Exercises: Number sets

Classify the following numbers by indicating which of the number sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$ or $\mathbb{R}$ they belong to (bear in mind that a number can belong to several sets at the same time):

$-5$, $3.14$, $\sqrt{16}$, $\sqrt{7}$, $0.\widehat{3}$, $\frac{4}{2}$.

Find the irreducible generating fraction of the following decimal numbers:

1. a) $2.45$
2. b) $1.\widehat{6}$
3. c) $0.1\widehat{25}$

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

1. a) Every integer is a rational number.
2. b) The number $0$ is an irrational number.
3. c) All decimal numbers can be expressed as a fraction.

Write as an interval and represent graphically on the real line the following set of numbers:

$A = \{x \in \mathbb{R} \mid -2 < x \le 5\}$

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

$3.14$, $\pi$, $\sqrt{10}$, $3.1\widehat{4}$, $\frac{22}{7}$

Find a rational number and an irrational number that are strictly between $1.5$ and $1.6$.

Given the intervals $A = [-3, 4)$ and $B = (0, 7]$, determine as an interval the result of the following operations:

1. a) $A \cup B$
2. b) $A \cap B$

Determine exactly which integers belong to the interval $(-2.5, 3]$. Express the result using set notation with curly brackets.

In a Physics class, different quantities are measured or used for an experiment: the mass of a particle $m = 2.5 \cdot 10^{-3}$ pounds, the period of a pendulum $T = 2\pi$ s, the number of oscillations $n = 15$, the change in temperature $\Delta \theta = -4^\circ$°F and the energy $E = \sqrt{2}$ J. Classify the exact numerical values of these measurements according to the most restrictive (smallest) number set they belong to ($\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{I}$).

A company divides a profit of $\$1000$ equally among $3$ partners. The accounting software truncates the result and transfers $\$333.33$ to each account.

1. a) Express the exact amount corresponding to each partner using fractions.
2. b) Calculate the total absolute error made by the company when using the decimal approximation for the three partners instead of the exact value.

Observe the Venn diagram representing the number sets. Place the following numbers in the most restrictive region that corresponds to them:

$a = \frac{\sqrt{18}}{\sqrt{2}}$, $b = -2.1\widehat{5}$, $c = \frac{\pi}{2}$, $d = \sqrt[3]{-8}$, $e = 1.010010001\dots$

In the design of a microchip, the allowable tolerance for the thickness of a silicon track is given by the set $T = \{x \in \mathbb{R} \mid |x - 1.25| \le 0.05\}$ in micrometers.

1. a) Express this set as an interval.
2. b) Determine whether the thicknesses $x_1 = 1.30$ and $x_2 = 1.25 + \frac{\sqrt{2}}{100}$ meet the required tolerance.

A chemical reaction is viable if the temperature $T$ (in $^\circ$°F) belongs to the interval $A = [-10, 45)$ and, simultaneously, the pressure forces a temperature control parameter $k = \frac{T}{10}$ to belong to the interval $B = (0, 5]$. Calculate the exact interval of temperatures $T$ for which the reaction is viable, satisfying both conditions at the same time.

An architect calculates the length of a wall by adding two sections whose theoretical measurements are $L_1 = 2.\widehat{3}$ ft and $L_2 = 1.1\widehat{6}$ ft.

1. a) Calculate the exact total length of the wall by operating with the generating fractions of both numbers.
2. b) Express the final result as a decimal number.

Without using a calculator, determine whether the result of the following operations is a rational or irrational number by expanding the algebraic expressions:

1. a) $(\sqrt{2} - 1)(\sqrt{2} + 1)$
2. b) $\sqrt{3} + \sqrt{12}$
3. c) $\frac{\sqrt{50}}{\sqrt{2}}$

The density property of rational numbers states that between any two distinct rational numbers there is always another rational number. Prove this property by finding three distinct rational numbers strictly between $\frac{4}{7}$ and $\frac{5}{7}$.

Prove, using proof by contradiction, that $\sqrt{3}$ is an irrational number. (Assume that $\sqrt{3} = \frac{p}{q}$ where $p$ and $q$ are coprime integers, and reach a contradiction).

Given the sets defined on the real line $A = (-\infty, 3k-1]$ and $B = [k+5, \infty)$, determine the value of the parameter $k \in \mathbb{R}$ for which the intersection $A \cap B$ consists of a single real number. Classify this number in the sets $\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{I}$.

The figure shows the geometric construction of a number on the real line using a right-angled triangle and an arc of a circle.

1. a) Identify which exact real number the point $P$ represents on the horizontal axis.
2. b) Classify this number in the number sets.
3. c) Prove its value analytically by applying Pythagoras' theorem.

The golden ratio is defined as $\phi = \frac{1+\sqrt{5}}{2}$.

1. a) Prove algebraically that $\phi$ satisfies the equation $\phi^2 - \phi - 1 = 0$.
2. b) From this result, deduce whether the number $\frac{1}{\phi}$ is rational or irrational and express it as a binomial in terms of $\phi$.

Consider the real number $x = \sqrt{7 - 4\sqrt{3}} + \sqrt{3}$.

Simplify the expression fully (hint: try to express the radicand $7 - 4\sqrt{3}$ as a perfect square binomial) and determine which number sets ($\mathbb{N}$, $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{I}$) the exact value of $x$ belongs to.

Given the sets of real numbers $A = \{x \in \mathbb{R} \mid x^2 - 5x + 6 \le 0\}$ and $B = \{x \in \mathbb{R} \mid |x - 2| < 1\}$, find the resulting sets of the operations $A \cup B$ and $A \cap B$. Express the final results as an interval.

A sports analyst models an athlete's optimal performance through the intersection of three reaction time intervals (in seconds) obtained from different biomechanical studies:

$I_1 = [0.15, 0.25]$

$I_2 = (0.18, 0.30)$

$I_3 = [0.20, \infty)$

1. a) Calculate the resulting interval of the intersection $I_1 \cap I_2 \cap I_3$.
2. b) If the absolute ideal reaction time must be an irrational number strictly contained within that intersection, propose an exact value (using roots) that meets the condition.

Knowing that between any two real numbers there are infinitely many rational and irrational numbers, construct an irrational number $x$ that simultaneously meets the following two conditions:

1. a) $0.01 < x < 0.02$
2. b) In its infinite decimal expansion, only the digits $0$ and $1$ appear.

Write at least the first 12 decimal places of your number to make clear the formation rule that guarantees its irrationality.