Exercises: Concept

In everyday life, we use integers to express opposite situations relative to a reference point or "zero point". Match each of the following statements with the integer (positive or negative) that best represents it:

• a) A diver is at a depth of $15$ metres below sea level.

• b) A hiker has climbed to the top of a mountain $2300$ metres high.

• c) The balance on your public transport card is $4$ euros in debt.

• d) An aeroplane is flying at an altitude of $9500$ metres.

• e) The temperature at a ski resort is $7$ degrees below zero.

Observe the following number line where several points are marked with letters. Indicate which integer corresponds to each of the letters represented:

Considering that the distance between two consecutive small marks is $1$ unit, determine the values of $A$, $B$, $C$ and $D$.

The absolute value (or modulus) of an integer is the distance separating it from zero on the number line, regardless of direction. Calculate the following absolute values:

• a) $|-8| =$

• b) $|+12| =$

• c) $|-35| =$

• d) $|0| =$

• e) $|+7| =$

Two integers are opposites if they are at the same distance from zero but on opposite sides of the number line. Write the opposite of each of the following numbers and represent them in pairs on a small number line:

• a) $+5$

• b) $-3$

• c) $-1$

• d) $+4$

In a city in northern Spain, the following minimum temperatures were recorded during the first week of January:

• Monday: $-4$ ºC

• Tuesday: $-7$ ºC

• Wednesday: $0$ ºC

• Thursday: $+2$ ºC

• Friday: $-1$ ºC

• Saturday: $+3$ ºC

• Sunday: $-5$ ºC

Order these days from coldest to warmest (that is, from lowest to highest temperature) using the symbol $<$ between the values.

A lift in a very large shopping centre has floors above and below ground level. If we consider the ground floor (street level) as $0$:

• a) What floor do we reach if we are in basement $-3$ and go up $5$ floors?

• b) What floor do we reach if we are on floor $+2$ and go down $4$ floors?

• c) If we are in basement $-2$ and want to go up to floor $+4$, how many floors must we go up?

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

• a) $-10 \quad \square \quad -2$

• b) $+5 \quad \square \quad -8$

• c) $-1 \quad \square \quad 0$

• d) $-15 \quad \square \quad -20$

• e) $|-12| \quad \square \quad +10$

Write all integers $x$ that satisfy the following conditions at the same time:

1. Their modulus is less than $6$ ($|x| < 6$).

2. They are numbers less than $2$ ($x < 2$).

3. They are numbers greater than or equal to $-3$ ($x \geq -3$).

Represent them on a number line.

In a weather broadcast, the minimum temperatures recorded by AEMET in three municipalities of the province of Teruel are analysed: Griegos ($-14\,^\circ\text{C}$), Cantavieja ($-6\,^\circ\text{C}$), and Bronchales ($-9\,^\circ\text{C}$). In class, Marcos states the following: "Since $|-14| = 14$, $|-9| = 9$, and $|-6| = 6$, and we know that $6 < 9 < 14$, the order from lowest to highest temperature must be $-6\,^\circ\text{C} < -9\,^\circ\text{C} < -14\,^\circ\text{C}$, because the smallest integer is always the one with the smallest absolute value."

1. a) Identify and explain in detail the conceptual error Marcos makes when comparing negative integers using their absolute values.
2. b) Using the positions of the numbers on the number line relative to zero, write the correct ordering of the three temperatures from lowest to highest using the symbol $<$.

A mystery point $P(x, y)$ has coordinates formed by integers, and you must find its exact position using the following clues:

1. Point $P$ is located in quadrant II of the Cartesian plane.

2. The distance from its x-coordinate $x$ to the origin on the number line is $5$ units (that is, $|x| = 5$).

3. Its y-coordinate $y$ satisfies $op(y) = -3$.

1. a) Reasonably deduce the value of the integer $x$ by combining clue 1 and clue 2.
2. b) Deduce the value of the integer $y$ from clue 3, applying the property of the opposite.
3. c) Write the final coordinates of point $P(x, y)$ and indicate in which quadrant its symmetric point $Q(op(x), op(y))$ would be located, justifying the sign of each of its coordinates.

During a scientific expedition, a weather balloon probe flies over a volcano at an altitude of $+125$ meters above sea level, while a robotic probe is exploring the bottom of a nearby lake at a depth of $-84$ meters.

1. a) Represent both positions on a vertical number line, placing sea level at the origin.

1. b) Calculate the total distance, in absolute value, between the balloon and the probe.

1. c) If the probe rises $30$ meters, what will its new position be? Express it as an integer.

A team of underwater archaeologists is exploring the remains of an ancient galleon. They use a small research submarine to document the find. Express the following situations using integers:

1. a) The mothership is at sea level.
2. b) The galleon is resting on the seabed, at a depth of $45$ metres.
3. c) The submarine descends from the surface to a crevice located $12$ metres below the galleon.
4. d) After exploring the crevice, the submarine rises $25$ metres to photograph the ship's mast.

