diff --git a/chapter_4/exercises.md b/chapter_4/exercises.md index 8b5fdff..0ce47d5 100644 --- a/chapter_4/exercises.md +++ b/chapter_4/exercises.md @@ -1857,3 +1857,241 @@ Thus there exists an nonnegative integer $n$ such that $2^{2n} + 1$ is not prime, and therefore the given statement is false. Q.E.D. + +--- + +**Exercise Set 4.3** + +Page 210 + +The numbers in 1-7 are all rational. Write each number as a ratio of two +integers. + +1. $-\dfrac{35}{6}$ + +2. $4.6037$ + +3. $\dfrac{4}{5} + \dfrac{2}{9}$ + +4. $0.37373737\dots$ + +5. $0.56565656\dots$ + +6. $320.5492492492\dots$ + +7. $52.4672167216721\dots$ + +8. The zero product property, says that if a product of two real numbers is $0$, + then one of the numbers must be $0$. + +a. Write this property formally using quantifiers and variables. + +b. Write the contrapositive of your answer to part (a). + +c. Write an informal version (without quantifier symbols or variables) for your +part to part (b). + +9. Assume that $a$ and $b$ are both integers and that $a \neq 0$ and $b \neq 0$. + Explain why $\dfrac{(b - a)}{(ab^2)}$ must be a rational number. + +10. Assume that $m$ and $n$ are both integers and that $n \neq 0$. Explain why + $\dfrac{(5m - 12n)}{(4n)}$ must be a rational number. + +11. Prove that every integer is a rational number. + +12. Let $S$ be the statement "The square of any rational number is rational." A + formal version of $S$ is "For every rational number $r$, $r^2$ is rational." + Fill in the blanks in the proof for $S$. + +**Proof:** + +Suppose that $r$ is __ (a) __. By definition of rational, $r = \dfrac{a}{b}$ for +some __ (b) __ with $b \neq 0$. By substitution, + +$$ r^2 = \text{\_\_ (c) \_\_} = \frac{a^2}{b^2} $$ + +Since $a$ and $b$ are both integers, so are the products $a^2$ and __ (d) __. +Also $b^2 \neq 0$ by the __ (e) __. Hence $r^2$ is a ratio of two integers with +a non-zero denominator,n and so __ (f) __ by definition of rational. + +13. Consider the following statement: The negative of any rational number is + rational. + +a. Write the statement formally using a quantifier and a variable. + +b. Determine whether the statement is true or false and justify your answer. + +14. Consider the statement: The cube of any rational number is a rational + number. + +a. Write the statement formally using a quantifier and a variable. + +b. Determine whether the statement is true or false and justify your answer. + +Determine which of the statements in 15-19 are true and which are false. Prove +each true statement directly from the definitions, and give a counterexample for +each false statement. For a statement that is false, determine whether a small +change would make it true. If so, make the change and prove the new statement. +Follow the directions for writing proofs on page 173. + +15. The product of any two rational numbers is a rational number. + +16. The quotient of any two rational numbers is a rational number. + +17. The difference of any two rational numbers is a rational number. + +18. If $r$ and $s$ are any two rational numbers, then $\dfrac{r + s}{2}$ is + rational. + +19. For all real numbers $a$ and $b$, if $a < b$ then + $a < \dfrac{a + b}{2} < b$. + + (You may use the properties of inequalities in T17-T27 of Appendix A.) + +20. Use the results of exercises 18 and 19 to prove that given any two rational + numbers $r$ and $s$ with $r < s$, there is another rational number between + $r$ and $s$. An important consequence is that there are infinitely many + rational numbers in between any two distinct rational numbers. See Section + 7.4. + +Use the properties of even and odd integers that are listed in Example 4.3.3 to +do exercises 21-23. Indicate which properties you use to justify your reasoning. + +21. True or false? If $m$ is any even integer and $n$ is any odd integer, then + $m^2 + 3n$ is odd. Explain. + +22. True or false? If $a$ is any odd integer, then $a^2 + a$ is even. Explain. + +23. True or false? If $k$ is any even integer and $m$ is any odd integer, then + $(k + 2)^2 - (m - 1)^2$ is even. Explain. + +Derive the statements in 24-26 as corollaries of Theorems 4.3.1, 4.3.2, and the +results of exercises 12, 13, 14, 15, and 17. + +24. For any rational numbers $r$ and $s$, $2r + 3s$ is rational. + +25. If $r$ is any rational number, then $3r^2 - 2r + 4$ is rational. + +26. For any rational number $s$, $5s^3 + 8s^2 - 7$ is rational. + +27. It is a fact that if $n$ is any nonnegative integer, then + +$$ 1 + \frac{1}{2} + \frac{1}{2^2} + \frac{1}{2^3} + \dots + \frac{1}{2^n} = \frac{1 - \left(\dfrac{1}{2^{n + 1}}\right)}{1 - \left(\dfrac{1}{2}\right)} $$ + +(A more general form of this statement is proved in Section 5.2.) Is the +right-hand side of this equation rational? If so, express it as a ratio of two +integers. + +28. Suppose $a$, $b$, $c$, and $d$ are integers and $a \neq c$. Suppose also + that $x$ is a real number that satisfies the equation + +$$ \frac{ax + b}{cs + d} = 1 $$ + +Must $x$ be rational? If so, express $x$ as a ratio of two integers. + +29. Suppose $a$, $b$, and $c$ are integers and $x$, $y$, and $z$ are nonzero + real numbers that satisfy the following equations: + +$$ \frac{xy}{x + y} = a \quad \text{ and } \quad \frac{xz}{x + z} = b \quad \text{ and } \quad \frac{yz}{y + z} = c $$ + +Is $x$ rational? If so, express it as ratio of two integers. + +30. Prove that if one solution for a quadratic equation of the form + $x^2 + bx + c = 0$ is rational (where $b$ and $c$ are rational), then the + other solution is also rational. (Use the fact that if the solutions of the + equation are $r$ and $s$, then $x^2 + bx + c = (x - r)(x - s)$.) + +31. Prove that if a real number $c$ satisfies a polynomial equation of the form + +$$ r_3x^3 + r_2x^2 + r_1x + r_0 = 0 $$ + +where $r_0$, $r_1$, $r_2$, and $r_3$ are rational numbers, then $c$ satisfies an +equation of the form + +$$ n_3x^3 + n_2x^2 + n_1x + n_0 = 0 $$ + +where $n_0$, $n_1$, $n_2$, and $n_3$ are integers. + +**Definition:** A number $c$ is called a **root** of a polynomial $p(x)$ if, and +only if, $p(c) = 0$. + +32. Prove that for every real number $c$, if $c$ is a root of a polynomial with + rational coefficients, then $c$ is a root of a polynomial with integer + coefficients. + +Use the properties of even and odd integers that are listed in Example 4.3.3 to +do exercises 33 and 34. + +33. When expressions of the form $(x - r)(x - s)$ are multiplied out, a + quadratic polynomial is obtained. For instance, + $(x - 2)(x - (-7)) = (x - 2)(x + 7) = x^2 + 5x - 14$. + +a. What can be said about the coefficients of the polynomial obtained by +multiplying out $(x - r)(x - s)$ when both $r$ and $s$ are odd integers? When +both $r$ and $s$ are even integers? When one of $r$ and $s$ is even and the +other odd? + +b. It follows from part (a) that $x^2 - 1253x + 255$ cannot be written as a +product of two polynomials with integer coefficients. Explain why this is so. + +34. Observe that + +$$ (x - r)(x -s)(x - t) = x^3 - (r + s + t)x^2 + (rs + rs + st)x - rst $$ + +a. Derive a result for cubic polynomials similar to the result in part (a) of +exercise 33 for quadratic polynomials. + +b. Can $x^3 + 7x^2 - 8x - 27$ be written as a product of three polynomials with +integer coefficients? Explain. + +In 35-39 find the mistakes in the "proofs" that the sum of any two rational +numbers is a rational number. + +35. + +**"Proof:** Any two rational numbers produce a rational number when added +together. So if $r$ and $s$ are particular but arbitrarily chosen rational +numbers, then $r + s$ is rational." + +36. + +**"Proof:** Let rational numbers $r = \dfrac{1}{4}$ and $s = \dfrac{1}{2}$ be +given. Then $r + s = \dfrac{1}{4} + \dfrac{1}{2} = \dfrac{3}{4}$, which is a +rational number. This is what was to be shown." + +37. + +**"Proof:** Suppose $r$ and $s$ are rational numbers. By definition of rational, +$r = \dfrac{a}{b}$ for some integers $a$ and $b$ with $b \neq 0$, and +$s = \dfrac{a}{b}$ for some integers $a$ and $b$ with $b \neq 0$. Then + +$$ r + s = \frac{a}{b} + \frac{a}{b} = \frac{2a}{b} $$ + +Let $p = 2a$. Then $p$ is an integer since it is a product of integers. Hence +$r + s = \dfrac{p}{b}$, where $p$ and $b$ are integers and $b \neq 0$. Thus +$r + s$ is a rational number by definition of rational. This is what was to be +shown." + +38. + +**"Proof:** Suppose $r$ and $s$ are rational numbers. Then $r = \dfrac{a}{b}$ +and $s = \dfrac{c}{d}$ for some integers $a$, $b$, $c$, and $d$ with $b \neq 0$ +and $d \neq 0$ (by definition of rational.) Then + +$$ r + s = \frac{a}{b} + \frac{c}{d} $$ + +But this is a sum of two fractions, which is a fraction. So $r - s$ is a +rational number since a rational number is a fraction." + +39. + +**"Proof:** Suppose $r$ and $s$ are rational numbers. If $r + s$ is rational, +then by definition of rational $r + s = \dfrac{a}{b}$ for some integers $a$ and +$b$ with $b \neq 0$. Also since $r$ and $s$ are rational, $r = \dfrac{i}{j}$ and +$s = \dfrac{m}{n}$ for some integers $i$, $j$, $m$, and $n$ with $j \neq 0$ and +$n \neq 0$. It follows that + +$$ r + s = \frac{i}{j} + \frac{m}{n} = \frac{a}{b} $$ + +which is a quotient of two integers with a nonzero denominator. Hence it is a +rational number. This is what is to be shown. diff --git a/chapter_4/notes.md b/chapter_4/notes.md index 499bb45..726a991 100644 --- a/chapter_4/notes.md +++ b/chapter_4/notes.md @@ -111,3 +111,67 @@ Page 196 Personal Note: The entirety of 4.2 is extremely helpful in breaking down in exactly how to write proofs (for beginners). I'd advise revisiting this entire section frequently. + +--- + +Page 206 + +**Definition** + +A real number $r$ is **rational** if, and only if, it can be expressed as a +quotient of two integers with a nonzero denominator. A real number that is not +rational is **irrational**. More formally, if $r$ is a real number, then + +$$ r \text{ is rational } \Leftrightarrow \exists \text{ integers } a \text{ and } b \text{ such that } r = \frac{a}{b} \text{ and } b \neq 0 $$ + +--- + +Page 207 + +**Zero Product Property** + +If neither of two real numbers is zero, then their product is also not zero. + +--- + +Page 208 + +**Theorem 4.3.1** + +Every integer is a rational number. + +--- + +Page 209 + +**Theorem 4.3.2** + +The sum of any two rational numbers is rational. + +**Proof:** + +Suppose $r$ and $s$ are any rational numbers. _[We must show that $r + s$ is +rational.]_ Then, by definition of rational, $r = \dfrac{a}{b}$ and +$s = \dfrac{c}{d}$ for some integers $a$, $b$, $c$, and $d$ with $b \neq 0$ and +$d \neq 0$. Thus + +$$ r + s = \frac{a}{b} + \frac{c}{d} \quad \text{ by substitution} $$ + +$$ \quad = \frac{ad + bc}{bd} \quad \text{ by basic algebra} $$ + +Let $p = ad + bc$ and $q = bd$. Then $p$ and $q$ are integers because products +and sums of integers are integers and because $a$, $b$, $c$, and $d$ are +integers. Also $q \neq 0$ by the zero product property. Thus + +$$ r + s = \frac{p}{q} \text{ where } p \text{ and } q \text{ are integers and } q \neq 0 $$ + +Therefore, $r + s$ is rational by the definition of a rational number _[as was +to be shown]_. + +--- + +Page 210 + +**Corollary 4.2.3** + +The double of a rational number is rational. diff --git a/chapter_4/test_yourself.md b/chapter_4/test_yourself.md index 7f96957..29759e9 100644 --- a/chapter_4/test_yourself.md +++ b/chapter_4/test_yourself.md @@ -82,3 +82,16 @@ arguing from examples; using the same letter to mean two different things; jumping to a conclusion; assuming what is to be proved; confusion between what is known and what is still to be shown; use of _any_ when the correct word is _some_; misuse of the word _if_ + +--- + +**Test Yourself** + +Page 210 + +1. To show that a real number is rational, we must show that we can write it as + ______. + +2. An irrational number is a ______ that is ______. + +3. Zero is a rational number because ______.