Exercises: 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 $15$ € to a friend.
3. c) A diver is at a depth of $20$ metres.
4. d) An aeroplane flies at an altitude of $8,500$ metres.
5. e) You have won $50$ points in a level of your favourite video game.
6. f) The shopping centre car park is on the third basement floor.

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) $op(-1) =$

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

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

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:

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

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: $0$ $^\circ C$

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 Mathematics exam, Aoife has to compare several integers and writes on her answer sheet: "Since $12$ is greater than $5$, then $-12 > -5$, as both have the same digits and preserve the same order relation". Furthermore, she states that "the absolute value of $-12$ is less than that of $-5$ because $-12$ is located further to the left on the number line".

1. a) Identify the two conceptual mistakes Aoife made and explain in your own words why her reasoning is incorrect, relying on the position of the numbers relative to zero.
2. b) Write the correct comparison between $-12$ and $-5$ using the appropriate symbol ($>$ or $<$) and calculate the absolute value of both numbers to justify which of the two absolute values is greater.

During a winter early morning, Met Éireann, the Irish Meteorological Service, recorded the following minimum temperatures at four weather stations in Ireland: Straide ($-11\ ^\circ\text{C}$), Mount Dillon ($-7\ ^\circ\text{C}$), Birr ($-4\ ^\circ\text{C}$), and Valentia Island ($+2\ ^\circ\text{C}$).

1. a) Model this situation by ordering the four locations from lowest to highest temperature, using the appropriate mathematical symbols ($<$ or $>$).
2. b) Which weather station recorded the temperature with the greatest distance from the freezing point ($0\ ^\circ\text{C}$)? Justify your answer formally using the concept of absolute value.
3. c) If we plot the point $P = (T_{\text{Valentia}}, T_{\text{Birr}})$ on a Cartesian plane, determine which quadrant it is in and interpret what the sign of each of its coordinates means.

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 $0\,^{\circ}\text{C}$?

A delivery driver gets into the lift of a shopping centre 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 car park, and finally goes up $2$ floors to make a delivery.

1. a) Which floor is the car park 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 travelled throughout his route? (Hint: use the concept of modulus).

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.

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.

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.

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.

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

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

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.

We are looking for two distinct integers, $a$ and $b$, that simultaneously satisfy the following four conditions:

1. The point $P(a, b)$ is located in the second quadrant of the Cartesian plane.

2. The absolute value of $a$ is exactly $5$.

3. The value $|b|$ is a prime number less than $5$, but $b$ does not equal the opposite of $a$.

4. The distance between $a$ and $b$ on the number line is strictly less than $8$ units.

1. a) Deduce the sign that $a$ and $b$ must necessarily have based on the quadrant where point $P$ is located.
2. b) Find all possible values of $a$ and $b$, justifying step by step why you discard the options that do not simultaneously meet all the conditions.
3. c) Write the final coordinates of all possible points $P(a, b)$ that solve the problem.

Consider the integers $m = -14$ and $n = +6$ represented on a number line.

1. a) Before doing any exact calculations, make a reasoned estimate: will the exact midpoint between $m$ and $n$ be located to the left or to the right of zero? Will the total distance between the two be greater or less than $|m|$? Explain what you base your intuition on.
2. b) Now calculate the exact numerical value of the distance separating $m$ from $n$ on the line, relying on the meaning of their absolute values.
3. c) Determine which integer represents the midpoint between $m$ and $n$. Check whether your initial estimate in part a) was correct and explain why the sign of the midpoint matches the sign of the number with the greater absolute value.

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?

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

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

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).

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 ($y$).

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.

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.

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 centre point where the diagonals of the square intersect?

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

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 $|a + b|$.

To find out in which quadrant of the Cartesian plane the point $A(x, y)$ is located, we know with total certainty that the opposite of the abscissa is strictly negative ($op(x) < 0$) and that the ordinate satisfies $|y| = -y$, with $y \neq 0$. Two 1st Year students propose two different ways to figure it out:

• Cian's method: He chooses specific numbers that meet the clues (for example, he tests $x = 3$ and $y = -2$), checks that they fit the given conditions, and locates the resulting point in the quadrants.

• Aoife's method: She algebraically deduces the sign that $x$ must have from the definition of opposite and, then, deduces the sign of $y$ applying the properties of absolute value, without making up any particular number.

1. a) Apply Cian's method with that specific example and determine which quadrant the point falls into.
2. b) Apply Aoife's method by rigorously reasoning the signs of $x$ and $y$ from the theoretical definitions to reach the general conclusion.
3. c) Compare both approaches: does Cian's method guarantee that the conclusion holds for any pair of numbers satisfying the statement, or could there be unforeseen cases? Explain why Aoife's method provides a superior and more reliable mathematical justification.

During an oceanographic campaign in the Cantabrian Sea, the vertical position of three measuring probes, $S_1$, $S_2$ and $S_3$, is monitored. Their depths relative to sea level are expressed as integers in meters (where sea level is $0\text{ m}$):

1. Probe $S_1$ is submerged and its elevation is a negative integer whose absolute value is a multiple of $4$ strictly between $10$ and $18$, and its absolute value is also a multiple of $3$.

2. The elevation of probe $S_2$ is lower than that of $S_1$, and its distance to the surface exceeds that of $S_1$ by exactly $6$ units.

3. Probe $S_3$ is at an elevation such that $|S_3|$ is exactly half of $|S_1|$. Furthermore, it is known that the opposite of the elevation of $S_3$ is a positive number.

1. a) Reconstruct, with reasoning and justifying each deduction, the integer value of the elevation at which each of the three probes operates.

1. b) Model the relative position of the probes by ordering them from the one located at the greatest depth to the one closest to the surface, using the corresponding chain of inequalities with the symbol $<$.

1. c) If we construct the points of the Cartesian plane $P_1 = (S_1, S_2)$ and $P_2 = (op(S_3), S_1)$, determine in which quadrant each point lies, justifying it according to the signs of its coordinates.