Exercises: Real numbers

Represent the fractions $\frac{7}{3}$ and $-\frac{5}{4}$ exactly on the real line by dividing the unit into the corresponding parts.

State which of the following numbers belong to the interval $[-1, \pi)$: $-1,5$, $-1$, $0$, $3,14$, $3,15$, $\sqrt{10}$.

Classify the following numbers by indicating which of the number sets ($\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{I}, \mathbb{R}$) they belong to. Note that a number can belong to more than one set.

• $A = -8$

• $B = 3,14159...$ (with infinitely many non-recurring digits)

• $C = \sqrt{25}$

• $D = 4,3\overline{5}$

• $E = \frac{\pi}{2}$

Express the following sets of real numbers using inequalities and visualize them on the real line:

• The interval $[-4, 7)$

• The ray $(-\infty, 3]$

• The neighbourhood $E(0, 5)$

Express the following sets of numbers defined by inequalities in interval notation:

• $\{x \in \mathbb{R} \mid -2 < x \leq 9\}$

• $\{x \in \mathbb{R} \mid x \geq -5\}$

• $\{x \in \mathbb{R} \mid |x| < 4\}$

Work out the distance on the real line between the following pairs of points using the absolute value:

• $A(-9)$ and $B(3)$

• $C(4,5)$ and $D(-2,1)$

• $E\left(\frac{1}{2}\right)$ and $F\left(\frac{7}{2}\right)$

State the lengths that the legs of a right-angled triangle must have to represent the following irrational numbers exactly on the real line, using Pythagoras' theorem:

• $\sqrt{10}$

• $\sqrt{17}$

• $\sqrt{2}$

Real numbers are dense, which means that between any two real numbers there are always infinitely many real numbers.

Find two rational numbers and two irrational numbers strictly between $5,1$ and $5,2$.

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

• The union of the set of rational numbers and irrational numbers forms the set of real numbers ($\mathbb{Q} \cup \mathbb{I} = \mathbb{R}$).

• Every decimal number can be expressed as a fraction.

• The set of integers is included in the set of natural numbers ($\mathbb{Z} \subset \mathbb{N}$).

Given the irrational number $\sqrt{11} \approx 3,31662479...$:

• Write down its approximation by truncation to three decimal places.

• Write down its approximation by rounding to three decimal places.

• Work out the absolute error made in the rounded approximation.

On a stretch of the A-6 dual carriageway with a speed limit set at $120\text{ km/h}$, mobile speed cameras of the Directorate-General for Traffic (DGT) apply a tolerance margin of $7\,\%$. This means that the speed camera does not issue a fine if the actual speed $v$ of the vehicle (in $\text{km/h}$) does not deviate from $120$ by more than $8,4\text{ km/h}$, which is expressed as $|v - 120| \le 8,4$.

1. a) Write as an interval the range of speeds at which a vehicle travels at or above the maximum permitted speed ($v \ge 120$), but without receiving a fine. Specify whether this interval is open, closed, or half-open and explain why.
2. b) Express the set of all speeds at which the driver would indeed receive a fine for speeding using ray notation.
3. c) If a van's tachograph records that it traveled at $128,1\text{ km/h}$, calculate the distance from this value to $120$ and reason whether the driver will be fined or not.

Lucía solved the following exercise in her Mathematics A exam: "Express using an absolute value inequality and as an interval the set of real numbers whose distance to the number $-4$ is less than or equal to $5$."

On her exam sheet, she wrote:

"Step 1: The distance to the number $-4$ is written as $|x - 4| \le 5$.

Step 2: I subtract and add $5$ to the number: $4 - 5 = -1$ and $4 + 5 = 9$.

Step 3: Therefore, the solution is the open interval $(-1, 9)$, since the distance cannot exceed that margin."

1. a) Identify and explain the THREE conceptual mistakes made by Lucía: one in the absolute value expression, another when calculating the endpoints due to the sign, and another in the type of interval chosen (parentheses vs. brackets).
2. b) Write the corrected absolute value inequality and calculate the correct closed interval, checking whether or not the number $-8$ belongs to the final solution by calculating its distance to $-4$.

An architect designs a rectangular window whose base measures $\sqrt{12} \text{ m}$ and whose height measures $\sqrt{3} \text{ m}$. Work out the perimeter and the exact area of the window, expressing the results in simplest form.

Express the set of real numbers whose distance from the origin is strictly greater than $5$ as the union of two intervals.

Given the following intervals and rays of real numbers:

$A = [-5, 4)$

$B = (1, 8]$

$C = (-\infty, 2]$

Work out and express as a single interval or ray the result of the following operations:

• $A \cup B$

• $A \cap B$

• $B \cap C$

In a biology laboratory, an enzyme only remains active if the temperature $T$ (in degrees Celsius) of the incubator strictly satisfies the condition of being at a distance from $37,5^\circ\text{C}$ that is less than $1,2^\circ\text{C}$.

