots \quad -3$
4. d) $-1 \quad \dots \quad +1$

A group of friends is measuring the temperature in different European cities on a winter day:

• Berlin: $-4 \text{ °C}$

• Madrid: $+9 \text{ °C}$

• Oslo: $-12 \text{ °C}$

• Rome: $+15 \text{ °C}$

• Zurich: $-1 \text{ °C}$

Order the cities from coldest to warmest.

Draw a number line in your exercise book and represent the following integers on it:

$$-5, \quad +3, \quad -2, \quad 0, \quad +1, \quad -4$$

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

State whether the following statements are true or false:

1. a) The number $-7$ is a natural number ($\mathbb{N}$).
2. b) All natural numbers are also integers ($\mathbb{Z}$).
3. c) The absolute value of a number is always positive or zero.
4. d) The opposite of a negative number is always negative.

You are in a very large shopping center that has floors above and below street level (floor $0$).

1. a) If you are on floor $-3$ and go up $5$ floors, which floor are you on?

1. b) If you are on floor $2$ and go down $3$ floors, which floor do you reach?

Lucas has the point $A(-4, 6)$ located in a Cartesian coordinate system and wants to find its reflection across the $x$-axis. In his notebook, he writes the following explanation:

"Since the $x$-axis is the horizontal line, to reflect across it I have to change the sign of the first coordinate, which is the horizontal one. Therefore, the reflected point is $A'(4, 6)$".

1. a) Explain with reasons what conceptual mistake Lucas is making, relying on the distance from the point to the horizontal axis of symmetry.
2. b) Write the correct coordinates of the reflection of $A$ across the $x$-axis and also those of its reflection across the $y$-axis.
3. c) If a point $B(0, -5)$ lies on the $y$-axis, what is its reflection across that same axis? Justify your answer using the concept of opposite number.

The State Meteorological Agency (AEMET) recorded the following temperature changes at the mountain station of Puerto de Navacerrada during a winter day: at 6:00 h the thermometer read $5$ degrees below zero; at 13:00 h the temperature had risen $7$ ºC compared to the morning; finally, at 22:00 h the thermometer dropped $4$ ºC compared to midday.

1. a) Model the situation by assigning an integer to the temperature recorded at each of the three indicated times ($6:00$ h, $13:00$ h, and $22:00$ h).
2. b) Classify each of the three integers obtained, indicating which set they belong to: to the natural numbers $\mathbb{N}$, only to the set $\mathbb{Z} \setminus \mathbb{N}$, or to both sets? Justify your classification by recalling the relation $\mathbb{N} \subset \mathbb{Z}$.
3. c) A television reporter states: "At 13:00 h the temperature was an integer, but not a natural number". Reason whether the reporter's statement is true or false.

Associate an integer with each of the following situations taken from a teenager's daily life:

1. a) The balance on your travel card after spending $5$ € on trips and having a previous top-up of $2$ €.
2. b) The position of an exploration submarine in a video game that is $150$ metres below sea level.
3. c) The temperature at a ski resort reading $4$ degrees below zero.
4. d) The year in which the abacus is estimated to have been invented, approximately $500$ years BC.
5. e) Going down in a shopping centre lift from the cinema floor (floor $3$) to the car park located two floors below.

Look at the following number line where several points have been marked with letters. Identify the integer corresponding to each letter and then calculate the absolute value of each of them.

Write down all the integers $x$ that satisfy both of the following conditions simultaneously:

1. Their absolute value is less than or equal to $4$: $|x| \le 4$.

2. They are greater than the opposite of $2$: $x > Op(+2)$.

In a high-mountain town, the weather service records the following temperatures throughout a winter's day:

• At 06:00: $-8$ ºC.

• At 12:00: The temperature has risen by $10$ ºC compared to 06:00.

• At 20:00: The temperature has fallen by $5$ ºC compared to 12:00.

• At 00:00: The temperature is the opposite of the one recorded at 12:00.

Represent these four temperatures on a number line and order them from lowest to highest using the $<$ symbol.

