Rational Numbers on a Number Line

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Questions and Answers

Consider a mathematical system where the 'absolute value' of a number, denoted by $||x||$, is defined as the square root of the number squared. Which of the following implications is universally true?

If $||x|| = ||y||$, then $x = y$.
For all $x$ and $y$, $||x + y|| = ||x|| + ||y||$. This assertion is true if and only if $x$ and $y$ are either both zero or of opposite signs.
If $x < 0$, then $||x|| > x$.
For all $x$, $||x|| \geq 0$, and if $||x|| = 0$, then $x = 0$. Furthermore, if $||x + y|| = 0$, then $x = -y$. (correct)

Let $Q$ be the set of rational numbers. Suppose a new set $S$ is constructed by applying the transformation $f(x) = \frac{ax + b}{cx + d}$, where $a, b, c, d \in Q$ and $ad - bc \neq 0$, to every element $x$ in $Q$. Under what conditions is $S$ guaranteed to be a subset of $Q$?

$S$ is always a subset of $Q$ because the transformation involves only rational coefficients and operations.
$S$ is a subset of $Q$ if and only if for all $x \in Q$, $cx + d \neq 0$.
$S$ is a subset of $Q$ if the transformation $f$ is injective and surjective over $Q$.
$S$ is guaranteed to be a subset of $Q$ regardless of the values of $a, b, c, d$, as long as $ad - bc \neq 0$ and $cx + d \neq 0$ for all $x \in Q$ (correct)

Given the properties of rational numbers under standard arithmetic operations, which of the following statements is NOT necessarily true?

Every rational number has a multiplicative inverse that is also a rational number. (correct)
Between any two distinct rational numbers, there exists another rational number.
The set of rational numbers is closed under addition, subtraction, multiplication, and division (excluding division by zero).
The set of rational numbers is an ordered field.

Consider a scenario where two irrational numbers, $a$ and $b$, are subjected to the operation $a \star b = (a - b)^2$. Which of the following statements is most accurate regarding the result being a rational number?

The result can be rational even if $a \neq b$, contingent on specific values of $a$ and $b$ (B) Signup and view all the answers

Suppose a student is asked to represent the solution set of $|x| < a$ on a number line, where $a$ is a positive rational number. If the student incorrectly shades the regions $x < -a$ and $x > a$, what conceptual misunderstanding is most likely the cause?

The student confuses the 'and' and 'or' conditions in compound inequalities, interpreting $|x| < a$ as $x < -a$ or $x > a$ instead of $-a < x < a$. (A) Signup and view all the answers

In the context of rational numbers and their decimal representations, which of the following statements is universally true regarding the decimal expansion of a rational number $\frac{p}{q}$ (where $p$ and $q$ are integers and $q \neq 0$)?

The decimal expansion of $\frac{p}{q}$ either terminates or eventually repeats, and its repeating block can be at most $q - 1$ digits long. (D) Signup and view all the answers

Let $S$ be a set of rational numbers defined recursively as follows: $0 \in S$, and if $x \in S$, then $\frac{x}{2}$ and $x + 1$ are also in $S$. Which of the following statements accurately describes the nature of the elements in $S$?

$S$ contains only integers and dyadic rationals (rational numbers that can be expressed in the form $\frac{a}{2^b}$, where $a$ and $b$ are integers). (A) Signup and view all the answers

A student is asked to order the following rational numbers from least to greatest: $\frac{a}{b}, \frac{c}{d}, \frac{e}{f}$, where $a, c, e$ are positive integers and $b, d, f$ are distinct positive integers. If the student uses a common denominator approach but makes a mistake in calculating the least common multiple (LCM), what is the most likely consequence of this error?

The student may incorrectly order the numbers if their calculated common denominator is not a multiple of the true LCM, leading to inaccurate scaling of the numerators. (C) Signup and view all the answers

Consider the equation $|ax + b| = cx + d$, where $a, b, c, d$ are rational numbers. Which of the following conditions must be satisfied to guarantee that the equation has at least one real solution?

$c \geq |a|$ or $cx + d \geq 0$ for some $x$ that satisfies either $ax + b = cx + d$ or $ax + b = -(cx + d)$ (C) Signup and view all the answers

Given a set of rational numbers $S = {\frac{p}{q} \mid p, q \in \mathbb{Z}, q \neq 0}$, consider a binary operation $\oplus$ defined as $\frac{a}{b} \oplus \frac{c}{d} = \frac{ad + bc}{bd}$. Under what condition does $(S, \oplus)$ form a group?

