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Questions and Answers
Consider the set $S$ of all rational numbers that can be expressed in the form $\frac{3m+1}{5n+2}$, where $m$ and $n$ are integers. Which of the following statements is necessarily true regarding the closure of $S$ under the standard operations of addition and multiplication?
Consider the set $S$ of all rational numbers that can be expressed in the form $\frac{3m+1}{5n+2}$, where $m$ and $n$ are integers. Which of the following statements is necessarily true regarding the closure of $S$ under the standard operations of addition and multiplication?
- $S$ is closed under multiplication but not under addition.
- $S$ is closed under neither addition nor multiplication. (correct)
- $S$ is closed under addition but not under multiplication.
- $S$ is closed under both addition and multiplication.
Given two rational numbers $a = \frac{p}{q}$ and $b = \frac{r}{s}$ where $p, q, r, s \in \mathbb{Z}$ and $q, s \neq 0$, under what condition is the expression $\frac{a+b}{a-b}$ undefined within the field of rational numbers?
Given two rational numbers $a = \frac{p}{q}$ and $b = \frac{r}{s}$ where $p, q, r, s \in \mathbb{Z}$ and $q, s \neq 0$, under what condition is the expression $\frac{a+b}{a-b}$ undefined within the field of rational numbers?
- $ps = -qr$
- $pr = qs$
- $pq = rs$
- $ps = qr$ (correct)
Let $f(x) = ax + b$ and $g(x) = cx + d$ be two linear functions where $a, b, c, d$ are non-zero rational numbers. Determine the condition under which the composition $f(g(x))$ results in a function whose range, when evaluated over all rational $x$, is a singleton set.
Let $f(x) = ax + b$ and $g(x) = cx + d$ be two linear functions where $a, b, c, d$ are non-zero rational numbers. Determine the condition under which the composition $f(g(x))$ results in a function whose range, when evaluated over all rational $x$, is a singleton set.
- $ac = 1$
- $a = 0$ (correct)
- $c = 0$
- $a = 1$
Suppose we define a novel operation $\star$ on rational numbers such that for any two rational numbers $x$ and $y$, $x \star y = \frac{x + y}{1 + xy}$. Under what condition, involving rational numbers $a$, $b$, and $c$, does the associative property hold for this operation, i.e., $(a \star b) \star c = a \star (b \star c)$?
Suppose we define a novel operation $\star$ on rational numbers such that for any two rational numbers $x$ and $y$, $x \star y = \frac{x + y}{1 + xy}$. Under what condition, involving rational numbers $a$, $b$, and $c$, does the associative property hold for this operation, i.e., $(a \star b) \star c = a \star (b \star c)$?
Consider the set $Q$ of rational numbers with the binary operation $\ast$ defined as $a \ast b = a + b - ab$ for all $a, b \in Q$. Which of the following statements accurately describes the algebraic structure $(Q, \ast)$?
Consider the set $Q$ of rational numbers with the binary operation $\ast$ defined as $a \ast b = a + b - ab$ for all $a, b \in Q$. Which of the following statements accurately describes the algebraic structure $(Q, \ast)$?
Define an equivalence relation $\sim$ on the set of ordered pairs of integers $\mathbb{Z} \times (\mathbb{Z} \setminus {0})$ such that $(a, b) \sim (c, d)$ if and only if $ad = bc$. Prove that this relation is transitive by showing that if $(a, b) \sim (c, d)$ and $(c, d) \sim (e, f)$, then $(a, b) \sim (e, f)$. However, which subtle assumption about zero is crucial to ensure the transitivity.
Define an equivalence relation $\sim$ on the set of ordered pairs of integers $\mathbb{Z} \times (\mathbb{Z} \setminus {0})$ such that $(a, b) \sim (c, d)$ if and only if $ad = bc$. Prove that this relation is transitive by showing that if $(a, b) \sim (c, d)$ and $(c, d) \sim (e, f)$, then $(a, b) \sim (e, f)$. However, which subtle assumption about zero is crucial to ensure the transitivity.
