Rational and Irrational Number Operations - Algebra 1

1. What special case makes $ri$ rational when $r$ is rational and $i$ is irrational?

   If $r=0$, then $ri=0$ is rational. Zero times any number equals zero, which is rational.

2. Identify the correct classification of $\sqrt{5}+\sqrt{5}$ written as $2\sqrt{5}$: rational or irrational?

   Irrational. $2\sqrt{5}$ is irrational.

3. Identify the correct conclusion: if $r$ is rational and $i$ is irrational, what is $i-r$?

   $i-r$ is irrational. Irrational minus rational is still irrational.

4. Identify the correct classification of $\left(\frac{7}{3}\right)\left(\frac{6}{14}\right)$: rational or irrational?

   Rational. Product of two rational numbers.

5. What is the definition of a rational number in fraction form?

   A number that can be written as $\frac{a}{b}$ with $a,b$ integers and $b \ne 0$. This is the standard mathematical definition using integers.

6. Which statement is always true about $\frac{m}{n}+\frac{p}{q}$ for integers $m,n,p,q$ with $nq \ne 0$?

   $\frac{m}{n}+\frac{p}{q}$ is rational. Sum of rational fractions is rational.

7. Identify the correct classification of $\left(\frac{1}{3}\right)\sqrt{6}$: rational or irrational?

   Irrational. Nonzero rational times irrational is irrational.

8. Identify the correct classification of $\sqrt{2}+\sqrt{2}$: rational or irrational?

   Irrational. $2\sqrt{2}$ is still irrational.

9. What is the closure property of rational numbers under multiplication?

   If $r$ and $s$ are rational, then $rs$ is rational. Rational numbers are closed under multiplication.

10. Identify the correct classification of $\frac{9}{10}-\sqrt{11}$: rational or irrational?

    Irrational. Rational minus irrational is irrational.

11. Which statement is always true: if $r$ and $s$ are rational, what is $rs$?

    $rs$ is rational. Product of two rationals is always rational.

12. Identify the correct classification of $\left(-\frac{3}{4}\right)+\sqrt{8}$: rational or irrational?

    Irrational. Rational plus irrational is irrational.

13. Identify the correct classification of $\frac{1}{4}+\sqrt{2}$: rational or irrational?

    Irrational. Rational plus irrational is irrational.

14. Which option must be true if $r$ is nonzero rational and $i$ is irrational: $ri$ is what type?

    Irrational. By the fundamental property for such products.

15. Which option must be true if $r$ is rational and $i$ is irrational: $r+i$ is what type?

    Irrational. By the fundamental property for such sums.

16. Identify the correct classification of $\left(\frac{3}{4}\right)\left(\sqrt{8}\right)$: rational or irrational?

    Irrational. Nonzero rational times irrational is irrational.

17. What is always true about $r+i$ if $r$ is rational and $i$ is irrational?

    $r+i$ is irrational. Adding rational to irrational gives irrational.

18. What is the correct conclusion if $r$ is rational and $r+i$ is irrational?

    $i$ can be irrational (this is consistent with the rule). This confirms the rule holds.

19. What is the fraction form of $\frac{a}{b}+\frac{c}{d}$ that shows it is rational?

    $\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}$. Shows sum as fraction with integer parts.

20. Identify the correct classification of $\pi+3$: rational or irrational?

    Irrational. Rational plus irrational is irrational.

21. Identify the correct classification of $\left(\frac{2}{3}\right)+\left(\frac{4}{9}\right)$: rational or irrational?

    Rational. Sum of two rational numbers.

22. Identify the correct conclusion: if $r$ is rational and $i$ is irrational, what is $-r+i$?

    $-r+i$ is irrational. Negative rational plus irrational is irrational.

23. Identify the correct classification of $\sqrt{13}+\left(-\frac{4}{9}\right)$: rational or irrational?

    Irrational. Irrational plus rational is irrational.

24. What is the fraction form of $\frac{a}{b}\cdot\frac{c}{d}$ that shows it is rational?

    $\frac{a}{b}\cdot\frac{c}{d}=\frac{ac}{bd}$. Shows product as fraction with integer parts.

25. What is the correct conclusion if $r$ is rational and $r+i$ is rational for some $i$?

    Then $i$ must be rational. If $r+i$ is rational, then $i = (r+i)-r$ is rational.

