Exercises: Irrational numbers

Classify the following numbers as rational or irrational, justifying your answer based on their decimal expansion:

• $A = 3,14159265\dots$ (without any repeating pattern)

• $B = -2,33333\dots$

• $C = \sqrt{16}$

• $D = \sqrt{15}$

• $E = 2,121121112\dots$

State whether the following statements are true or false. If they are false, give a counterexample to prove it.

1. a) The sum of two irrational numbers is always an irrational number.
2. b) All numbers with infinitely many decimal places are irrational.
3. c) The number $\pi$ can be written exactly as a fraction of two integers.
4. d) The set of real numbers $\mathbb{R}$ is formed by the union of the rational numbers $\mathbb{Q}$ and the irrational numbers $\mathbb{I}$.

Invent and write down the decimal expansion of two distinct irrational numbers that lie strictly between the values $7,4$ and $7,5$.

Classify the exact result of the following operations as a rational or irrational number, carrying out the calculation first:

1. a) $\sqrt{2} \cdot \sqrt{8}$
2. b) $\sqrt{3} + \sqrt{3}$
3. c) $\frac{\sqrt{12}}{\sqrt{3}}$
4. d) $\sqrt{5} \cdot \sqrt{2}$

Match each of the following famous irrational numbers with its mathematical symbol and its approximate value to three decimal places:

• Numbers: Golden ratio, Pi, Euler's number.

• Symbols: $\pi$, $e$, $\phi$.

• Values: $2,718\dots$, $1,618\dots$, $3,141\dots$.

Determine which number sets ($\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{I}$, $\mathbb{R}$) the number $-\sqrt{81}$ belongs to. Write down all the sets it is a member of.

Write down an irrational number expressed as a square root that lies between the fractions $\frac{1}{2}$ and $\frac{3}{4}$.

Without using a calculator, find out whether the number $\sqrt{250}$ is rational or irrational. Justify your reasoning by bounding it between the nearest perfect squares.

In a 4th-year ESO class, Marcos states out loud during homework correction: “The number $\sqrt{18}$ is rational because $18$ is an even number and, moreover, if you calculate it with the calculator it gives $4,242640687$, which ends in $7$, so it is a terminating decimal.” Identify the two conceptual mistakes made by Marcos, explain in detail why his reasoning is incorrect in each case, and classify $\sqrt{18}$ into the corresponding set ($\mathbb{Q}$ or $\mathbb{I}$), justifying your decision based on the concept of perfect square.

The City Council of Seville has marked out square plots for community urban gardens in the Parque del Alamillo. Two adjacent plots have areas of $36\text{ m}^2$ and $20\text{ m}^2$, respectively. To order the perimeter fencing, the technicians need to know the exact side length of each plot.

1. a) Express the exact side length of each plot using a radical and classify each of the two values as rational ($\mathbb{Q}$) or irrational ($\mathbb{I}$), justifying your answer based on whether they are perfect squares or not.
2. b) The maintenance supervisor claims that he can buy a standard tape measure and measure the side of the $20\text{ m}^2$ plot so that the reading matches the end of the fence exactly, with nothing lacking or left over. Explain with sound reasoning why this is impossible from a mathematical point of view.

Plot the number $\sqrt{10}$ accurately on the real number line.

1. a) Use the Pythagorean theorem to find two natural numbers that can serve as the legs of a right triangle whose hypotenuse has length $\sqrt{10}$.
2. b) Describe the steps of the geometric construction using a ruler and compass to locate $\sqrt{10}$ on the real number line.

The standard format of credit cards and ID cards is close in design to a "golden rectangle". If the shorter side of an ID card measures $53,98$ mm, calculate the exact length of the longer side using the golden ratio $\phi = \frac{1+\sqrt{5}}{2}$. Is the exact result a rational or an irrational number?

The period $T$ of a simple pendulum (the time it takes to complete one full oscillation back and forth) is given by the physics formula $T = 2\pi \sqrt{\frac{L}{g}}$, where $L$ is the length of the string and $g$ is the acceleration due to gravity. Calculate the exact period of a pendulum $10$ metres long assuming $g = 10 \text{ m/s}^2$. Is the time obtained a rational or an irrational number?

Find a rational number and an irrational number that lie strictly between the decimal approximations $1,41$ and the exact value of $\sqrt{2}$. Justify your answer.

Carry out the following algebraic calculation using standard identities and classify the final result by stating which number sets it belongs to:

$(\sqrt{7} - \sqrt{3})(\sqrt{7} + \sqrt{3}) + (\sqrt{2})^2$

Plot the irrational number $1 + \sqrt{5}$ on the real number line. To do this:

1. a) Detail which right triangle you use for your construction, justifying the length of its hypotenuse using the Pythagorean theorem.
2. b) Indicate from which point on the number line you should begin drawing the arc to represent the sum $1 + \sqrt{5}$.
3. c) Draw the graphical representation on the real number line.

You can do it on paper and upload a photo of your answer.

A football pitch has official dimensions of $105$ m long by $68$ m wide. Calculate the exact length of its diagonal in metres. Is it a rational or an irrational number? If it is irrational, state between which two consecutive integers it lies.

A regular hexagon is inscribed in a circle of radius $r = 5$ cm. Calculate the exact perimeter of the hexagon and the exact circumference of the circle. Prove that the ratio of the circumference of the circle to the perimeter of the hexagon is an irrational number independent of the radius.

According to the official regulations of the Spanish Basketball Federation (FEB), the regulatory court is a rectangle $28\text{ m}$ long by $15\text{ m}$ wide. During a physical test, referees must sprint in a straight line from one corner to the opposite corner (along the diagonal).

