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):

$-5$, $3.14$, $\sqrt{16}$, $\sqrt{7}$, $0,\widehat{3}$, $\frac{4}{2}$.

In a Physics class, different quantities are measured or used for an experiment: the mass of a particle $m = 2.5 \times 10^{-3}$ kg, the period of a pendulum $T = 2\pi$ s, the number of oscillations $n = 15$, the temperature change $\Delta \theta = -4^\circ$ºC and the energy $E = \sqrt{2}$ J. Classify the exact numerical values of these measurements according to the most restrictive (smallest) number set to which they belong ($\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{I}$).

Look at the Venn diagram representing the number sets. Place the following numbers in the most restrictive region corresponding to them:

$a = \frac{\sqrt{18}}{\sqrt{2}}$, $b = -2.1\widehat{5}$, $c = \frac{\pi}{2}$, $d = \sqrt[3]{-8}$, $e = 1.010010001\dots$

The figure shows the geometric construction of a number on the number line using a right-angled triangle and a circular arc.

1. a) Identify which exact real number is represented by point $P$ on the horizontal axis.
2. b) Classify this number into the number sets.
3. c) Prove its value analytically by applying Pythagoras' theorem.

Find the irreducible generating fraction of the following decimal numbers:

1. a) $2.45$
2. b) $1,\widehat{6}$
3. c) $0.1\widehat{25}$

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.

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.

A company divides a profit of $1000$ pesos equally among $3$ partners. The accounting software truncates the result and transfers $333.33$ pesos to each account.

1. a) Express the exact amount corresponding to each partner using fractions.
2. b) Calculate the total absolute error made by the company when using the decimal approximation for the three partners instead of the exact value.

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) $\frac{\sqrt{50}}{\sqrt{2}}$

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).

The golden ratio is defined as $\phi = \frac{1+\sqrt{5}}{2}$.

1. a) Prove algebraically that $\phi$ satisfies the equation $\phi^2 - \phi - 1 = 0$.
2. b) From this result, deduce whether the number $\frac{1}{\phi}$ is rational or irrational and express it as a binomial in terms of $\phi$.

Consider the real number $x = \sqrt{7 - 4\sqrt{3}} + \sqrt{3}$.

Fully simplify the expression (hint: try to express the radicand $7 - 4\sqrt{3}$ as a perfect square binomial) and determine which number sets ($\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{I}$) the exact value of $x$ belongs to.

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.

A chemical reaction is viable if the temperature $T$ (in $^\circ$ºC) belongs to the interval $A = [-10, 45)$ and, simultaneously, the pressure requires a temperature control parameter $k = \frac{T}{10}$ to belong to the interval $B = (0, 5]$. Calculate the exact temperature interval $T$ for which the reaction is viable, satisfying both conditions at the same time.

In the design of a microchip, the allowable tolerance for the thickness of a silicon track is given by the set $T = \{x \in \mathbb{R} \mid |x - 1.25| \le 0.05\}$ in micrometres.

1. a) Express this set as an interval.
2. b) Determine whether the thicknesses $x_1 = 1.30$ and $x_2 = 1.25 + \frac{\sqrt{2}}{100}$ meet the required tolerance.

Given the sets of real numbers $A = \{x \in \mathbb{R} \mid x^2 - 5x + 6 \le 0\}$ and $B = \{x \in \mathbb{R} \mid |x - 2| < 1\}$, find the resulting sets from the operations $A \cup B$ and $A \cap B$. Express the final results as intervals.

Given the sets $A = (-\infty, 3k-1]$ and $B = [k+5, \infty)$ defined on the number line, determine the value of the parameter $k \in \mathbb{R}$ for which the intersection $A \cap B$ consists of a single real number. Classify this number into the sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{I}$.

A sports analyst models an athlete's optimal performance using the intersection of three reaction time intervals (in seconds) obtained from different biomechanical studies:

$I_1 = [0.15; 0.25]$

$I_2 = (0.18; 0.30)$

$I_3 = [0.20; \infty)$

1. a) Calculate the interval resulting from the intersection $I_1 \cap I_2 \cap I_3$.
2. b) If the absolute ideal reaction time must be an irrational number contained strictly within that intersection, suggest an exact value (using surds) that satisfies the condition.

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 $\frac{5}{7}$.

Knowing that between any two real numbers there are infinitely many rational and irrational numbers, construct an irrational number $x$ that simultaneously satisfies the following two conditions:

1. a) $0.01 < x < 0.02$
2. b) Only the digits $0$ and $1$ appear in its infinite decimal expansion.

Write at least the first 12 decimal places of your number to make clear the rule of formation that guarantees its irrationality.