Approximate Irrational Numbers - 8th Grade Math
Card 1 of 30
Which inequality correctly orders the numbers $1.7$, $sqrt{3}$, and $1.8$?
Which inequality correctly orders the numbers $1.7$, $sqrt{3}$, and $1.8$?
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$1.7<\sqrt{3}<1.8$. Since $1.7^2=2.89<3<3.24=1.8^2$.
$1.7<\sqrt{3}<1.8$. Since $1.7^2=2.89<3<3.24=1.8^2$.
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Estimate $sqrt{3}$ to the nearest tenth by comparing $1.7^2=2.89$ and $1.8^2=3.24$.
Estimate $sqrt{3}$ to the nearest tenth by comparing $1.7^2=2.89$ and $1.8^2=3.24$.
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$\sqrt{3} \approx 1.7$. Since $2.89$ is closer to $3$ than $3.24$, round to $1.7$.
$\sqrt{3} \approx 1.7$. Since $2.89$ is closer to $3$ than $3.24$, round to $1.7$.
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Identify the correct placement: is $sqrt{2}$ closer to $1$ or to $2$ on a number line?
Identify the correct placement: is $sqrt{2}$ closer to $1$ or to $2$ on a number line?
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Closer to $1$. Since $1.41$ is less than $1.5$, it's closer to $1$.
Closer to $1$. Since $1.41$ is less than $1.5$, it's closer to $1$.
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What is the approximate value of $sqrt{2}$ to the nearest hundredth?
What is the approximate value of $sqrt{2}$ to the nearest hundredth?
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$\sqrt{2} \approx 1.41$. Standard approximation memorized for common calculations.
$\sqrt{2} \approx 1.41$. Standard approximation memorized for common calculations.
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Which number is larger: $sqrt{3}$ or $sqrt{5}$?
Which number is larger: $sqrt{3}$ or $sqrt{5}$?
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$\sqrt{5}$. Since $5>3$, we have $sqrt{5}>sqrt{3}$.
$\sqrt{5}$. Since $5>3$, we have $sqrt{5}>sqrt{3}$.
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Which number is larger: $sqrt{18}$ or $sqrt{20}$?
Which number is larger: $sqrt{18}$ or $sqrt{20}$?
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$\sqrt{20}$. Larger number under the radical gives larger square root.
$\sqrt{20}$. Larger number under the radical gives larger square root.
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Estimate $sqrt{50}$ to the nearest tenth using $7.1^2=50.41$ and $7.0^2=49$.
Estimate $sqrt{50}$ to the nearest tenth using $7.1^2=50.41$ and $7.0^2=49$.
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$\sqrt{50} \approx 7.1$. Since $50.41$ is very close to $50$, round down to $7.1$.
$\sqrt{50} \approx 7.1$. Since $50.41$ is very close to $50$, round down to $7.1$.
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Estimate $sqrt{50}$ using the nearest perfect squares.
Estimate $sqrt{50}$ using the nearest perfect squares.
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$7<\sqrt{50}<8$. Since $7^2=49$ and $8^2=64$, and $49<50<64$.
$7<\sqrt{50}<8$. Since $7^2=49$ and $8^2=64$, and $49<50<64$.
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Estimate $2pi$ using $pi \approx 3.14$.
Estimate $2pi$ using $pi \approx 3.14$.
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$2\pi \approx 6.28$. Multiply: $2 imes 3.14 = 6.28$.
$2\pi \approx 6.28$. Multiply: $2 imes 3.14 = 6.28$.
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Identify the best first integer bounds for $\sqrt{10}$: which inequality is correct?
Identify the best first integer bounds for $\sqrt{10}$: which inequality is correct?
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$3<\sqrt{10}<4$. Since $3^2=9$ and $4^2=16$, and $9<10<16$.
$3<\sqrt{10}<4$. Since $3^2=9$ and $4^2=16$, and $9<10<16$.
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Estimate $pi^2$ using $pi \approx 3.14$.
Estimate $pi^2$ using $pi \approx 3.14$.
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$\pi^2 \approx 9.86$. Square the approximation: $3.14^2 = 9.8596 approx 9.86$.
$\pi^2 \approx 9.86$. Square the approximation: $3.14^2 = 9.8596 approx 9.86$.
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What is the definition of an irrational number in terms of its decimal form?
What is the definition of an irrational number in terms of its decimal form?
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A number with a nonterminating, nonrepeating decimal. Irrational decimals go on forever without repeating patterns.
A number with a nonterminating, nonrepeating decimal. Irrational decimals go on forever without repeating patterns.
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Which statement is always true about squaring positive numbers: if $a<b$, then what about $a^2$ and $b^2$?
Which statement is always true about squaring positive numbers: if $a<b$, then what about $a^2$ and $b^2$?
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If $0<a<b$, then $a^2<b^2$. Squaring preserves inequality for positive numbers.
If $0<a<b$, then $a^2<b^2$. Squaring preserves inequality for positive numbers.
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Identify the best first integer bounds for $sqrt{2}$: which inequality is correct?
Identify the best first integer bounds for $sqrt{2}$: which inequality is correct?
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$1<\sqrt{2}<2$. Since $1^2=1$ and $2^2=4$, and $1<2<4$.
$1<\sqrt{2}<2$. Since $1^2=1$ and $2^2=4$, and $1<2<4$.
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What inequality shows $sqrt{2}$ is between $1.4$ and $1.5$ using squares?
What inequality shows $sqrt{2}$ is between $1.4$ and $1.5$ using squares?
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$1.4^2<\sqrt{2}^2<1.5^2$. Since $1.4^2=1.96$ and $1.5^2=2.25$, and $1.96<2<2.25$.