1. a) Apply the Pythagorean theorem to set up the exact expression for the length of that diagonal and justify why this distance is an irrational number ($\mathbb{I}$).
2. b) Without calculating the square root with decimals, estimate with sound reasoning between which two consecutive integer numbers of metres the length of that diagonal lies. Show the reference squares you use to bound the result.
3. c) A GPS heart rate monitor records that the diagonal covered measures $31,76\text{ m}$. Assess whether this result is consistent with your previous estimate and explain why the device's reading is necessarily a rational approximation and not the exact real distance.

Solve the following questions by constructing the requested numbers and justifying your procedure:

1. a) Write the decimal expression of an irrational number strictly between the rational numbers $2,45$ and $2,46$. To guarantee that it is irrational, describe the infinite non-repeating rule or pattern with which you generate its decimal digits.
2. b) Find a natural number $n$ between $40$ and $55$ such that $\sqrt{n}$ is a rational number, and another natural number $m$ in that same interval such that $\sqrt{m}$ is irrational. Justify both choices based on the definition of a perfect square.
3. c) Design a multiplication operation between two different irrational numbers, $x$ and $y$, whose result is an integer. Explain why the product of two irrationals is not always irrational.

Prove by contradiction that $\sqrt{2}$ is an irrational number. To do this, assume that it can be written as an irreducible fraction $\frac{a}{b}$ (where $a$ and $b$ have no common factors) and reach a contradiction by analysing the parity (whether they are even or odd) of $a$ and $b$.

A golden rectangle has the mathematical property that if a square formed by its shorter side is removed, the remaining rectangle is similar to the original rectangle. If the shorter side measures $1$ and the longer side measures $x$, set up the similarity equation $\frac{x}{1} = \frac{1}{x-1}$. Solve it and show that the positive solution is exactly the irrational number $\phi$.

Find the rational values of the parameters $a$ and $b$ so that the following equality holds:

$(a + 2\sqrt{5})(3 - b\sqrt{5}) = 11 - \sqrt{5}$

Plot the irrational number $\sqrt{7}$ accurately on the real number line. Since $7$ cannot be expressed as the sum of two perfect squares of integers, the Pythagorean theorem cannot be applied directly with integer legs. Instead, use the Geometric Mean Theorem (Altitude Theorem) by constructing a semicircle on the real number line, as illustrated in the conceptual diagram in the figure.

1. a) Justify how to decompose $7$ into the product of two positive integer factors, $m$ and $n$, so that the altitude to the hypotenuse has length $h = \sqrt{7}$.
2. b) If you place the junction point between both segments at the origin ($0$), determine the coordinates of the two endpoints of the diameter, the position of the centre of the semicircle on the axis, and the length of its radius.
3. c) Describe in detail the geometric steps required to draw the semicircle, construct the altitude, and transfer the exact measure of $\sqrt{7}$ onto the positive semi-axis using a compass.

Prove using counterexamples that the set of irrational numbers ($\mathbb{I}$) is not closed under the four basic arithmetic operations. That is, find two irrational numbers such that their sum, subtraction, multiplication and division each yield a rational number.

Find two irrational numbers $x$ and $y$ such that the power $x^y$ is a rational number.

The space diagonal of a cube (the segment joining two opposite vertices passing through the centre of the cube) measures exactly $\sqrt{12}$ cm. Calculate the exact total surface area and volume of the cube. Classify the final results obtained as rational or irrational numbers.

Determine for which positive integer values of $n$ the expression $\sqrt{n^2 + 5n + 6}$ is a rational number.

To find out whether the real number $N = \sqrt{11 - 6\sqrt{2}} + \sqrt{2}$ is rational or irrational, two 4th-year ESO students propose different strategies:

• David's strategy: He enters the expression into the scientific calculator, gets the value $3$ on the screen, and concludes that the number is rational because the result shows no decimal digits.

• Sara's strategy: She suspects that the radicand $11 - 6\sqrt{2}$ comes from expanding the square of a difference. She algebraically calculates $(3 - \sqrt{2})^2$ to check it and simplifies the nested radical before classifying the number.

1. a) Critically analyse both strategies: explain why David's calculator does not serve as a rigorous mathematical proof in this type of problem and what conceptual advantage Sara's method provides.
2. b) Work out the calculations for Sara's strategy to obtain the exact and simplified value of $N$. Classify it definitively into $\mathbb{Q}$ or $\mathbb{I}$.
3. c) Based on what you observed, state a general conclusion about the risk of classifying a number with radicals before completely simplifying its expression.

A company installs triangular supports for solar panels in an industrial warehouse in Toledo following the tilt guidelines of the IDAE (Institute for the Diversification and Saving of Energy). Each side support is shaped like a right triangle, where the horizontal base measures exactly $1,6\text{ m}$ and the inclined beam (hypotenuse) measures $2,0\text{ m}$.

1. a) Model the situation by calculating the exact vertical height $h$ of the support in metres (in radical form). Express the radicand as an irreducible fraction and prove with sound reasoning whether this height $h$ is a rational ($\mathbb{Q}$) or irrational ($\mathbb{I}$) number.
2. b) A worker states: “Since the starting data $1,6$ and $2,0$ are terminating decimals with a single decimal digit, the height $h$ must also be a terminating decimal: exactly $1,2\text{ m}$”. Prove algebraically whether $h = 1,2\text{ m}$ is the exact value or merely an incorrect approximation.
3. c) The municipal urban planning regulations require that the vertical height of the structure must not exceed $\sqrt{1,5}\text{ m}$. Without resorting to an approximate decimal calculation with a calculator, determine with sound reasoning whether the structure complies with the regulation by directly comparing the squares of both measurements.