Exercises: Real numbers

Classify the following real numbers, indicating which number sets they belong to: natural numbers ($\mathbb{N}$), integers ($\mathbb{Z}$), rational numbers ($\mathbb{Q}$), or irrational numbers ($\mathbb{I}$). Briefly justify your answer for decimal numbers and roots:

• $a) \ - \sqrt{16}$

• $b) \ \frac{15}{3}$

• $c) \ 2,131313...$

• $d) \ \sqrt{7}$

• $e) \ \frac{\pi}{2}$

• $f) \ -4,25$

Represent the following intervals and rays on the real number line, and express them using set notation (using inequalities). For $I_3$, which already appears with inequalities, also write it in interval notation.

• $I_1 = (-3, 2]$

• $I_2 = [0, +\infty)$

• $I_3 = \{x \in \mathbb{R} \mid -1 < x < 4\}$

Calculate the distance between the following pairs of points on the number line using the concept of modulus:

• $a)$ $x = 5$ and $y = -2$

• $b)$ $x = -8$ and $y = -3$

• $c)$ $x = \frac{1}{2}$ and $y = \frac{3}{4}$

Express the following neighborhoods as open intervals and represent them graphically on the real number line:

1. a) $E(2; 3)$ (neighborhood with center 2 and radius 3)

1. b) $E(-1; 0,5)$ (neighborhood with center $-1$ and radius $0,5$)

1. c) $E^*(0; 4)$ (punctured neighborhood with center 0 and radius 4)

Find the values of $x$ that satisfy the following equations and inequalities involving absolute values. Express the result as an interval:

• $a) \ |x| = 7$

• $b) \ |x - 2| = 5$

• $c) \ |x| < 3$

• $d) \ |x + 1| \leq 2$

A temperature sensor on a data server indicates that the CPU is at $42,6^{\circ}C$. If the actual value measured with a precision thermometer is $42,45^{\circ}C$, calculate:

• $a)$ The absolute error of the sensor measurement.

• $b)$ The relative error made, expressed as a percentage.

Simplify the following expressions using the properties of powers and roots, giving the result in the form of a power with a fractional exponent:

• $a) \ \sqrt[3]{x^2} \cdot \sqrt{x}$

• $b) \ \frac{x^3}{\sqrt[4]{x}}$

• $c) \ (\sqrt[5]{x^2})^3$

In an electronics components store, the price of a racing drone model fluctuates depending on market supply. The price $p$, in euros, is known to satisfy the condition $|p - 150| \leq 20$.

1. a) Determine the interval of possible prices for this drone.

1. b) What are the minimum and maximum prices a user could pay?

Mateo states the following in a 2nd year of Bachillerato class:

"Given that $(x - 3)^2 = x^2 - 6x + 9$, we can ensure that $\sqrt{x^2 - 6x + 9} = x - 3$ for any value of $x \in \mathbb{R}$. For example, if we evaluate at $x = 1$, we have $\sqrt{1 - 6 + 9} = 1 - 3$, which proves that $\sqrt{4} = -2$".

1. a) Identify and explain in detail the two conceptual errors made by Mateo (one regarding the arithmetic definition of the square root in $\mathbb{R}$ and another concerning the algebraic simplification of $\sqrt{A^2}$).
2. b) Correct the equality by expressing the correct result of $\sqrt{x^2 - 6x + 9}$ using absolute value, and expand it as a piecewise-defined function according to the sign of the radicand.
3. c) Check your correction by evaluating the expression at $x = 1$ and at $x = 5$, verifying that no contradiction arises.

From the following sets of real numbers, carry out the reverse process to construct their analytic expressions:

1. a) Write the set of numbers whose distance to the midpoint $-1,5$ on the real line is strictly less than $2,5$ using:

1. An absolute value inequality.

2. Symmetric neighbourhood notation $E(a, r)$.

3. A bounded interval in $\mathbb{R}$.

1. b) Given the union of rays $(-\infty; -2] \cup [8; +\infty)$, determine its centre of symmetry and its exclusion radius. From these, write a single absolute value inequality whose solution is precisely that set.
2. c) Given the set $(-3; 5) \setminus \{1\}$, express it in deleted neighbourhood notation $E^*(a, r)$ and state the meaning of excluding the centre with respect to the formal definition of limit at a point.

In the manufacturing process of microchips for satellites, the length of a copper track must be $12,5$ microns. For thermal reasons, a margin of error (tolerance) is allowed such that the difference between the actual length $x$ and the theoretical one is at most $0,05$ microns.

a) Express this condition by means of an inequality with absolute value.

b) Determine the interval of acceptable values for the length of the track using interval notation.

c) Graphically represent this interval on the real number line.

