Irrational numbers
You know that $\sqrt{4} = 2$ is exact. But what is the value of $\sqrt{2}$? The calculator gives $1,4142135\ldots$ and it never ends or repeats. This is not a limitation of the calculator: it can be proved that $\sqrt{2}$ can never be written as a fraction.
What an irrational number is
There are three typical families of irrational numbers at this level: inexact roots ($\sqrt{2}$, $\sqrt{7}$), constants such as $\pi$ or Euler's number $e$, and decimals deliberately constructed with no period, such as $0,1010010001\ldots$
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: a)…
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:
- a) $3,14159$
- b) $\sqrt{7}$
- c) $\frac{22}{7}$
- d) $1,1010010001...$
- e) $\sqrt{25}$
Identify the pattern followed by the decimal digits of the following number and explain why it is considered an irrational number: $x = 0,12345678910111213...$
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 irrationality of $\sqrt{2}$: a proof
This is one of the oldest proofs in mathematics (the ancient Greeks already knew it). It uses the method of proof by contradiction: we suppose the opposite of what we want to prove and arrive at a contradiction.
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…
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.
Geometric construction using Pythagoras' theorem
Although $\sqrt{2}$ is not a fraction, it does occupy an exact point on the number line: it can be constructed with a right-angled triangle and a pair of compasses.
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…
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.
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…
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?
Operating with irrational numbers
Operations between irrational numbers do not always give an irrational result: it depends on whether they "cancel" each other out.
Calculate the result of the following operations and classify the final result as rational or irrational: a) $\sqrt{2} \cdot \sqrt{8}$ b) $\sqrt{3} + \sqrt{3}$ c)…
Calculate the result of the following operations and classify the final result as rational or irrational:
- a) $\sqrt{2} \cdot \sqrt{8}$
- b) $\sqrt{3} + \sqrt{3}$
- c) $(\sqrt{5})^2$
- d) $\pi - \pi$
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…
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.
$\mathbb{R}$: rational and irrational numbers together
Just as $\mathbb{Q}$ is dense, so are the irrational numbers: between any two real numbers there are always infinitely many rational numbers and infinitely many irrational numbers. This is why the number line has no gaps: every point, rational or irrational, corresponds to a unique real number, and vice versa.
Order the following real numbers from smallest to largest. To do so, you may use their decimal approximations: $3,14$ , $\pi$ , $\sqrt{10}$ , $\frac{22}{7}$ , $3,1416$
Order the following real numbers from smallest to largest. To do so, you may use their decimal approximations:
$3,14$ , $\pi$ , $\sqrt{10}$ , $\frac{22}{7}$ , $3,1416$
Back to the beginning: what is the exact value of $\sqrt{2}$? Its exact value is the radical $\sqrt{2}$ itself (or the hypotenuse of a triangle with legs $1$ and $1$); any value with decimal digits such as $1{,}4142$ is merely an approximation.