The Concept of Integers

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 $15$ meters below sea level.

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

• c) The balance on your public transport card is $240$ pesos in debt.

• d) An airplane is flying at an altitude of $9500$ meters.

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

An elevator in a very large shopping mall 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?

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 $45$ meters.
3. c) The submarine descends from the surface to a crevice located $12$ meters below the galleon.
4. d) After exploring the crevice, the submarine rises $25$ meters to photograph the ship's mast.

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

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) $+4$

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.

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.

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 $D$.

In Constanza, 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.

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 scientific expedition, a weather balloon 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.
2. b) Calculate the total distance, as an absolute value, separating the balloon from the probe.
3. c) If the probe ascends $30$ meters, what will its new position be? Express it as an integer.

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

$A = |−12|$

$B = op(+7)$

$C = −|−5|$

$D = op(−2)$

$E = −10$

$F = |op(−4)|$

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.

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.

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 $B$?

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.

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.

An underwater reconnaissance drone sets off from a platform situated $15$ meters above sea level ($+15$ m). It makes the following consecutive movements:

• Descends $28$ meters to take samples.

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

• Finally, 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.

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?

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.