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

A biotechnology laboratory maintains an enzyme sample in a climate chamber. To prevent the enzymes from denaturing, the temperature $T$ (in degrees Celsius) must satisfy the following condition:

Solve the inequality and determine the temperature range (interval) in which researchers can work without damaging the sample.

To determine the domain of a future function, we need to solve the following quadratic inequality:

a) Find the roots of the associated polynomial.

b) Determine the solution set in interval form.

c) Justify the result by analysing the sign of the polynomial in each section of the number line.

In astrophysics, the distance between two celestial bodies is often measured in astronomical units (AU). Suppose two measurements of the distance to an exoplanet have been made: $d_1 = 4,56 \cdot 10^8$ km and $d_2 = 4,562 \cdot 10^8$ km.

a) Calculate the absolute error and the relative error made if we take the first measurement as an approximation of the second.

b) Express the result of the relative error as a percentage and comment on whether the approximation is acceptable for a scientific context.

Simplify the following expression involving radicals and powers, expressing the final result as a single power of base $a$ with a rational exponent:

Note: Assume that $a > 0$.

The design of a triangular metal part requires its sides to measure $\sqrt{18}$ cm, $\sqrt{50}$ cm, and $\sqrt{98}$ cm.

a) Simplify each of the radicals by extracting factors.

b) Calculate the exact perimetre of the part, expressing it as a single radical.

c) If the material used costs $2,5$ € per centimetre of perimetre, calculate the total cost rounded to two decimal places.

Using the properties of logarithms, expand the following expression in terms of $\log x$, $\log y$, and $\log z$:

Then, given that $\log x = 0,5$, $\log y = 1,2$, and $\log z = 0,8$, calculate the numerical value of $L$.

Given the following subsets of the real line:

$A = \{ x \in \mathbb{R} : |x - 2| < 3 \}$

$B = [ -1, 4 ]$

a) Express the set $A$ using interval notation.

b) Calculate and express using interval notation the set resulting from the intersection $A \cap B$.

c) Calculate and express using interval notation the set resulting from the union $A \cup B$.

The State Meteorological Agency (AEMET) monitors extreme conditions at two very different Spanish observatories. At the high-mountain station of Puerto de Navacerrada (Madrid), the thermometer records a minimum temperature of $-4,0\text{ }^\circ\text{C}$ with an absolute error bound of the sensor of $0,2\text{ }^\circ\text{C}$. On the other hand, the Sevilla-Aeropuerto station records a maximum temperature of $40,0\text{ }^\circ\text{C}$ in the summer season with an absolute error bound of $0,8\text{ }^\circ\text{C}$.

1. a) Model the interval of admissible real temperatures for each observatory using an absolute value inequality and using symmetric neighbourhood notation $E(a, r)$.
2. b) Calculate the percentage relative error bound for each of the two measurements.
3. c) Interpret the results: although the Navacerrada sensor has an absolute error bound four times smaller than that of Sevilla ($0,2\text{ }^\circ\text{C}$ versus $0,8\text{ }^\circ\text{C}$), which of the two measurements is of better relative quality? Justify why relative error is the objective criterion to compare precisions between quantities of different scales.

Consider the real number $K = \log_3(80) - \sqrt{17} + \log_2\left(\frac{1}{10}\right)$.

1. a) Estimate: Without using a calculator:

1. Bound $\log_3(80)$ between two consecutive integers, justifying your answer using powers of base $3$.

2. Bound $\sqrt{17}$ between two consecutive integers by comparing it with perfect squares.

3. Bound $\log_2\left(\frac{1}{10}\right)$ between two consecutive integers knowing that $\frac{1}{10} = 10^{-1}$ and using powers of base $2$.

From these three bounds, obtain an interval for the value of $K$ and predict with reasoning whether its sign must be positive or negative.

1. b) Check and refine: Knowing that $80$ is extremely close to $81$ and that $17$ is extremely close to $16$, mentally approximate the value of the difference $\log_3(80) - \sqrt{17}$. Next, taking into account that $2^3 = 8$ and $2^{3,5} = \sqrt{128} \approx 11,31 > 10$, check whether the overall value of $K$ is closer to $-3$ or to $-4$. Justify whether this check confirms or refutes the prediction from your initial estimate.

Determine, giving reasons, the set of values of $x \in \mathbb{R}$ satisfying the following compound inequality, expressing the final result as a union of intervals:

Take into account domain restrictions and analyse the different cases depending on the sign of the expression inside the modulus.

In designing a high-precision part for an aerospace engine, the radius of a cylinder, denoted by $x$, must be sufficiently close to the nominal radius of $5\text{ cm}$ so that the error in its cross-sectional area, $A(x) = \pi x^2$, does not exceed $0,01\text{ cm}^2$.