26. Identify the correct classification of $\sqrt{3}-\frac{11}{6}$: rational or irrational?

    Irrational. Irrational minus rational is irrational.

27. What is always true about $r-i$ if $r$ is rational and $i$ is irrational?

    $r-i$ is irrational. Subtracting irrational from rational gives irrational.

28. Identify the correct classification of $-2\sqrt{5}$: rational or irrational?

    Irrational. Nonzero rational times irrational is irrational.

29. Identify the correct classification of $\pi-\pi$: rational or irrational?

    Rational. Any number minus itself equals zero.

30. What is the closure property of rational numbers under addition?

    If $r$ and $s$ are rational, then $r+s$ is rational. Rational numbers are closed under addition.

31. What is the definition of an irrational number?

    A real number that cannot be written as $\frac{a}{b}$ with integers $a,b$ and $b \ne 0$. Cannot be expressed as a fraction of integers.

32. Identify the correct classification of $\left(\frac{1}{2}\right)\pi$: rational or irrational?

    Irrational. Nonzero rational times irrational is irrational.

33. Identify the correct classification of $\sqrt{2}-\sqrt{2}$: rational or irrational?

    Rational. Any number minus itself equals zero.

34. Identify the correct classification of $\frac{7}{9}+\frac{2}{3}$: rational or irrational?

    Rational. Sum of two rational numbers.

35. Identify the correct classification of $\left(\frac{3}{5}\right)\sqrt{7}$: rational or irrational?

    Irrational. Nonzero rational times irrational is irrational.

36. Which property justifies that $bd \ne 0$ if $b \ne 0$ and $d \ne 0$?

    Nonzero numbers have a nonzero product: $bd \ne 0$. Product of nonzero numbers is nonzero.

37. Identify the correct classification of $\frac{5}{6}\cdot 0$: rational or irrational?

    Rational. Any number times zero equals zero.

38. Identify the correct classification of $\left(\frac{-5}{8}\right)\left(\frac{12}{7}\right)$: rational or irrational?

    Rational. Product of two rational numbers.

39. Identify the correct classification of $\left(\frac{2}{3}\right)\left(\frac{4}{9}\right)$: rational or irrational?

    Rational. Product of two rational numbers.

40. Which statement is always true: if $r$ and $s$ are rational, what is $r+s$?

    $r+s$ is rational. Sum of two rationals is always rational.

41. What is the correct conclusion if $r$ is nonzero rational and $ri$ is irrational?

    $i$ can be irrational (this is consistent with the rule). This confirms the rule holds.

42. Which statement is always true about $\frac{m}{n}\cdot\frac{p}{q}$ for integers $m,n,p,q$ with $nq \ne 0$?

    $\frac{m}{n}\cdot\frac{p}{q}$ is rational. Product of rational fractions is rational.

43. Identify the correct classification of $0\cdot\sqrt{10}$: rational or irrational?

    Rational. Zero times any number equals zero.

44. What is always true about $ri$ if $r$ is nonzero rational and $i$ is irrational?

    $ri$ is irrational. Nonzero rational times irrational is irrational.

45. Which statement is always true: if $r$ is rational, what is $-r$?

    $-r$ is rational. Negative of a rational is rational.

46. Identify the correct classification of $\sqrt{6}+\left(\frac{1}{3}\right)$: rational or irrational?

    Irrational. Irrational plus rational is irrational.

47. Which statement is always true: if $i$ is irrational, what is $-i$?

    $-i$ is irrational. Negative of an irrational is irrational.

48. Identify the correct classification of $\frac{2}{7}+\left(-\frac{2}{7}\right)$: rational or irrational?

    Rational. Sum equals zero, which is rational.

49. Identify the correct classification of $\left(\frac{-7}{3}\right)\sqrt{2}$: rational or irrational?

    Irrational. Nonzero rational times irrational is irrational.

50. Identify the correct classification of $\left(\sqrt{2}\right)\left(\sqrt{2}\right)$: rational or irrational?

    Rational. $\sqrt{2} \cdot \sqrt{2} = 2$, which is rational.

51. What is the correct conclusion if $r$ is nonzero rational and $ri$ is rational for some $i$?

    Then $i$ must be rational. If $ri$ is rational, then $i = \frac{ri}{r}$ is rational.

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