$(S, \oplus)$ does not form a group because the operation is not associative. (B) Signup and view all the answers

Suppose you have a field $(F, +, )$ where $F$ is a set of rational numbers and $+$ and $$ are standard addition and multiplication. If a new operation $\star$ is defined such that for any $a, b \in F$, $a \star b = a + b + ab$, does $(F, +, \star)$ still maintain the properties of a field?

No, because distributivity of $\star$ over $+$ fails, violating a critical field axiom. (B) Signup and view all the answers

Consider two rational numbers, $x$ and $y$, such that $x \neq y$ and both are between 0 and 1. A new number $z$ is created by averaging their numerators and denominators separately; that is, if $x = \frac{a}{b}$ and $y = \frac{c}{d}$, then $z = \frac{a+c}{b+d}$. What can be definitively said about the relationship between $x, y,$ and $z$?

$z$ is always between $x$ and $y$: $\min(x, y) < z < \max(x, y)$. (D) Signup and view all the answers

A particular infinite series is defined as $\sum_{i=1}^{\infty} a_i$, where each $a_i$ is a positive rational number. Which of the following conditions is sufficient to guarantee that this series converges to a rational number?

Each $a_i$ is a term of a geometric sequence with a common ratio $r$ such that $0 < r < 1$ and $r$ is a rational number. (A) Signup and view all the answers

Consider a real number $x$ expressed in binary form as $x = (b_n b_{n-1} ... b_0 . b_{-1} b_{-2} ...)_2$, where each $b_i$ is either 0 or 1. Under what condition can you definitively conclude that $x$ is a rational number?

If the binary representation of $x$ is either finite or eventually periodic (repeating). (B) Signup and view all the answers

Define a "peculiar" rational number as one that can be expressed as a fraction $\frac{a}{b}$ where $a$ and $b$ are positive integers and $a + b = ab$. What is the sum of all such "peculiar" rational numbers?

2 (A) Signup and view all the answers

Let $f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0$ be a polynomial with rational coefficients $a_i$. If $x = r$ is a rational root of this polynomial, what condition relating the coefficients must hold true according to the Rational Root Theorem?

If $r = \frac{p}{q}$ where $p$ and $q$ are coprime integers, then $p$ must divide $a_0$ and $q$ must divide $a_n$. (D) Signup and view all the answers

Suppose a machine performs calculations using a floating-point representation with a finite number of bits. When adding two rational numbers, $x$ and $y$, the result is rounded to the nearest representable floating-point number. Under what conditions is the result of this operation guaranteed to be the same as the exact rational sum?

When the sum $x + y$ can be represented exactly within the machine's floating-point precision. (B) Signup and view all the answers

A student attempts to prove that the set of irrational numbers is closed under addition, by arguing that the sum of two irrational numbers is always irrational. Which of the following counterexamples definitively disproves the student's claim?

$\sqrt{2} + (-\sqrt{2}) = 0$, which is rational. (C) Signup and view all the answers

Let two rational numbers $x$ and $y$ be represented by repeating decimals $0.\overline{a}$ and $0.\overline{b}$ respectively, where $a$ and $b$ are single digits. If we define an operation $\star$ as the concatenation of the repeating blocks (i.e., $x \star y = 0.\overline{ab}$), is the result necessarily a rational number?

Yes, because any repeating decimal represents a rational number. (A) Signup and view all the answers

Consider a game where two players alternately choose distinct rational numbers from the interval $(0, 1)$. The game ends when the sum of all chosen numbers equals or exceeds 1. The player who chooses the last number wins. If both players play optimally, who will always win, and why?

The first player will always win, since they can choose a number arbitrarily close to 1 on their first turn. (C) Signup and view all the answers

Suppose a function $f : Q \rightarrow Q$ is defined such that $f(x + y) = f(x) + f(y)$ for all rational numbers $x$ and $y$. If $f(1) = c$, where $c$ is a rational number, what general form must $f(x)$ take?

$f(x) = cx$ (D) Signup and view all the answers

Given a construction using straightedge and compass, is it possible to construct a line segment whose length is $(\sqrt[3]{2})$ times the length of a given unit segment? Additionally, is $\sqrt[3]{2}$ a rational number and why?

No, it is impossible because doubling the cube is not constructible and $(\sqrt[3]{2})$ is irrational. $\sqrt[3]{2}$ is an irrational number because it cannot be expressed as a ratio of two integers. (C) Signup and view all the answers

Flashcards

Absolute value

Distance from zero on a number line, disregarding direction.

Absolute Value Equation

An equation that includes the absolute value of a variable.