Represent these four positions on a scaled vertical line where the origin is sea level.

Order the following values from smallest to largest using the symbols $<$ or $>$. We recommend calculating the numerical value of each expression first:

$A = |−12|$

$B = op(+7)$

$C = −|−5|$

$D = op(−2)$

$E = −10$

$F = |op(−4)|$

Observe the following number line where three points, $X$, $Y$ and $Z$, have been marked.

If each division on the number line represents $2$ units:

1. a) Write the coordinates of the points $X$, $Y$ and $Z$.
2. b) Calculate the distance between point $X$ and point $Z$.
3. c) Write the opposite of $X$ and represent it on the line.

On a set of Cartesian coordinate axes, we have drawn three vertices of a rectangle: $A(−3, 2)$, $B(4, 2)$ and $C(4, −1)$.

1. a) Plot the points on the plane.
2. b) Determine the coordinates of the fourth vertex $D$ to complete the rectangle.
3. c) Which quadrant is each of the vertices in?
4. d) Calculate the perimetre of the rectangle by counting the grid units.

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

State whether the following statements are true or false. If a statement is false, give an example (counterexample) to prove it:

1. a) The modulus of a number is always greater than the number itself.
2. b) If two numbers are negative, the one with the smaller modulus is greater.
3. c) The opposite of an integer is always a negative number.
4. d) The distance between a number and its opposite is always double its modulus.

Complete the following magic integer pyramid. The rule is very simple: the value of each box is the sum of the two numbers directly below it.

A platform video game scores the movements of a character along a horizontal line.

• For each gold coin collected, the character moves forward $+5$ positions.

• For each obstacle hit, the character moves back $op(+3)$ positions.

• If they fall into a trap, they are placed at position $-|−10|$.

Calculate the final position of the character if:

1. They start at the origin ($0$).

2. They collect $4$ gold coins.

3. They hit $2$ obstacles.

4. They fall into a trap.

5. Finally, they move forward $8$ more positions.

The National Geographic Institute (IGN) takes mean sea level in Alicante as elevation zero ($0\text{ m}$) to measure altitudes in Spain. A team of oceanographers and surveyors monitors four points in a coastal area:

• Point $A$ is a buoy anchored exactly at sea level ($0\text{ m}$).

• Point $B$ is a beacon on a cliff located at an altitude of $+42\text{ m}$.

• Point $C$ is the entrance to an underwater cave whose elevation is given by $op(|-28|)\text{ m}$.

• Point $D$ is the bottom of a marine trench located at an elevation of $-42\text{ m}$.

1. a) Determine the integer that expresses the elevation of each of the four points.
2. b) Justify which of these four elevations belong to the set of natural numbers $\mathbb{N}$ and which belong to $\mathbb{Z}$ but not to $\mathbb{N}$.
3. c) Compare the positions of $B$ and $D$: do they have the same elevation? Are they at the same distance from sea level? Justify your answer using the concepts of opposite number and absolute value.
4. d) Order the four elevations from lowest to highest using the symbol $<$.

Consider the following two integers defined in terms of opposites and absolute values:

$$A = op(|-18|)$$ $$B = op(op(-11))$$

1. a) Estimate: Before performing any direct numerical calculation, reason based exclusively on the definitions of absolute value and opposite number what the final sign of $A$ and what the final sign of $B$ must be. From this deduction of signs and by comparing the absolute values, estimate which of the two numbers must necessarily be smaller on the number line.
2. b) Calculate: Apply the properties step by step to simplify $A$ and $B$, obtaining the exact integer that each represents.
3. c) Check: Check whether the obtained numerical order confirms your previous estimate. Next, calculate the distance separating both numbers on the number line and justify whether it is reasonable for this distance to be less than $10$ units.

A content creator analyses the evolution of their subscribers over the course of a week. On Monday, they started with a baseline figure. The daily changes were recorded as follows: Tuesday ($+150$), Wednesday (twice the loss of Thursday), Thursday ($-200$), Friday (an increase equal to the absolute value of Thursday's loss), and Saturday (an increase that left the total weekly balance at exactly $0$ compared to Monday).

Calculate the integer representing Wednesday's change and the integer representing Saturday's change. Represent the daily changes on a number line, placing $0$ as Monday's initial state.

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

1. Their modulus is less than $10$, that is: $|x| < 10$.

2. It is a number strictly less than its opposite: $x < Op(x)$.

3. Adding $-3$ to it results in a number whose modulus is greater than $5$.

List the numbers obtained and represent them on the number line.

An underwater reconnaissance drone starts from a platform located $15$ meters above sea level ($+15$ m). It performs the following consecutive movements:

• It descends $27$ meters to take samples.

