Number sets

Classify the following numbers by indicating which of the number sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ or $\mathbb{R}$ they belong to (note that a number can belong to several sets at the same time):

$-5$, $3,14$, $\sqrt{16}$, $\sqrt{7}$, $0,\widehat{3}$, $\frac{4}{2}$.

State whether the following statements are true or false, briefly justifying your answer:

1. a) Every integer is a rational number.
2. b) The number $0$ is an irrational number.
3. c) All decimal numbers can be expressed as a fraction.

Find the irreducible generating fraction of the following decimal numbers:

1. a) $2,45$
2. b) $1,\widehat{6}$
3. c) $0,1\widehat{25}$

An architect calculates the length of a wall by adding two sections whose theoretical measurements are $L_1 = 2,\widehat{3}$ m and $L_2 = 1,1\widehat{6}$ m.

1. a) Calculate the exact total length of the wall by working with the generating fractions of both numbers.
2. b) Express the final result as a decimal number.

Prove, using proof by contradiction, that $\sqrt{3}$ is an irrational number. (Assume that $\sqrt{3} = \frac{p}{q}$ where $p$ and $q$ are coprime integers, and reach a contradiction).

Without using a calculator, determine whether the result of the following operations is a rational or irrational number by expanding the algebraic expressions:

1. a) $(\sqrt{2} - 1)(\sqrt{2} + 1)$
2. b) $\sqrt{3} + \sqrt{12}$
3. c) $\frac{\sqrt{50}}{\sqrt{2}}$

Write the following set of numbers as an interval and represent it graphically on the number line:

$A = \{x \in \mathbb{R} \mid -2 < x \le 5\}$

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

Given the intervals $A = [-3, 4)$ and $B = (0, 7]$, determine the result of the following operations as an interval:

1. a) $A \cup B$
2. b) $A \cap B$

Determine exactly which integers belong to the interval $(-2,5; 3]$. Express the result using set notation with curly brackets.

Order the following real numbers from smallest to largest. To do so, you may use their approximate decimal expansion:

$3,14$, $\pi$, $\sqrt{10}$, $3,1\widehat{4}$, $\frac{22}{7}$

Find a rational number and an irrational number strictly between $1,5$ and $1,6$.

The density property of rational numbers states that between any two distinct rational numbers there is always another rational number. Prove this property by finding three distinct rational numbers strictly between $\frac{4}{7}$ and $\frac{5}{7}$.