Consider two integers $a$ and $b$. It is known that $a$ is a negative number and $b$ is a positive number. State whether the following statements are always true, sometimes true, or always false, justifying your answer with an example:

1. a) $|a|$ is always greater than $b$.
2. b) The opposite of $b$ is always less than $a$.
3. c) The absolute value of $a$ is always a positive number.
4. d) $b$ is always greater than $a$.

A toy drone moves along a straight corridor marked as a number line. The take-off point is $0$. The drone makes the following consecutive moves:

1. Moves forward $+6$ metres.

2. Moves back $10$ metres.

3. Moves to the point opposite its current position.

4. Finally, advances to a point whose absolute value is $2$ and which is to the left of the origin.

At what point on the line is the drone at the end of its journey? Draw it in a diagram.

Order the following numbers from smallest to largest, placing the negative values first and using the corresponding inequality signs:

$$|-7|, \quad +3, \quad Op(-5), \quad -4, \quad 0, \quad Op(+8), \quad -1, \quad |+2|$$

Complete the following table in your exercise book, filling in the blanks so that each row is consistent with the definition of an integer, its opposite, and its absolute value:

Number ($z$)	Opposite ($Op(z)$)	Absolute Value ($|z|$)
$-12$
	$+7$
		$5$ (If $z$ is positive)
		$9$ (If $z$ is negative)
$Op(-3)$

We want to define a set of unknown integers using several simultaneous conditions, such that the only numbers in all of $\mathbb{Z}$ that satisfy all of them at once are exactly $\{-6, -4, -2\}$.

Design three conditions that an integer $x$ must satisfy to obtain this result:

• The first condition must define a numerical range using the absolute value $|x|$.

• The second condition must refer to belonging or not belonging to the sets $\mathbb{N}$ and $\mathbb{Z}$.

• The third condition must require a property regarding the divisibility or parity of the number.

Once you have written your three conditions:

1. a) Check step by step that the numbers $-6$, $-4$, and $-2$ satisfy all three simultaneously.
2. b) Write an integer that satisfies the first two conditions but not the third, showing that your third condition is essential so that other solutions do not slip in.

A submersible robot from the Spanish Institute of Oceanography departs from a research platform located at $+16\text{ m}$ above sea level. To study a marine fault, it performs three consecutive maneuvers:

1. It descends $38\text{ m}$ to calibrate its cameras.

2. It descends another $42\text{ m}$ to collect samples from the seabed.

3. It ascends a distance equal to half the sum of the meters it descended in the two previous maneuvers.

1. a) Estimate: Before performing combined operations step by step, reason whether the final position of the robot will be above the sea surface (positive integer), exactly at the surface ($0$), or below sea level (negative integer). Briefly explain what you look at to make your estimate.
2. b) Calculate: Write the mathematical expression with integers that represents all the movements and calculate the exact final elevation at which the robot ends up.
3. c) Check: Compare your initial estimate with the value obtained. Is the result consistent? Does the final position belong to the set $\mathbb{N}$ or only to the set $\mathbb{Z}$?

Determine all the integers $x$ that simultaneously satisfy the following three conditions:

• Its absolute value is greater than $5$ and less than or equal to $9$.

• The number $x$ does not belong to the set of natural numbers $\mathbb{N}$.

• $x$ is an even number.

Graphically represent on the number line the set of solutions you obtained. You can use this unmarked number line as a guide:

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

A research submarine is at a depth of $-450$ metres relative to sea level. It launches a vertical probe that ascends $125$ metres to take a sample, and subsequently descends three times the distance it had ascended.

1. a) Express each movement and the final position of the probe using integers.
2. b) Calculate the absolute distance between the initial position of the submarine and the final position of the probe.

On a coordinate grid, three of the vertices of a rectangle are the points $A(-4, 3)$, $B(2, 3)$ and $C(2, -1)$.

1. a) Determine the coordinates of the fourth vertex $D$ to complete the rectangle.
2. b) Calculate the perimetre of the rectangle knowing that each grid unit is equivalent to $1$ metre.
3. c) What is the symmetric point of vertex $A$ with respect to the x-axis (horizontal axis)? State its coordinates using integers.

