Understand Irrational Numbers - 8th Grade Math
Card 1 of 25
Convert the repeating decimal $0.1\overline{6}$ to a fraction in simplest form.
Convert the repeating decimal $0.1\overline{6}$ to a fraction in simplest form.
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$\frac{1}{6}$. $0.1\overline{6} = \frac{1}{10} + \frac{6}{90} = \frac{15}{90} = \frac{1}{6}$
$\frac{1}{6}$. $0.1\overline{6} = \frac{1}{10} + \frac{6}{90} = \frac{15}{90} = \frac{1}{6}$
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What is the key decimal-expansion fact for rational numbers?
What is the key decimal-expansion fact for rational numbers?
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A rational number s decimal terminates or repeats eventually. This distinguishes rational from irrational numbers.
A rational number s decimal terminates or repeats eventually. This distinguishes rational from irrational numbers.
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Convert the repeating decimal $0.\overline{6}$ to a fraction in simplest form.
Convert the repeating decimal $0.\overline{6}$ to a fraction in simplest form.
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$\frac{2}{3}$. Let $x = 0.\overline{6}$; then $10x - x = 6$, so $x = \frac{2}{3}$
$\frac{2}{3}$. Let $x = 0.\overline{6}$; then $10x - x = 6$, so $x = \frac{2}{3}$
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What is the decimal form of $\frac{1}{4}$?
What is the decimal form of $\frac{1}{4}$?
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$0.25$. Divide: $1 \div 4 = 0.25$.
$0.25$. Divide: $1 \div 4 = 0.25$.
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Convert the repeating decimal $1.\overline{2}$ to a fraction in simplest form.
Convert the repeating decimal $1.\overline{2}$ to a fraction in simplest form.
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$\frac{11}{9}$. Let $x = 1.\overline{2}$; then $10x - x = 11$, so $x = \frac{11}{9}$
$\frac{11}{9}$. Let $x = 1.\overline{2}$; then $10x - x = 11$, so $x = \frac{11}{9}$
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Convert the repeating decimal $0.\overline{12}$ to a fraction in simplest form.
Convert the repeating decimal $0.\overline{12}$ to a fraction in simplest form.
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$\frac{4}{33}$. Let $x = 0.\overline{12}$; then $100x - x = 12$, so $x = \frac{4}{33}$.
$\frac{4}{33}$. Let $x = 0.\overline{12}$; then $100x - x = 12$, so $x = \frac{4}{33}$.
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Convert the repeating decimal $3.1\overline{2}$ to a fraction in simplest form.
Convert the repeating decimal $3.1\overline{2}$ to a fraction in simplest form.
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$\frac{281}{90}$. $3.1\overline{2} = 3 + \frac{1}{10} + \frac{2}{90} = \frac{281}{90}$
$\frac{281}{90}$. $3.1\overline{2} = 3 + \frac{1}{10} + \frac{2}{90} = \frac{281}{90}$
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Convert the repeating decimal $0.0\overline{7}$ to a fraction in simplest form.
Convert the repeating decimal $0.0\overline{7}$ to a fraction in simplest form.
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$\frac{7}{90}$. Let $x = 0.0\overline{7}$; then $10x - x = 0.7$, so $x = \frac{7}{90}$
$\frac{7}{90}$. Let $x = 0.0\overline{7}$; then $10x - x = 0.7$, so $x = \frac{7}{90}$
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Convert the terminating decimal $0.6$ to a fraction in simplest form.
Convert the terminating decimal $0.6$ to a fraction in simplest form.
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$\frac{3}{5}$. $0.6 = \frac{6}{10} = \frac{3}{5}$ after simplifying.
$\frac{3}{5}$. $0.6 = \frac{6}{10} = \frac{3}{5}$ after simplifying.
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Which option is irrational: $0.125$ or $0.101001000100001\ldots$?
Which option is irrational: $0.125$ or $0.101001000100001\ldots$?
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$0.101001000100001\ldots$ is irrational. The pattern doesn't repeat, making it irrational.
$0.101001000100001\ldots$ is irrational. The pattern doesn't repeat, making it irrational.
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Identify whether $\sqrt{2}$ is rational or irrational.
Identify whether $\sqrt{2}$ is rational or irrational.
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$\sqrt{2}$ is irrational. Cannot be expressed as $\frac{p}{q}$ with integers $p, q$.
$\sqrt{2}$ is irrational. Cannot be expressed as $\frac{p}{q}$ with integers $p, q$.
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Identify whether $\pi$ is rational or irrational.
