Exercises: 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):

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

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.

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.

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 $1,6$.

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.

Hugo has to find the generating fraction of the decimal number $2,1\widehat{45}$ and writes the following in his notebook:

"Let $x = 2,1454545\ldots$

Since the repetend has two digits, I multiply by $100$:

$100x = 214,54545\ldots$

Now I subtract both expressions to remove the decimals:

$100x - x = 214,54545\ldots - 2,1454545\ldots = 212,4$

Therefore, $99x = 212,4 \implies x = \frac{212,4}{99} = \frac{2124}{990} = \frac{118}{55}$.

Since I have obtained a fraction of integers, the result is correct".

1. a) Explain the conceptual error in the method Hugo followed. Look closely at the subtraction he performs: does the infinite repeating part actually cancel out, or does it become distorted?
2. b) Correct his solution, explaining which powers of $10$ must be used to accurately align the repetends, and write down the step-by-step procedure.
3. c) Find the correct irreducible generating fraction and compare it with Hugo's. Do they happen to match by chance, or was Hugo's result incorrect?

Consider the real number $N = \sqrt{10} + \sqrt{26}$.

1. a) Estimate: without using a calculator, bound each of the roots between two consecutive integers using the closest perfect squares. With that information, deduce between which two integers the value of $N$ must lie.
2. b) Check: using the three-decimal-place approximations $\sqrt{10} \approx 3,162$ and $\sqrt{26} \approx 5,099$, calculate the approximate value of $N$. Is the result obtained consistent with your previous estimate?
3. c) Reasonably justify whether $N$ belongs to the closed interval $[8, 9]$. Then, classify $N$ into the most restrictive number set possible ($\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, or $\mathbb{I}$), justifying why neither $\sqrt{10}$ nor $\sqrt{26}$ can be expressed as rational numbers.

In a Physics class, different quantities are measured or used for an experiment: the mass of a particle $m = 2,5 \cdot 10^{-3}$ 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 ($\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{I}$).

A company divides a profit of $1000$ euros equally among $3$ partners. The accounting software truncates the result and transfers $333,33$ euros 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.

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

$a = \frac{\sqrt{18}}{\sqrt{2}}$, $b = -2,1\widehat{5}$, $c = \frac{\pi}{2}$, $d = \sqrt[3]{-8}$, $e = 1,010010001\dots$

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.

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.

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.

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

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 $\frac{5}{7}$.

The density property of real numbers states that between any two distinct real numbers there are infinitely many rationals and infinitely many irrationals. Consider the open interval $(0,45\,;\, 0,46)$.

1. a) Construct a rational number $r$ that strictly belongs to that interval. First write its terminating decimal expansion, and then find its irreducible generating fraction $\frac{p}{q}$ with $p, q \in \mathbb{Z}$.
2. b) Construct an irrational number $s$ that belongs to that interval. Clearly explain the formation rule of its infinite decimal digits to ensure that it has no repeating period.
3. c) Find two distinct irrational numbers, $x$ and $y$, both belonging to the interval $(1, 2)$, such that their product is exactly a natural number: $x \cdot y = 2$. Prove that both $x$ and $y$ are indeed irrational.

In an agroclimatic monitoring station in the Guadalquivir valley, soil moisture percentage ($H$) is measured in three olive grove plots to automate the drip irrigation system. The control center follows this protocol according to the value of $H$:

• Severe deficit state: $H \in [0, 15)$ (triggers emergency irrigation).

• Optimal state: $H \in [15, 28]$ (keeps irrigation deactivated).

• Saturation state: $H \in (28, 100]$ (triggers preventive drainage).

The sensors record the following exact values in three plots:

• Plot 1: $H_1 = \sqrt{200}\,\%$

• Plot 2: $H_2 = \frac{85}{6}\,\%$

• Plot 3: $H_3 = 14,\widehat{9}\,\%$

1. a) Classify each of the values $H_1$, $H_2$, and $H_3$ into the most restrictive number set to which they belong ($\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, or $\mathbb{I}$).
2. b) Determine the operational state of each of the plots. For Plot 3, calculate its irreducible generating fraction, explain what mathematical peculiarity the decimal $14,\widehat{9}$ presents, and conclude what action the system must execute in that plot.
3. c) Express as a union of intervals the set of moisture values for which the system must actively intervene (emergency irrigation or preventive drainage).

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).

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 $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{I}$.

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.

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.

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.

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.

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.

You want to rigorously prove that the number $x = \sqrt{2} + \sqrt{5}$ is an irrational number. In your work group, three different proposals arise:

• Method 1 (Decimal approximation): Compute $\sqrt{2} \approx 1,4142\ldots$ and $\sqrt{5} \approx 2,2360\ldots$, add the obtained digits, and observe that the resulting decimal part shows no repeating period.

• Method 2 (Proof by contradiction by isolating a radical): Assume that $\sqrt{2} + \sqrt{5} = q$ with $q \in \mathbb{Q}$, transform the equality into $\sqrt{5} = q - \sqrt{2}$, square both sides, and solve for $\sqrt{2}$ to analyze the nature of the resulting equality.

• Method 3 (Direct squaring): Square the sum directly to obtain $(\sqrt{2} + \sqrt{5})^2 = 7 + 2\sqrt{10}$, and state that since the sum contains $\sqrt{10}$, the original number must be irrational.

1. a) Critically analyze Method 1: does it constitute a valid mathematical proof? Justify why the use of finite approximations can never prove the absence of periodicity in the infinite decimal places of a number.
2. b) Evaluate Method 3: explain what logical implication error it makes (is it true that if $x^2$ is irrational, then $x$ is irrational?, and has it already been formally proved that $\sqrt{10}$ is irrational?).
3. c) Apply and fully develop Method 2: write out the formal proof by contradiction, detailing each algebraic step and explaining with mathematical precision the final contradiction reached.

On a written test, the following statement is posed to determine whether it is true or false: "The sum of any two irrational numbers is always an irrational number". Marcos answers:

"It is true. An irrational number has infinitely many decimals that follow no repeating pattern. When adding two numbers whose decimal digits are non-repeating, it is impossible for the infinite digits to magically synchronize to form a repeating cycle. Therefore, the sum will always have infinitely many non-repeating decimals and will belong to $\mathbb{I}$".

1. a) Identify and explain Marcos's conceptual error. Why is his intuition about the "sum of digits that cannot synchronize" mathematically invalid?
2. b) Provide two explicit numerical counterexamples that disprove the statement:

1. Two numbers $x, y \in \mathbb{I}$ such that their sum $x + y$ is an integer $\mathbb{Z}$.

2. Two numbers $u, v \in \mathbb{I}$ such that their product $u \cdot v$ is a non-integer rational number (that is, belonging to $\mathbb{Q} \setminus \mathbb{Z}$).

1. c) Formally prove, using proof by contradiction, the property that does always hold: "The sum of any rational number $q \in \mathbb{Q}$ and any irrational number $r \in \mathbb{I}$ is always an irrational number ($q + r \in \mathbb{I}$)".