• It ascends a distance equal to half the absolute value of its current position.

• Finally, it goes down to an abyssal zone located at $-120$ meters.

Determine the integer representing the drone's position after the second movement and calculate the total distance traveled by the drone during the three movements.

Consider two integers $a$ and $b$ for which we know the following:

• The modulus of $a$ is $12$.

• The opposite of $b$ is $-15$.

• The distance between $a$ and $b$ on the number line is as small as possible.

Based on these data:

1. a) Identify the values of $a$ and $b$.
2. b) What is the largest integer $k$ such that $a < k < b$?
3. c) If we multiply $a$ by a negative number, will the result be greater or less than $b$? Give reasons for your answer.

In a strategy video game, the coordinates of your base are $A(-3, 4)$. You must place an outpost at a point $B$ that meets the following conditions:

1. Its abscissa ($x$-coordinate) is the opposite of the abscissa of $A$.

2. Its ordinate ($y$-coordinate) is a negative integer whose absolute value is $2$.

Use the grid to plot both points (each small square measures one unit):

1. a) Determine the coordinates of point $B$.

1. b) Plot $A$ and $B$ on the Cartesian plane and state which quadrant each one is in.

1. c) What are the horizontal and vertical displacements needed to go from $A$ to $B$?

There are three consecutive integers whose sum is a negative number. If the absolute value of the middle number is $1$:

1. a) What are these three numbers?
2. b) Order the absolute values of these numbers from least to greatest.
3. c) What happens to the sign of the sum if we choose the "greatest" three consecutive integers (furthest to the right on the number line) that are still all negative?

Analyse the following statements and determine whether they are true or false, providing a counterexample for any you consider false:

1. If $|a| > |b|$, then $a > b$ always holds.

2. The modulus of the sum of two integers is always equal to the sum of their moduli: $|a + b| = |a| + |b|$.

3. If $n$ is a negative integer, then its opposite $Op(n)$ is always greater than $n$.

4. The distance between two opposite numbers is always twice the modulus of one of them.

In a cryogenic laboratory, a sample is at $-18$ ºC. The control system allows two types of adjustments:

• Adjustment A: Add $-5$ ºC to the current temperature.

• Adjustment B: Change the temperature to its opposite and then subtract $10$ ºC.

If we first apply Adjustment B and then Adjustment A, what will the final temperature be? Would the result be the same if we first applied Adjustment A and then Adjustment B? Justify your answer by doing both calculations step by step.

The goal is to determine in which quadrant of the Cartesian plane any point $P$ is located whose coordinates are given by the algebraic expressions $P(op(|a|), op(op(b)))$, knowing only as initial information that $a$ is a negative integer ($a < 0$) and that $b$ is a positive integer ($b > 0$).

Two students propose two different solution methods:

• Method 1 (Lucía - Verification with specific values): She chooses specific numerical values that satisfy the conditions, for example $a = -5$ and $b = +2$. She substitutes these values into the coordinates of $P$, calculates the resulting integers, and deduces the quadrant from the signs obtained.

• Method 2 (Mateo - General deduction by properties): He argues that for any $a \ne 0$ its absolute value is strictly positive ($|a| > 0$), so its opposite $op(|a|)$ must be negative; and that by the property $op(op(b)) = b$, the second coordinate keeps the positive sign of $b$.

1. a) Apply Lucía's Method 1 showing all calculation steps and determine the quadrant she obtains.
2. b) Apply Mateo's Method 2 rigorously justifying the final sign of the x-coordinate and the y-coordinate.
3. c) Critically compare both methods: does Lucía's method guarantee that the conclusion is valid for all possible integers satisfying the condition? What risks does this approach have compared to Mateo's deduction? Justify which of the two methods is mathematically rigorous.

Two unknown integers, $x$ and $y$, simultaneously satisfy the following four conditions:

1. Both belong to $\mathbb{Z}$, but neither belongs to $\mathbb{N}$, and neither is equal to $0$.

2. The integer $x$ is strictly greater than the integer $y$ ($x > y$).

3. The sum of their distances to zero on the number line is equal to ten: $|x| + |y| = 10$.

4. The opposite of $x$ is equal to three: $op(x) = +3$.

1. a) Deduce the value of the integer $x$ from condition 4. Justify whether this value satisfies condition 1.
2. b) Substitute the absolute value of $x$ into condition 3 to obtain the value of $|y|$. Which two integers have that absolute value as their distance to the origin?
3. c) Use conditions 1 and 2 to rule out one of the two possible values and determine the only possible value of $y$. Justify ruling it out by comparing the absolute values of the negative numbers involved.
4. d) With the values found for $x$ and $y$, reasonably determine in which quadrant of the Cartesian plane the point $R(op(x), op(y))$ is located.