Identify whether $\pi$ is rational or irrational.
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$\pi$ is irrational. Its decimal expansion never terminates or repeats.
$\pi$ is irrational. Its decimal expansion never terminates or repeats.
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What decimal expansion pattern indicates a number is irrational?
What decimal expansion pattern indicates a number is irrational?
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A decimal that is nonterminating and nonrepeating. This pattern cannot be expressed as $\frac{p}{q}$.
A decimal that is nonterminating and nonrepeating. This pattern cannot be expressed as $\frac{p}{q}$.
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Which decimal type always represents a rational number: terminating, repeating, or nonrepeating nonterminating?
Which decimal type always represents a rational number: terminating, repeating, or nonrepeating nonterminating?
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Terminating and repeating decimals are rational. Both can be expressed as fractions $\frac{p}{q}$.
Terminating and repeating decimals are rational. Both can be expressed as fractions $\frac{p}{q}$.
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Which option is rational: $0.\overline{27}$ or $0.2710010001\ldots$?
Which option is rational: $0.\overline{27}$ or $0.2710010001\ldots$?
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$0.\overline{27}$ is rational. Repeating decimals are always rational.
$0.\overline{27}$ is rational. Repeating decimals are always rational.
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What is the definition of an irrational number?
What is the definition of an irrational number?
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A number that cannot be written as $p/q$ with integers $p$, $q
eq 0$. Irrational numbers have no fraction representation.
A number that cannot be written as $p/q$ with integers $p$, $q eq 0$. Irrational numbers have no fraction representation.
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Convert the repeating decimal $2.\overline{45}$ to a fraction in simplest form.
Convert the repeating decimal $2.\overline{45}$ to a fraction in simplest form.
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$\frac{27}{11}$. Let $x = 2.\overline{45}$; then $100x - x = 243$, so $x = \frac{27}{11}$.
$\frac{27}{11}$. Let $x = 2.\overline{45}$; then $100x - x = 243$, so $x = \frac{27}{11}$.
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Convert the repeating decimal $0.\overline{3}$ to a fraction in simplest form.
Convert the repeating decimal $0.\overline{3}$ to a fraction in simplest form.
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$\frac{1}{3}$. Let $x = 0.\overline{3}$; then $10x - x = 3$, so $x = \frac{1}{3}$
$\frac{1}{3}$. Let $x = 0.\overline{3}$; then $10x - x = 3$, so $x = \frac{1}{3}$
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What is the definition of a rational number in terms of integers $p$ and $q$?
What is the definition of a rational number in terms of integers $p$ and $q$?
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A number that can be written as $ p / q $ with integers $p$, $q \neq 0$. This is the fundamental definition of rational numbers.
A number that can be written as $ p / q $ with integers $p$, $q \neq 0$. This is the fundamental definition of rational numbers.
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Convert the terminating decimal $2.75$ to a fraction in simplest form.
Convert the terminating decimal $2.75$ to a fraction in simplest form.
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$\frac{11}{4}$. $2.75 = \frac{275}{100} = \frac{11}{4}$ simplified.
$\frac{11}{4}$. $2.75 = \frac{275}{100} = \frac{11}{4}$ simplified.
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What is an irrational number, stated using the definition of rational numbers?
What is an irrational number, stated using the definition of rational numbers?
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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.
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.
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Choose the correct classification for $\frac{22}{7}$: rational or irrational?
Choose the correct classification for $\frac{22}{7}$: rational or irrational?
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Rational. It's a fraction of two integers, so it's rational.
Rational. It's a fraction of two integers, so it's rational.
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Convert the terminating decimal $0.125$ to a fraction in simplest form.
Convert the terminating decimal $0.125$ to a fraction in simplest form.
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$\frac{1}{8}$. $0.125 = \frac{125}{1000} = \frac{1}{8}$ after simplifying.
$\frac{1}{8}$. $0.125 = \frac{125}{1000} = \frac{1}{8}$ after simplifying.
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Choose the correct classification for $ \pi$: rational or irrational?
Choose the correct classification for $ \pi$: rational or irrational?
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Irrational. $\pi$ is proven to have no fraction representation.
Irrational. $\pi$ is proven to have no fraction representation.
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Which decimal type represents an irrational number: terminating, repeating, or nonrepeating nonterminating?
Which decimal type represents an irrational number: terminating, repeating, or nonrepeating nonterminating?
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Nonterminating, nonrepeating decimals are irrational. These decimals go on forever without a pattern.
Nonterminating, nonrepeating decimals are irrational. These decimals go on forever without a pattern.
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