$1.4^2<\sqrt{2}^2<1.5^2$. Since $1.4^2=1.96$ and $1.5^2=2.25$, and $1.96<2<2.25$.
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Which is closer to $sqrt{2}$: $1.41$ or $1.42$?
Which is closer to $sqrt{2}$: $1.41$ or $1.42$?
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$1.41$. Since $1.41^2=1.9881$ is closer to $2$ than $1.42^2=2.0164$.
$1.41$. Since $1.41^2=1.9881$ is closer to $2$ than $1.42^2=2.0164$.
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Which is closer to $sqrt{5}$: $2.23$ or $2.24$?
Which is closer to $sqrt{5}$: $2.23$ or $2.24$?
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$2.24$. Since $2.24^2=5.0176$ is closer to $5$ than $2.23^2=4.9729$.
$2.24$. Since $2.24^2=5.0176$ is closer to $5$ than $2.23^2=4.9729$.
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What is a common rational approximation for $pi$ to the nearest hundredth?
What is a common rational approximation for $pi$ to the nearest hundredth?
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$3.14$. Standard approximation of $pi$ rounded to hundredths.
$3.14$. Standard approximation of $pi$ rounded to hundredths.
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What is the definition of a rational number in terms of fractions?
What is the definition of a rational number in terms of fractions?
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A number that can be written as $\frac{a}{b}$ with integers $a,b$ and $b \neq 0$. Any fraction of integers (with non-zero denominator) is rational.
A number that can be written as $\frac{a}{b}$ with integers $a,b$ and $b \neq 0$. Any fraction of integers (with non-zero denominator) is rational.
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Which inequality is true: $6<\sqrt{40}<7$ or $7<\sqrt{40}<8$?
Which inequality is true: $6<\sqrt{40}<7$ or $7<\sqrt{40}<8$?
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$6<\sqrt{40}<7$. Since $6^2=36$ and $7^2=49$, and $40$ is between them.
$6<\sqrt{40}<7$. Since $6^2=36$ and $7^2=49$, and $40$ is between them.
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Which inequality is true: $1.41<\sqrt{2}<1.42$ or $1.42<\sqrt{2}<1.43$?
Which inequality is true: $1.41<\sqrt{2}<1.42$ or $1.42<\sqrt{2}<1.43$?
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$1.41<\sqrt{2}<1.42$. Since $1.41^2=1.9881$ and $1.42^2=2.0164$, and $2$ is between them.
$1.41<\sqrt{2}<1.42$. Since $1.41^2=1.9881$ and $1.42^2=2.0164$, and $2$ is between them.
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What is an irrational number (in terms of its decimal form)?
What is an irrational number (in terms of its decimal form)?
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A number with a decimal that is nonterminating and nonrepeating. Cannot be expressed as a fraction of integers.
A number with a decimal that is nonterminating and nonrepeating. Cannot be expressed as a fraction of integers.
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What is a rational approximation of an irrational number?
What is a rational approximation of an irrational number?
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A nearby rational number, often a truncated or rounded decimal. Used to estimate and compare irrational values.
A nearby rational number, often a truncated or rounded decimal. Used to estimate and compare irrational values.
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Which inequality is true: $1<\sqrt{2}<2$ or $2<\sqrt{2}<3$?
Which inequality is true: $1<\sqrt{2}<2$ or $2<\sqrt{2}<3$?
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$1<\sqrt{2}<2$. Since $1^2=1$ and $2^2=4$, and $2$ is between them.
$1<\sqrt{2}<2$. Since $1^2=1$ and $2^2=4$, and $2$ is between them.
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Which inequality is true: $1.4<\sqrt{2}<1.5$ or $1.5<\sqrt{2}<1.6$?
Which inequality is true: $1.4<\sqrt{2}<1.5$ or $1.5<\sqrt{2}<1.6$?
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$1.4<\sqrt{2}<1.5$. Since $1.4^2=1.96$ and $1.5^2=2.25$, and $2$ is between them.
$1.4<\sqrt{2}<1.5$. Since $1.4^2=1.96$ and $1.5^2=2.25$, and $2$ is between them.
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What is $\sqrt{2}$ rounded to the nearest tenth?
What is $\sqrt{2}$ rounded to the nearest tenth?
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$1.4$. $\sqrt{2} \approx 1.414$, which rounds to $1.4$.
$1.4$. $\sqrt{2} \approx 1.414$, which rounds to $1.4$.
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What is $\sqrt{2}$ rounded to the nearest hundredth?
What is $\sqrt{2}$ rounded to the nearest hundredth?
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$1.41$. $\sqrt{2} \approx 1.414$, which rounds to $1.41$.
$1.41$. $\sqrt{2} \approx 1.414$, which rounds to $1.41$.
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Which is larger: $\sqrt{5}$ or $2.2$?
Which is larger: $\sqrt{5}$ or $2.2$?
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$\sqrt{5}$. $\sqrt{5} \approx 2.236$, which is greater than $2.2$.
$\sqrt{5}$. $\sqrt{5} \approx 2.236$, which is greater than $2.2$.
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Which is larger: $\sqrt{10}$ or $3.1$?
Which is larger: $\sqrt{10}$ or $3.1$?
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$\sqrt{10}$. $\sqrt{10} \approx 3.162$, which is greater than $3.1$.
$\sqrt{10}$. $\sqrt{10} \approx 3.162$, which is greater than $3.1$.
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Which is larger: $\pi$ or $3.14$?
Which is larger: $\pi$ or $3.14$?
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$\pi$. $\pi \approx 3.14159$, which is greater than $3.14$.
$\pi$. $\pi \approx 3.14159$, which is greater than $3.14$.
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