• Express this condition using an inequality with an absolute value.

• Express the range of acceptable temperatures as a neighbourhood $E(a, r)$.

• Express the range of temperatures as an interval.

Plot the irrational number $\sqrt{8}$ geometrically and accurately on the real number line using the given scale. To do this:

1. a) Decompose the radicand into a sum of two perfect squares and state the lengths of the legs of the right triangle that you need to construct.

1. b) Detail the steps of your construction, explaining how you transfer the length of the hypotenuse onto the real number line using a compass.

A precision parts factory produces metal rods. The ideal length of the rod is $L = 150 \text{ mm}$. The quality control department rejects any rod whose length differs from the ideal by more than $0,8 \text{ mm}$.

• Express the acceptance condition using an absolute value inequality, where $x$ is the actual length of the rod.

• Write the set of acceptable lengths in interval form.

• If a rod measures $149,3 \text{ mm}$, will it be accepted or rejected?

Convert the following intervals into neighbourhood notation $E(a, r)$, determining their centre $a$ and radius $r$:

• $I_1 = (2, 10)$

• $I_2 = [-3,5; 7,5]$

• $I_3 = \left( \frac{1}{4}, \frac{5}{4} \right)$

Justify whether the following statements are always true, sometimes true or never true, providing an example in each case:

• The sum of two irrational numbers results in an irrational number.

• The product of a non-zero rational number and an irrational number is irrational.

• The square of an irrational number is a rational number.

The golden ratio, denoted by the Greek letter $\phi$ (phi), is an irrational number that frequently appears in art, architecture and nature. Its exact value is $\phi = \frac{1+\sqrt{5}}{2}$.

• Given that $\sqrt{5} \approx 2,23606$, work out the approximate value of $\phi$ rounded to four decimal places.

• Explain in detail the geometric steps you would follow to represent the number $\phi$ exactly on the real line.

Solve the following absolute value equation and represent its set of solutions on the real line:

$|3x - 6| = 12$

The State Meteorological Agency (AEMET) records in Molina de Aragón (Guadalajara) one of the greatest thermal ranges of the peninsula. For a day in January, the weather forecast sets a minimum temperature of $-5\text{ }^\circ\text{C}$ and a maximum of $9\text{ }^\circ\text{C}$.

1. a) Estimate: Without doing written calculations, estimate mentally what temperature $a$ represents the average value (center) of the day and by how many degrees $r$ the temperature rises or falls at most from that center. Write your prediction and explain the intuitive reasoning you followed.
2. b) Check and model: Check your estimate by calculating exactly the center $a$ and the radius $r$ of the interval $[-5, 9]$. Express the forecasted temperature range using neighborhood notation $E(a, r)$ and using an absolute value inequality of the form $|T - a| \le r$.
3. c) AEMET issues a severe frost warning if the temperature belongs to the ray $(-\infty; -3]\text{ }^\circ\text{C}$. Reason whether this warning will be activated during the day and calculate how many degrees of margin there are between the forecasted minimum and the warning threshold.

Design and justify the solutions to the following inverse challenges on the real number line:

1. a) Write an absolute value inequality of the form $|x - a| > r$ whose solution set coincides exactly with the union of rays $(-\infty; -3) \cup (7; +\infty)$. Explain how you deduced the value of the centre $a$ and the radius $r$ from the position of the endpoints on the number line.
2. b) Find a rational number (written as an irreducible fraction) and an irrational number (showing its expression with a root or its non-terminating, non-repeating decimal pattern) that both lie strictly between the decimal numbers $3,1$ and $3,2$. Justify why the first is rational and the second is not.
3. c) Write a quadratic equation of the form $x^2 = k$, where $k$ is an integer, whose positive solution is an irrational real number located strictly within the interval $(4, 5)$. Prove why that solution does not belong to the set $\mathbb{Q}$ of rational numbers.

Represent the irrational value $1 + \sqrt{5}$ exactly on the number line. Detail the step-by-step geometric construction process, justifying the measurements used.

Solve the following absolute value inequality and express the final solution as an interval or union of intervals on the real line: $|2x - 5| \leq 7$.

Prove by contradiction that the sum of a rational number $q$ and an irrational number $i$ always results in an irrational number. Apply this reasoning to formally justify why $2 + \sqrt{3}$ belongs to the set of irrational numbers.

In an electronic circuit, the equivalent resistance $R_{eq}$ of two resistors $R_1$ and $R_2$ connected in parallel is given by the formula $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2}$. If the values of the resistors are $R_1 = \sqrt{2} \text{ }\Omega$ and $R_2 = \sqrt{8} \text{ }\Omega$, work out the exact value of $R_{eq}$ by rationalising and fully simplifying the result.