Reason whether the following statements are true or false, providing a counterexample if they are false:

1. a) If an integer $a$ is less than $b$, then the modulus of $a$ is always less than the modulus of $b$ ($|a| < |b|$).
2. b) The opposite of the modulus of any integer (other than zero) is always a negative number.
3. c) There is no integer whose modulus is equal to its opposite.

Given the integers $a = -12$ and $b = +5$:

1. a) Calculate the distance between them on the number line.
2. b) Find the number $c$ that is at exactly the same distance from $a$ as from $b$. Is $c$ an integer? Justify your answer.
3. c) If $d$ is a number such that $|d| = |a| + |b|$, what values can $d$ take?

Write the name of a historical figure who satisfies the following conditions:

• They were born in the year $x$, where $x$ is the opposite of $356$.

• They died in the year $y$, where $y$ is an integer such that $|y| = 323$ and $y < 0$.

• Calculate how many years this figure lived, knowing that there is no year zero in the Gregorian calendar (it goes from 1 BC to 1 AD).

A sequence of integers is defined as follows:

1. The first term is $a_1 = -5$.

2. The second term is the opposite of the modulus of $a_1$, increased by $2$ units.

3. The third term is the modulus of the second term, decreased by $10$ units.

Calculate the fourth term if it is the opposite of the sum of the three previous terms.

Imagine that a smart thermometer records the maximum and minimum temperature of a city. One day, the minimum temperature was $-8$ $^\circ C$. The maximum temperature was an integer $T$ satisfying:

• $|T| < 15$

• The difference between the maximum and minimum (temperature range) is $12$ degrees.

• $T$ is a number greater than its opposite.

What was the maximum recorded temperature? State all possibilities, if any.

In a 2nd year ESO class, a question arises about how to find the coordinates of the resulting point after applying two consecutive reflections to a generic point $P(x, y)$ with non-zero integer coordinates: first across the $x$-axis and then, to the resulting point, across the $y$-axis. Two students propose different approaches:

• Sergio's method: He chooses a specific point, for example $P(-3, 5)$. He applies the reflection across the $x$-axis, obtaining $P_1(-3, -5)$, and then the reflection across the $y$-axis to $P_1$, obtaining $P_2(3, -5)$. Since he observes that both coordinates have changed sign compared to $P$, he concludes that this rule works for any point in the plane.

• Lucía's method: She applies algebraic rules with letters: across the $x$-axis, the point $(x, y)$ transforms into $(x, op(y)) = (x, -y)$; then, across the $y$-axis, a point $(a, b)$ transforms into $(op(a), b) = (-a, b)$, so applying it to $(x, -y)$ results in $(-x, -y)$.

1. a) Evaluate both methods: which of the two demonstrates with mathematical rigor that the property is true for all integers in the plane, and why does the other method only provide a particular conjecture?
2. b) Apply the general conclusion to directly find the final coordinates of $Q(15, -42)$ after these two successive reflections, without calculating the intermediate point.
3. c) If a third reflection were then applied, once again across the $x$-axis, what coordinates would the final point have in terms of the initial $(x, y)$? What single reflection are these three reflections together equivalent to?

During a debate about the properties of integers, Irene claims that the following three statements are always true for any integer:

• Statement 1: «If $a$ and $b$ are any two integers such that $a < b$, then necessarily $|a| < |b|$».

• Statement 2: «For any integer $n \in \mathbb{Z}$, its square $n^2$ is always a number that belongs to $\mathbb{N}$ and also satisfies $n^2 > n$».

• Statement 3: «If you subtract a negative integer from any negative integer, the result always belongs to $\mathbb{Z} \setminus \mathbb{N}$ (that is, it can never be a natural number)».

1. a) Remember that to prove that a statement about all integers is false, it is enough to find a single counterexample. Prove that all three of Irene's statements are false by giving a clear and specific numerical counterexample for each one.
2. b) Explain the conceptual mistake Irene makes in Statement 3 and write a precise condition on the absolute values of the two negative numbers so that their subtraction does give a positive integer as a result.