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.

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) $op(-1) =$

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.

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 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.

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?

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 ($y$).