The following figure shows a geometric procedure on the real line. Identify which exact irrational number is represented at point $P$ and justify your answer detailing each step of the construction.

A logistics distribution centre is to be located on a straight road connecting two cities $A$ and $B$, located at the kilometre markers $10$ and $40$ respectively. The sum of the distances from the centre to both cities must be exactly $40\text{ km}$. Set up an absolute value equation and determine at which kilometre markers $x$ the centre could be located.

Prove analytically that the intersection of the neighbourhoods $E(2; 3)$ and $E(4; 2)$ is another neighbourhood. Work out the centre and radius of this new resulting neighbourhood.

Find an irreducible fraction $\frac{a}{b}$ that approximates the irrational number $\sqrt{7}$ such that the absolute error made is strictly less than $0,05$. Express the error condition using absolute value inequalities.

Prove by contradiction that $\sqrt{3}$ is an irrational number. To do this, assume that $\sqrt{3}$ is rational, that is, it can be expressed as an irreducible fraction $\frac{p}{q}$ (where $p$ and $q$ are integers with no common factors). Square both sides and analyse the parity or divisibility by 3 of $p$ and $q$ to reach a contradiction.

Find the set of real numbers $x$ that simultaneously satisfy the following two conditions:

1. Their distance from the point $4$ is less than or equal to $5$.

2. Their distance from the point $-2$ is strictly greater than $3$.

Solve the corresponding absolute value inequalities and express the final solution as the union of intervals.

Represent the irrational number $\sqrt{13} - 2$ geometrically and accurately on the number line.

1. a) Explain the construction process, indicating at which point you would place the vertex of the right angle of the right-angled triangle and what the lengths of its legs would be.
2. b) Draw the graphical representation on the calibrated number line.

Consider the number $a = 1,9999...$ (with infinitely many nines in its decimal part) and the number $b = 2$.

• Express the number $a$ as a generating fraction.

• Based on the previous result, is there any real number strictly between $a$ and $b$? Justify your answer by reasoning about the density of real numbers.

An olive oil bottling plant in Jaén is programming its optical quality control system. Each bottle has a nominal volume of $750\text{ ml}$, and Spanish regulations tolerate a maximum deviation of $9\text{ ml}$ (above or below). Bottles that exceed this tolerance are automatically discarded.

To program the software filter, two engineers propose different criteria:

• Criterion A (Direct interval): Accept the bottle if its volume $V$ satisfies $V \in [741, 759]$; discard otherwise.

• Criterion B (Absolute value): Accept the bottle if $|V - 750| \le 9$; discard otherwise.

1. a) Prove with reasoning that both criteria are mathematically equivalent for any bottle analyzed.
2. b) If a sensor detects a bottle with $V = 759,2\text{ ml}$, describe how each criterion evaluates whether the bottle is accepted or rejected, interpreting the result according to the distance to the nominal value.
3. c) The factory decides to introduce a new $2000\text{ ml}$ jug format while maintaining the same absolute tolerance margin of $9\text{ ml}$. Which of the two criteria requires modifying fewer parameters in the program code and reduces the possibility of human calculation error when calibrating the machine? Justify your answer by analyzing the concepts of center and radius.

In a landscaping project in the Jardines del Turia (Valencia), a technical team must lay out two metal pergolas with millimeter precision on the ground. The technical specifications require exact lengths involving irrational numbers:

• The first pergola requires a diagonal beam of exact length $\sqrt{13}\text{ meters}$.

• The second pergola requires a diagonal beam of exact length $\sqrt{14}\text{ meters}$.

1. a) The team only has conventional measuring tapes and set squares. Explain how they can mark out on the ground the exact length of $\sqrt{13}\text{ m}$ using a single right triangle whose legs have integer lengths in meters. Justify the process applying the Pythagorean theorem.
2. b) For the $\sqrt{14}\text{ m}$ beam, a worker proposes looking for two integers whose squares sum to $14$. Prove with reasoning that it is impossible to find two positive integers $a$ and $b$ such that $a^2 + b^2 = 14$ by analyzing all perfect squares less than $14$. Then, explain how to construct the exact length $\sqrt{14}\text{ m}$ by chaining a second right triangle starting from the hypotenuse of $\sqrt{13}\text{ m}$.
3. c) For pedestrian clearance reasons, the $\sqrt{14}\text{ m}$ beam must fit into a usable space measuring between $3,7\text{ m}$ and $3,8\text{ m}$ (that is, in the interval $(3,7; 3,8)$). Without using the square root key on the calculator, check by squaring the endpoints whether the exact length $\sqrt{14}$ falls within that acceptable interval.