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 $\mathbb{I}$.

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.

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

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.

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

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.

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?

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 $b$.