Algebra Flashcards: Rational And Irrational Number Operations

What this deck covers

This deck focuses on Rational And Irrational Number Operations, giving you a quick way to review the definitions, rules, and examples that matter most for Algebra.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

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

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

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

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

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

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

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

Answer: Rational. Product of two rational numbers.

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

Answer: 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.

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

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

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

Answer: Irrational. Nonzero rational times irrational is irrational.

Flashcard 8: Identify the correct classification of $\left(\frac{2}{3}\right)\left(\frac{4}{9}\right)$: rational or irrational?

Answer: Rational. Product of two rational numbers.

Flashcard 9: Identify the correct conclusion: if $r$ is rational and $i$ is irrational, what is $-r+i$?

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

Flashcard 10: Identify the correct classification of $\sqrt{2}+\sqrt{2}$: rational or irrational?

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

Flashcard 11: What is the closure property of rational numbers under multiplication?

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

Flashcard 12: Identify the correct classification of $\frac{9}{10}-\sqrt{11}$: rational or irrational?

Answer: Irrational. Rational minus irrational is irrational.

Flashcard 13: Which statement is always true: if $r$ and $s$ are rational, what is $rs$?

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

Flashcard 14: Identify the correct classification of $\left(-\frac{3}{4}\right)+\sqrt{8}$: rational or irrational?

Answer: Irrational. Rational plus irrational is irrational.

Flashcard 15: What is the fraction form of $\frac{a}{b}+\frac{c}{d}$ that shows it is rational?

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

Flashcard 16: Identify the correct classification of $\frac{1}{4}+\sqrt{2}$: rational or irrational?

Answer: Irrational. Rational plus irrational is irrational.

Flashcard 17: What is the definition of an irrational number?

Answer: 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.

Flashcard 18: Which option must be true if $r$ is nonzero rational and $i$ is irrational: $ri$ is what type?

Answer: Irrational. By the fundamental property for such products.

Flashcard 19: Which option must be true if $r$ is rational and $i$ is irrational: $r+i$ is what type?

Answer: Irrational. By the fundamental property for such sums.

Flashcard 20: Identify the correct classification of $\left(\frac{3}{4}\right)\left(\sqrt{8}\right)$: rational or irrational?

Answer: Irrational. Nonzero rational times irrational is irrational.

Flashcard 21: What is always true about $r+i$ if $r$ is rational and $i$ is irrational?

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

Flashcard 22: What is the correct conclusion if $r$ is rational and $r+i$ is irrational?

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

Flashcard 23: What is the fraction form of $\frac{a}{b}+\frac{c}{d}$ that shows it is rational?

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

Flashcard 24: Identify the correct classification of $\pi+3$: rational or irrational?

Answer: Irrational. Rational plus irrational is irrational.

Flashcard 25: Identify the correct classification of $\left(\frac{2}{3}\right)+\left(\frac{4}{9}\right)$: rational or irrational?

Answer: Rational. Sum of two rational numbers.

Flashcard 26: Identify the correct conclusion: if $r$ is rational and $i$ is irrational, what is $-r+i$?

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

Flashcard 27: Identify the correct classification of $\sqrt{13}+\left(-\frac{4}{9}\right)$: rational or irrational?

Answer: Irrational. Irrational plus rational is irrational.

Flashcard 28: What is the fraction form of $\frac{a}{b}\cdot\frac{c}{d}$ that shows it is rational?

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

Flashcard 29: What is the correct conclusion if $r$ is rational and $r+i$ is rational for some $i$?

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

Flashcard 30: Identify the correct classification of $\sqrt{3}-\frac{11}{6}$: rational or irrational?

Answer: Irrational. Irrational minus rational is irrational.

Flashcard 31: What is always true about $r-i$ if $r$ is rational and $i$ is irrational?

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

Flashcard 32: Identify the correct classification of $\left(\frac{-7}{3}\right)\sqrt{2}$: rational or irrational?

Answer: Irrational. Nonzero rational times irrational is irrational.

Flashcard 33: Identify the correct classification of $-2\sqrt{5}$: rational or irrational?

Answer: Irrational. Nonzero rational times irrational is irrational.

Flashcard 34: Identify the correct classification of $\pi-\pi$: rational or irrational?

Answer: Rational. Any number minus itself equals zero.

Flashcard 35: What is the closure property of rational numbers under addition?

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

Flashcard 36: What is the definition of an irrational number?

Answer: 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.

Flashcard 37: Identify the correct classification of $\left(\frac{1}{2}\right)\pi$: rational or irrational?

Answer: Irrational. Nonzero rational times irrational is irrational.

Flashcard 38: Identify the correct classification of $\sqrt{2}-\sqrt{2}$: rational or irrational?

Answer: Rational. Any number minus itself equals zero.

Flashcard 39: Identify the correct classification of $\frac{7}{9}+\frac{2}{3}$: rational or irrational?

Answer: Rational. Sum of two rational numbers.

Flashcard 40: Identify the correct classification of $\left(\frac{3}{5}\right)\sqrt{7}$: rational or irrational?

Answer: Irrational. Nonzero rational times irrational is irrational.

Flashcard 41: Which property justifies that $bd \ne 0$ if $b \ne 0$ and $d \ne 0$?

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

Flashcard 42: Identify the correct classification of $\frac{5}{6}\cdot 0$: rational or irrational?

Answer: Rational. Any number times zero equals zero.

Flashcard 43: Identify the correct classification of $\left(\frac{-5}{8}\right)\left(\frac{12}{7}\right)$: rational or irrational?

Answer: Rational. Product of two rational numbers.

Flashcard 44: Identify the correct classification of $\left(\frac{2}{3}\right)\left(\frac{4}{9}\right)$: rational or irrational?

Answer: Rational. Product of two rational numbers.

Flashcard 45: Which statement is always true: if $r$ and $s$ are rational, what is $r+s$?

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

Flashcard 46: What is the correct conclusion if $r$ is nonzero rational and $ri$ is irrational?

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

Flashcard 47: Which statement is always true about $\frac{m}{n}\cdot\frac{p}{q}$ for integers $m,n,p,q$ with $nq \ne 0$?

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

Flashcard 48: Identify the correct classification of $0\cdot\sqrt{10}$: rational or irrational?

Answer: Rational. Zero times any number equals zero.

Flashcard 49: What is always true about $ri$ if $r$ is nonzero rational and $i$ is irrational?

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

Flashcard 50: What special case makes $ri$ rational when $r$ is rational and $i$ is irrational?

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

Flashcard 51: What is the definition of a rational number in fraction form?

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

Flashcard 52: Which statement is always true: if $r$ is rational, what is $-r$?

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

Flashcard 53: Identify the correct classification of $\sqrt{6}+\left(\frac{1}{3}\right)$: rational or irrational?

Answer: Irrational. Irrational plus rational is irrational.

Flashcard 54: Which statement is always true: if $i$ is irrational, what is $-i$?

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

Flashcard 55: Identify the correct classification of $\frac{2}{7}+\left(-\frac{2}{7}\right)$: rational or irrational?

Answer: Rational. Sum equals zero, which is rational.

Flashcard 56: Identify the correct classification of $\left(\frac{-7}{3}\right)\sqrt{2}$: rational or irrational?

Answer: Irrational. Nonzero rational times irrational is irrational.

Flashcard 57: Identify the correct classification of $\left(\sqrt{2}\right)\left(\sqrt{2}\right)$: rational or irrational?

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

Flashcard 58: What is the correct conclusion if $r$ is nonzero rational and $ri$ is rational for some $i$?

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