Let $S$ be the set of all positive rational numbers. A function $f: S \rightarrow S$ satisfies the conditions $f(x + 1) = f(x) + 1$ and $f(x^2) = (f(x))^2$ for all $x \in S$. Determine a general form of $f(x)$.
Let $S$ be the set of all positive rational numbers. A function $f: S \rightarrow S$ satisfies the conditions $f(x + 1) = f(x) + 1$ and $f(x^2) = (f(x))^2$ for all $x \in S$. Determine a general form of $f(x)$.
Consider the Stern-Brocot tree, a binary tree whose nodes contain rational numbers. Each node is generated by taking the mediant of its nearest ancestors. What is the fundamental property of the Stern-Brocot tree that ensures it contains all positive rational numbers exactly once in their lowest terms?
Consider the Stern-Brocot tree, a binary tree whose nodes contain rational numbers. Each node is generated by taking the mediant of its nearest ancestors. What is the fundamental property of the Stern-Brocot tree that ensures it contains all positive rational numbers exactly once in their lowest terms?
Let $a$ and $b$ be positive rational numbers. Which of the following statements correctly describes the sufficient and necessary conditions for $\sqrt{a} + \sqrt{b}$ to also be a rational number?
Let $a$ and $b$ be positive rational numbers. Which of the following statements correctly describes the sufficient and necessary conditions for $\sqrt{a} + \sqrt{b}$ to also be a rational number?
Define a 'denomophobic' rational number as one whose denominator, when expressed in lowest terms, is a prime number greater than 100. What is the probability that the sum of two randomly chosen rational numbers between 0 and 1 (inclusive) is denomophobic, assuming a uniform distribution of rational numbers?
Define a 'denomophobic' rational number as one whose denominator, when expressed in lowest terms, is a prime number greater than 100. What is the probability that the sum of two randomly chosen rational numbers between 0 and 1 (inclusive) is denomophobic, assuming a uniform distribution of rational numbers?
Flashcards
Rational Numbers
Rational Numbers
Numbers expressible as a fraction p/q, where p and q are integers and q ≠0.
Adding Rationals (same denominator)
Adding Rationals (same denominator)
Add numerators and keep the denominator the same: a/c + b/c = (a+b)/c.
Adding Rationals (different denominators)
Adding Rationals (different denominators)
Find a common denominator, then add numerators: a/b + c/d = (ad + bc)/bd.
Subtracting Rationals (same denominator)
Subtracting Rationals (same denominator)
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Subtracting Rationals (different denominators)
Subtracting Rationals (different denominators)
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Multiplying Rational Numbers
Multiplying Rational Numbers
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Dividing Rational Numbers
Dividing Rational Numbers
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Distributive Property
Distributive Property
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Simplest Form
Simplest Form
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Order of Operations
Order of Operations
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Study Notes
- Rational numbers can be expressed as a fraction p/q, where p and q are integers and q ≠0.
- Operations on rational numbers include addition, subtraction, multiplication, and division.
Addition
- To add rational numbers sharing a denominator, add the numerators while keeping the denominator: a/c + b/c = (a+b)/c.
- To add rational numbers with differing denominators, identify a common denominator, convert each fraction to this denominator, then add the numerators: a/b + c/d = (ad + bc)/bd.
- Example: 1/5 + 2/5 = (1+2)/5 = 3/5.
- Example: 1/2 + 1/3 = (3+2)/6 = 5/6.
Subtraction
- To subtract rational numbers sharing a denominator, subtract the numerators while keeping the denominator: a/c - b/c = (a-b)/c.
- To subtract rational numbers with differing denominators, identify a common denominator, convert each fraction to this denominator, and then subtract the numerators: a/b - c/d = (ad - bc)/bd.
