Real numbers: a first look

Classify the following numbers by indicating which of the number sets ($\mathbb{N}, \mathbb{Z}, \mathbb{Q}, \mathbb{I}, \mathbb{R}$) they belong to. Note that a number can belong to more than one set.

• $A = -8$

• $B = 3,14159...$ (with infinitely many non-recurring digits)

• $C = \sqrt{25}$

• $D = 4,3\overline{5}$

• $E = \frac{\pi}{2}$

State whether the following statements are true or false, briefly justifying your answer:

• The union of the set of rational numbers and irrational numbers forms the set of real numbers ($\mathbb{Q} \cup \mathbb{I} = \mathbb{R}$).

• Every decimal number can be expressed as a fraction.

• The set of integers is included in the set of natural numbers ($\mathbb{Z} \subset \mathbb{N}$).

Express the following sets of real numbers using inequalities and visualize them on the real line:

• The interval $[-4, 7)$

• The ray $(-\infty, 3]$

• The neighbourhood $E(0, 5)$

Express the following sets of numbers defined by inequalities in interval notation:

• $\{x \in \mathbb{R} \mid -2 < x \leq 9\}$

• $\{x \in \mathbb{R} \mid x \geq -5\}$

• $\{x \in \mathbb{R} \mid |x| < 4\}$

Given the following intervals and rays of real numbers:

$A = [-5, 4)$

$B = (1, 8]$

$C = (-\infty, 2]$

Work out and express as a single interval or ray the result of the following operations:

• $A \cup B$

• $A \cap B$

• $B \cap C$

Work out the distance on the real line between the following pairs of points using the absolute value:

• $A(-9)$ and $B(3)$

• $C(4,5)$ and $D(-2,1)$

• $E\left(\frac{1}{2}\right)$ and $F\left(\frac{7}{2}\right)$

In a biology laboratory, an enzyme only remains active if the temperature $T$ (in degrees Celsius) of the incubator strictly satisfies the condition of being at a distance from $37,5^\circ\text{C}$ that is less than $1,2^\circ\text{C}$.

• Express this condition using an inequality with an absolute value.

• Express the range of acceptable temperatures as a neighbourhood $E(a, r)$.

• Express the range of temperatures as an interval.

Solve the following absolute value equation and represent its set of solutions on the real line:

$|3x - 6| = 12$

Express the set of real numbers whose distance from the origin is strictly greater than $5$ as the union of two intervals.

State the lengths that the legs of a right-angled triangle must have to represent the following irrational numbers exactly on the real line, using Pythagoras' theorem:

• $\sqrt{10}$

• $\sqrt{17}$

• $\sqrt{2}$

Plot the irrational number $\sqrt{8}$ geometrically and accurately on the real number line using the given scale. To do this:

1. a) Decompose the radicand into a sum of two perfect squares and state the lengths of the legs of the right triangle that you need to construct.

1. b) Detail the steps of your construction, explaining how you transfer the length of the hypotenuse onto the real number line using a compass.

The golden ratio, denoted by the Greek letter $\phi$ (phi), is an irrational number that frequently appears in art, architecture and nature. Its exact value is $\phi = \frac{1+\sqrt{5}}{2}$.

• Given that $\sqrt{5} \approx 2,23606$, work out the approximate value of $\phi$ rounded to four decimal places.

• Explain in detail the geometric steps you would follow to represent the number $\phi$ exactly on the real line.

Real numbers are dense, which means that between any two real numbers there are always infinitely many real numbers.

Find two rational numbers and two irrational numbers strictly between $5,1$ and $5,2$.

Given the irrational number $\sqrt{11} \approx 3,31662479...$:

• Write down its approximation by truncation to three decimal places.

• Write down its approximation by rounding to three decimal places.

• Work out the absolute error made in the rounded approximation.

Consider the number $a = 1,9999...$ (with infinitely many nines in its decimal part) and the number $b = 2$.

• Express the number $a$ as a generating fraction.

• Based on the previous result, is there any real number strictly between $a$ and $b$? Justify your answer by reasoning about the density of real numbers.

Justify whether the following statements are always true, sometimes true or never true, providing an example in each case:

• The sum of two irrational numbers results in an irrational number.

• The product of a non-zero rational number and an irrational number is irrational.

• The square of an irrational number is a rational number.

Prove by contradiction that $\sqrt{3}$ is an irrational number. To do this, assume that $\sqrt{3}$ is rational, that is, it can be expressed as an irreducible fraction $\frac{p}{q}$ (where $p$ and $q$ are integers with no common factors). Square both sides and analyse the parity or divisibility by 3 of $p$ and $q$ to reach a contradiction.