Using the formal definition of a deleted neighbourhood $E^*(a, \delta)$, find the maximum value of the tolerance $\delta$ that the radius $x$ can have to ensure that the area remains within the neighbourhood $E(25\pi, 0,01)$. Express $\delta$ in terms of $\pi$ in exact form.

Prove, using exclusively the metric definition of a limit ($\varepsilon - \delta$), that:

To do this, given $\varepsilon > 0$, find an expression for $\delta$ in terms of $\varepsilon$ (you may assume an initial bound for $\delta$, for example $\delta \leq 1$).

Consider the sequence defined by the general term $a_n = \frac{3n^2 - 1}{n^2 + 2n}$.

1. a) Calculate the limit $L$ of the sequence as $n \to \infty$.
2. b) Determine from which index $n_0 \in \mathbb{N}$ all terms of the sequence are at a distance from $L$ of less than one thousandth ($\varepsilon = 10^{-3}$).

Find the set of real values satisfying the following inequality with double modulus:

Represent the solution on the number line and express it as a single open interval.

The function $f(x) = \sqrt{x^2 - kx + 9}$ is defined. Determine the set of values of the real parameter $k$ for which the domain of the function is the entire real line $\mathbb{R}$.

For the largest value in this set, determine whether there is any horizontal or oblique asymptote as $x \to +\infty$.

An investor analyses the growth of a financial asset whose value in thousands of euros is given by the function $V(t) = \frac{4t^2 + k}{t^2 + 1}$, where $t \geq 0$ is the time in years.

1. a) Given that the initial value of the asset was $1000$ euros, determine the value of the constant $k$.
2. b) What will be the value of the asset in the "very long term"? Justify your answer using the concept of a limit at infinity.
3. c) Will the value of the asset exceed $4000$ euros at any point? Prove it analytically using an inequality.

Given the function $f(x) = \ln(|x^2 - 4| - 5)$, determine its domain of definition in exact form. Express the result as a union of intervals, paying special attention to the values where the argument of the logarithm becomes zero or negative. Analyse the symmetry of this domain.

To solve the inequality $|x + 2| + |x - 4| \le 8$ in $\mathbb{R}$, three students from 2nd year of Bachillerato propose different strategies:

• Álvaro proposes squaring both sides directly to eliminate the absolute value bars.

• Berta proposes studying the sign of each binomial on the intervals determined by the roots $x = -2$ and $x = 4$, solving the inequality without absolute values on each of the three intervals.

• Clara proposes interpreting the expression geometrically as the sum of the distances from a point $x$ to the points $-2$ and $4$ on the real line, reasoning about the minimum distances and the remaining margin.

1. a) Analyse Álvaro's proposal: explain algebraically what cross term appears when squaring the left-hand side and why this method does not simplify the problem, but rather complicates it.
2. b) Apply Berta's method (case-by-case interval analysis) in a complete and orderly way, detailing the partial solutions for each case and the global solution as an interval.
3. c) Solve the inequality applying Clara's method using purely geometric reasoning: what is the minimum distance between $-2$ and $4$? How is the remaining distance of $8$ distributed to the left and to the right?
4. d) Justify which of the two valid methods (B or C) is more efficient in this exercise and propose a small modification to the inequality that would force you to abandon Clara's geometric method and necessarily resort to Berta's interval method.

In the design of an optical sensor for the Gran Telescopio Canarias (GTC), the light response is modelled by the function $f(x) = \dfrac{8}{x + 1}$, where $x > 0$ represents the thickness in millimetres of a calibration filter. The nominal reference value for the thickness is $a = 3\text{ mm}$, which corresponds to a nominal output $L = f(3) = 2$.

The engineering team requires that the light response does not deviate from $2$ by more than $\varepsilon = 0,1$, that is, it is required that $f(x) \in E(2; 0,1)$.

1. a) Inverse problem: Solve the double inequality $1,9 < \dfrac{8}{x + 1} < 2,1$ to determine the exact interval $(x_1, x_2)$ in which the thickness $x$ must be maintained.
2. b) Check that the interval $(x_1, x_2)$ obtained is not centred exactly at the nominal value $a = 3$. Calculate the distances $d_1 = |3 - x_1|$ and $d_2 = |x_2 - 3|$.
3. c) Determine the largest value of the radius $\delta > 0$ of a symmetric neighbourhood centred at $3$, $E(3, \delta)$, so that the implication is rigorously guaranteed:

Why must you necessarily choose $\delta = \min(d_1, d_2)$ and not the larger of the two radii?

1. d) Translate the entire previous condition into the formal absolute value formulation: $|x - 3| < \delta \implies |f(x) - 2| < 0,1$. Explain in a reasoned manner the direct connection between this result and the formal definition of the limit $\lim_{x \to 3} f(x) = 2$ using $\varepsilon$-$\delta$ neighbourhoods.