8th Grade Math Flashcards: Understand Irrational Numbers

What this deck covers

This deck focuses on Understand Irrational Numbers, giving you a quick way to review the definitions, rules, and examples that matter most for 8th Grade Math.

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: Convert the repeating decimal $0.1\overline{6}$ to a fraction in simplest form.

Answer: $\frac{1}{6}$. $0.1\overline{6} = \frac{1}{10} + \frac{6}{90} = \frac{15}{90} = \frac{1}{6}$

Flashcard 2: What is the key decimal-expansion fact for rational numbers?

Answer: A rational numbers decimal terminates or repeats eventually. This distinguishes rational from irrational numbers.

Flashcard 3: Convert the repeating decimal $0.\overline{6}$ to a fraction in simplest form.

Answer: $\frac{2}{3}$. Let $x = 0.\overline{6}$; then $10x - x = 6$, so $x = \frac{2}{3}$

Flashcard 4: What is the decimal form of $\frac{1}{4}$?

Answer: $0.25$. Divide: $1 \div 4 = 0.25$.

Flashcard 5: Convert the repeating decimal $1.\overline{2}$ to a fraction in simplest form.

Answer: $\frac{11}{9}$. Let $x = 1.\overline{2}$; then $10x - x = 11$, so $x = \frac{11}{9}$

Flashcard 6: Convert the repeating decimal $0.\overline{12}$ to a fraction in simplest form.

Answer: $\frac{4}{33}$. Let $x = 0.\overline{12}$; then $100x - x = 12$, so $x = \frac{4}{33}$.

Flashcard 7: Convert the repeating decimal $3.1\overline{2}$ to a fraction in simplest form.

Answer: $\frac{281}{90}$. $3.1\overline{2} = 3 + \frac{1}{10} + \frac{2}{90} = \frac{281}{90}$

Flashcard 8: Convert the repeating decimal $0.0\overline{7}$ to a fraction in simplest form.

Answer: $\frac{7}{90}$. Let $x = 0.0\overline{7}$; then $10x - x = 0.7$, so $x = \frac{7}{90}$

Flashcard 9: Convert the terminating decimal $0.6$ to a fraction in simplest form.

Answer: $\frac{3}{5}$. $0.6 = \frac{6}{10} = \frac{3}{5}$ after simplifying.

Flashcard 10: Which option is irrational: $0.125$ or $0.101001000100001\ldots$?

Answer: $0.101001000100001\ldots$ is irrational. The pattern doesn't repeat, making it irrational.

Flashcard 11: Identify whether $\sqrt{2}$ is rational or irrational.

Answer: $\sqrt{2}$ is irrational. Cannot be expressed as $\frac{p}{q}$ with integers $p, q$.

Flashcard 12: Identify whether $\pi$ is rational or irrational.

Answer: $\pi$ is irrational. Its decimal expansion never terminates or repeats.

Flashcard 13: What decimal expansion pattern indicates a number is irrational?

Answer: A decimal that is nonterminating and nonrepeating. This pattern cannot be expressed as $\frac{p}{q}$.

Flashcard 14: Which decimal type always represents a rational number: terminating, repeating, or nonrepeating nonterminating?

Answer: Terminating and repeating decimals are rational. Both can be expressed as fractions $\frac{p}{q}$.

Flashcard 15: Which option is rational: $0.\overline{27}$ or $0.2710010001\ldots$?

Answer: $0.\overline{27}$ is rational. Repeating decimals are always rational.

Flashcard 16: What is the definition of an irrational number?

Answer: A number that cannot be written as $p/q$ with integers $p$, $q eq 0$. Irrational numbers have no fraction representation.

Flashcard 17: Convert the repeating decimal $2.\overline{45}$ to a fraction in simplest form.

Answer: $\frac{27}{11}$. Let $x = 2.\overline{45}$; then $100x - x = 243$, so $x = \frac{27}{11}$.

Flashcard 18: Convert the repeating decimal $0.\overline{3}$ to a fraction in simplest form.

Answer: $\frac{1}{3}$. Let $x = 0.\overline{3}$; then $10x - x = 3$, so $x = \frac{1}{3}$

Flashcard 19: What is the definition of a rational number in terms of integers $p$ and $q$?

Answer: A number that can be written as p/q with integers $p$, $q \neq 0$. This is the fundamental definition of rational numbers.

Flashcard 20: Convert the terminating decimal $2.75$ to a fraction in simplest form.

Answer: $\frac{11}{4}$. $2.75 = \frac{275}{100} = \frac{11}{4}$ simplified.

Flashcard 21: What is an irrational number, stated using the definition of rational numbers?

Answer: A number that cannot be written as $\frac{a}{b}$ for integers $a,b$ with $b \neq 0$. Numbers that cannot be expressed as fractions are irrational.

Flashcard 22: Choose the correct classification for $\frac{22}{7}$: rational or irrational?

Answer: Rational. It's a fraction of two integers, so it's rational.

Flashcard 23: Convert the terminating decimal $0.125$ to a fraction in simplest form.

Answer: $\frac{1}{8}$. $0.125 = \frac{125}{1000} = \frac{1}{8}$ after simplifying.

Flashcard 24: Choose the correct classification for $\pi$: rational or irrational?

Answer: Irrational. $\pi$ is proven to have no fraction representation.

Flashcard 25: Which decimal type represents an irrational number: terminating, repeating, or nonrepeating nonterminating?

Answer: Nonterminating, nonrepeating decimals are irrational. These decimals go on forever without a pattern.

Flashcard 26: Convert the repeating decimal $1.2\overline{3}$ to a fraction in simplest form.

Answer: $\frac{37}{30}$. $1.2333... = 1.2 + \frac{0.1}{3} = \frac{6}{5} + \frac{1}{30} = \frac{37}{30}$

Flashcard 27: Convert the repeating decimal $0.\overline{12}$ to a fraction in simplest form.

Answer: $\frac{4}{33}$. Let $x = 0.121212...$; then $100x - x = 12$, so $x = \frac{12}{99} = \frac{4}{33}$.

Flashcard 28: Convert the repeating decimal $0.\overline{27}$ to a fraction in simplest form.

Answer: $\frac{3}{11}$. Let $x = 0.272727...$; then $100x - x = 27$, so $x = \frac{27}{99} = \frac{3}{11}$

Flashcard 29: Which decimal type always represents a rational number: terminating, repeating, or nonrepeating?

Answer: Terminating or repeating decimals are rational. Both types can be expressed as fractions.

Flashcard 30: Identify the decimal form of $\frac{1}{3}$: terminating or repeating?

Answer: Repeating. $\frac{1}{3} = 0.333...$, where 3 repeats forever.