- Example: 3/5 - 1/5 = (3-1)/5 = 2/5.
- Example: 1/2 - 1/3 = (3-2)/6 = 1/6.
Multiplication
- Multiply rational numbers by multiplying the numerators and the denominators: (a/b) * (c/d) = (ac)/(bd).
- Example: (1/2) * (2/3) = (12)/(23) = 2/6 = 1/3.
- Example: (3/4) * (1/5) = (31)/(45) = 3/20.
Division
- Divide rational numbers by multiplying by the reciprocal of the divisor: (a/b) / (c/d) = (a/b) * (d/c) = (ad)/(bc).
- Example: (1/2) / (2/3) = (1/2) * (3/2) = (13)/(22) = 3/4.
- Example: (3/4) / (1/5) = (3/4) * (5/1) = (35)/(41) = 15/4.
Properties of Operations
- Commutative Property: a + b = b + a and a * b = b * a, applicable to both addition and multiplication of rational numbers.
- Associative Property: (a + b) + c = a + (b + c) and (a * b) * c = a * (b * c), applicable to both addition and multiplication of rational numbers.
- Distributive Property: a * (b + c) = a * b + a * c, multiplication distributes over addition for rational numbers.
- Identity Property: a + 0 = a (0 is the additive identity), and a * 1 = a (1 is the multiplicative identity).
- Inverse Property: For addition, a + (-a) = 0; for multiplication, a * (1/a) = 1, where a ≠0.
- Closure Property: Performing addition, subtraction, multiplication, or division (excluding division by zero) on any two rational numbers results in another rational number.
Order of Operations
- Follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).
- Example: Evaluate (1/2 + 1/3) * (3/4 - 1/4).
- First, evaluate the expressions within the parentheses: (5/6) * (2/4).
- Then, multiply: (5/6) * (1/2) = 5/12.
Simplifying Rational Numbers
- A rational number is in its simplest form when the numerator and denominator share no common factors other than 1.
- To simplify a rational number, divide both the numerator and denominator by their greatest common divisor (GCD).
- Example: Simplify 4/6. The GCD of 4 and 6 is 2, so divide both by 2 to get 2/3.
- Example: Simplify 15/25. The GCD of 15 and 25 is 5, so divide both by 5 to get 3/5.
Converting Decimals to Rational Numbers
- Terminating decimals are converted to rational numbers by writing the decimal as a fraction with a power of 10 in the denominator.
- Simplify the fraction if simplification is possible.
- Example: 0.75 = 75/100 = 3/4.
- Example: 0.125 = 125/1000 = 1/8.
- Repeating decimals can also be converted to rational numbers using algebraic methods.
- Example: Convert 0.333... to a fraction.
- Let x = 0.333...
- Then 10x = 3.333...
- Subtract the first equation from the second: 10x - x = 3.333... - 0.333..., which simplifies to 9x = 3.
- Solving for x: x = 3/9 = 1/3.
Comparing Rational Numbers
- Compare rational numbers by finding a common denominator and comparing the numerators.
- For rational numbers a/b and c/d, where b and d are positive, then:
- ad > bc implies a/b > c/d.
- ad < bc implies a/b < c/d.
- ad = bc implies a/b = c/d.
- Example: Comparing 1/2 and 2/5, the common denominator is 10, so 1/2 = 5/10 and 2/5 = 4/10. Since 5 > 4, 1/2 > 2/5.
- Example: Comparing -1/4 and -1/3, the common denominator is 12, so -1/4 = -3/12 and -1/3 = -4/12. Since -3 > -4, -1/4 > -1/3.
Applications
- Rational numbers have several real-world applications:
- Measurement: Rational numbers are used to express length, weight, and time.
- Finance: Interest rates, fractions of a dollar, and stock prices use rational numbers.
- Cooking: Recipes use fractions to represent ingredient quantities.
- Construction, engineering and other fields use rational numbers for precision and accuracy.
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