The set of integers

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.

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.