Exercises: Irrational numbers

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

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 infinite 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.

To represent the irrational number $\sqrt{13}$ on the number line using the Pythagorean theorem, we need to find two integers whose squares add up to $13$. State what these two numbers are and draw a mental sketch of how you would form the right triangle.

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

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

$x = 0.12345678910111213...$

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 expression and write down the first six decimal digits. Is it a rational or an irrational number?

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

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

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

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

A soccer field has dimensions of $100$ ft long by $50$ ft 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 an irrational number? Approximate the result to two decimal places to find out how many feet of pipe they must buy.

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

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

Observe the following graphical representation on the number line. A right triangle has been constructed resting on the horizontal axis.

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

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

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

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

A technology company is designing a new television screen. They want the ratio of 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 happens with letter-sized paper). If the height of the screen is going to be $50$ in, what must its exact width be? Approximate the result to the nearest tenth of an inch.

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

In the design of a logo, an equilateral triangle inscribed in a circle of radius $R = 10$ in 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 the Pythagorean theorem to find the exact length of the side $L$. Show that the perimeter of the triangle is an irrational number.

A circular plot of area $100$ ft$^2$ and a square plot that also has an area of $100$ ft$^2$ are to be fenced.

1. a) Calculate the exact perimeter of both plots. (Remember that the area of a circle is $A = \pi r^2$).
2. b) If the fence costs $\$15$ per foot, which plot will be cheaper to fence? Use all the decimal places on your calculator to give the exact cost in dollars for each one.

Observe the following cube of side $a = 5$ in. The diagonal of one of its faces is called $d$, and the main diagonal that crosses the inside of the cube is called $D$.

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

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

Find two different irrational numbers whose sum is an integer rational number and show with simple algebraic operations that this is possible.

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

Calculate the exact value 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?

An investor analyzes the growth of a bacteria culture used to manufacture bioplastics. The population in millions follows the formula $P = 2 \cdot e^t$, where $t$ is the time in hours and $e$ is Euler's irrational number ($e \approx 2.71828$).

Calculate the estimated population at $3$ hours using the value of $e$ from your calculator. If by mistake the laboratory used the approximation $e = 2.7$, how many bacteria difference (in millions) will there be compared to the more precise calculation?

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

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

Consider the sequence of numbers formed by writing zeros and ones in the following way: we start with a $0.1$, then we 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 $N = 0.101001000100001...$

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