Irrational Numbers

Classify the following numbers, indicating which belong to the set of irrational numbers and which do not. Justify your answer based on their decimal expansion:

1. a) $3.14159$
2. b) $\sqrt{7}$
3. c) $\frac{22}{7}$
4. d) $1.1010010001...$
5. e) $\sqrt{25}$

Determine whether the following statements are true or false.

1. a) All decimal numbers with infinitely many digits are irrational.
2. b) The square root of any natural number is an irrational number.
3. c) The number $\pi$ can be written exactly as a fraction.
4. d) The sum of a rational number and an irrational number is always irrational.

Identify the pattern followed by the decimal digits of the following number and explain why it is considered an irrational number:

$x = 0.12345678910111213...$

Consider the sequence of digits formed by writing zeros and ones in the following manner: we start with a $0.1$, then add a $0$ and a $1$, then two $0$s and a $1$, then three $0$s and a $1$, and so on to infinity.

The resulting number is $N = 0.101001000100001...$

Analyse the structure of this number and justify with reasoning why $N$ belongs to the set of irrational numbers and cannot be expressed as a fraction $\frac{a}{b}$.

Truncate and round the irrational number $\sqrt{11} = 3.31662479...$ to two decimal places (hundredths) and three decimal places (thousandths). Organise the results in a table.

The golden ratio, represented by the Greek letter $\Phi$ (Phi), has an exact value of $\frac{1+\sqrt{5}}{2}$. Use your calculator to find its decimal expansion and write down the first six decimal places. Is it a rational or an irrational number?

In a machine parts factory, a circular plate with a radius of exactly $40$ cm must be cut. The automatic lathe requires the circumference of the circle to be entered to perform the cut.

1. a) What is the exact circumference of the part?
2. b) If the operator enters the approximate value using $\pi = 3.14$, what is the absolute error made compared to the actual value (using all digits of $\pi$ on the calculator)?

In a laboratory practical, the quantity of product obtained in a chemical reaction, in milligrams, follows the formula $P = 2 \times e^t$, where $t$ is the time in hours and $e$ is Euler's irrational number ($e \approx 2.71828$).

Calculate the estimated amount after $3$ hours using the value of $e$ on your calculator. If by mistake a student used the approximation $e = 2.7$, how many milligrams difference will there be compared to the more accurate calculation?

To represent the irrational number $\sqrt{13}$ on the number line using Pythagoras' theorem, we need to find two integers whose squares sum to $13$. State which two numbers these are and draw a mental sketch of how you would construct the right-angled triangle.

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

A soccer field has dimensions of $100$ m long by $50$ m wide. The maintenance team needs to lay an irrigation pipe that crosses the field diagonally. Calculate the exact length of the pipe using radicals. Is this length a rational or irrational number? Approximate the result to two decimal places to find out how many meters of pipe they should buy.

Look at the following graphical representation on the number line. A right-angled triangle has been constructed resting on the horizontal axis.

Calculate the exact value of the hypotenuse $h$. Which irrational number does the red point $X$ on the number line represent?

In baseball, the bases form a perfect square with sides of $90$ feet ($27.43$ meters). When the catcher throws the ball from home plate to second base to prevent a stolen base, the ball crosses the field diagonally. Calculate the exact distance in meters that the ball travels. Is it a rational or irrational quantity?

A technology company is designing a new television screen. They want the aspect ratio (width to height) to be exactly $\sqrt{2}$ so that, when dividing the screen in half, the two resulting rectangles maintain the same proportion (as with A4 paper sheets). If the height of the screen is to be $50$ cm, what must its exact width be? Approximate the result to the nearest millimeter.

In the design of a logo, an equilateral triangle inscribed in a circle of radius $R = 10$ cm is used, as shown in the figure.

Knowing that in an equilateral triangle the radius $R$ is twice the apothem $a$, calculate the exact value of the apothem $a$. Then, use Pythagoras' theorem to find the exact side length $L$. Show that the perimeter of the triangle is an irrational number.

Look at the following cube with side length $a = 5$ cm. The diagonal of one of its faces is called $d$, and the space diagonal that crosses through the inside of the cube is called $D$.

Calculate the exact value of the face diagonal of the base ($d$) and, using it as a leg together with the height of the cube, calculate the exact length of the space diagonal ($D$). Is it possible for a cube with integer edge length to have a space diagonal that is a rational number?

The famous "Spiral of Theodorus" is constructed by drawing adjacent right-angled triangles. The first has legs of length $1$. The hypotenuse of this triangle serves as a leg for the next triangle, whose other leg is again of length $1$, and so on.

Calculate the exact values of the hypotenuses $h_1$, $h_2$ and $h_3$. Following this logical pattern, what will be the exact value of the hypotenuse $h_8$ of the eighth triangle? Is it rational or irrational?

Calculate the result of the following operations and classify the final result as rational or irrational:

1. a) $\sqrt{2} \times \sqrt{8}$
2. b) $\sqrt{3} + \sqrt{3}$
3. c) $(\sqrt{5})^2$
4. d) $\pi - \pi$

The golden ratio is often used in art and architecture. A picture frame is designed to be a "golden rectangle". If the shorter side of the frame measures $20$ cm, the longer side must measure $20 \times \Phi$ cm, where $\Phi = \frac{1+\sqrt{5}}{2}$. Calculate the exact length of the longer side and its approximation in centimeters, rounded to one decimal place.

The period $T$ (in seconds) of a simple pendulum, that is, the time it takes to complete one full oscillation, is calculated using the formula $T = 2\pi\sqrt{\frac{L}{g}}$, where $L$ is the length of the string in meters and $g$ is the acceleration due to gravity ($9.8$ m/s$^2$). Calculate the exact period of a pendulum with a string length of $9.8$ meters. Is the result a rational or an irrational number?

To construct an artificial honeycomb, a beekeeper designs regular hexagonal cells. Knowing that the area of a regular hexagon of side length $L$ is calculated with the formula $A = \frac{3\sqrt{3}}{2} L^2$, calculate the exact area of a cell if its side length measures $4$ mm. Is it a rational or an irrational number?

In a mathematics class, the teacher sets the following challenge: "If we add another irrational number to an irrational number, is the result always irrational?".

Find two distinct irrational numbers whose sum is an integer, and show with simple algebraic operations that this is possible.

Order the following real numbers from smallest to largest. To do so, you may use their decimal approximations:

$3.14$ , $\pi$ , $\sqrt{10}$ , $\frac{22}{7}$ , $3.1416$

We want to fence a circular plot of land with an area of $100$ m$^2$ and a square plot that also has an area of $100$ m$^2$.

1. a) Calculate the exact perimeter of both plots. (Remember that the area of a circle is $A = \pi r^2$).
2. b) If fencing costs $300$ pesos per meter, which plot will be cheaper to fence? Use all decimal places on your calculator to give the exact cost in pesos for each one.

Knowing that the area of a circle is $A = \pi \times r^2$, imagine that we want to construct a square that has exactly the same area as a circle with a radius of $10$ cm. (This is the ancient mathematical problem of squaring the circle).

Find the exact expression for the side length of that square. Is it possible to measure that side length exactly in the real world using a ruler marked in millimeters? Justify your answer.