8th Grade Math Flashcards: Approximate Irrational Numbers

What this deck covers

This deck focuses on Approximate 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: Which inequality correctly orders the numbers $1.7$, $sqrt{3}$, and $1.8$?

Answer: $1.7<\sqrt{3}<1.8$. Since $1.7^2=2.89<3<3.24=1.8^2$.

Flashcard 2: Estimate $sqrt{3}$ to the nearest tenth by comparing $1.7^2=2.89$ and $1.8^2=3.24$.

Answer: $\sqrt{3} \approx 1.7$. Since $2.89$ is closer to $3$ than $3.24$, round to $1.7$.

Flashcard 3: Identify the correct placement: is $sqrt{2}$ closer to $1$ or to $2$ on a number line?

Answer: Closer to $1$. Since $1.41$ is less than $1.5$, it's closer to $1$.

Flashcard 4: What is the approximate value of $sqrt{2}$ to the nearest hundredth?

Answer: $\sqrt{2} \approx 1.41$. Standard approximation memorized for common calculations.

Flashcard 5: Which number is larger: $sqrt{3}$ or $sqrt{5}$?

Answer: $\sqrt{5}$. Since $5>3$, we have $sqrt{5}>sqrt{3}$.

Flashcard 6: Which number is larger: $sqrt{18}$ or $sqrt{20}$?

Answer: $\sqrt{20}$. Larger number under the radical gives larger square root.

Flashcard 7: Estimate $sqrt{50}$ to the nearest tenth using $7.1^2=50.41$ and $7.0^2=49$.

Answer: $\sqrt{50} \approx 7.1$. Since $50.41$ is very close to $50$, round down to $7.1$.

Flashcard 8: Estimate $sqrt{50}$ using the nearest perfect squares.

Answer: $7<\sqrt{50}<8$. Since $7^2=49$ and $8^2=64$, and $49<50<64$.

Flashcard 9: Estimate $2pi$ using $pi \approx 3.14$.

Answer: $2\pi \approx 6.28$. Multiply: $2 imes 3.14 = 6.28$.

Flashcard 10: Identify the best first integer bounds for $\sqrt{10}$: which inequality is correct?

Answer: $3<\sqrt{10}<4$. Since $3^2=9$ and $4^2=16$, and $9<10<16$.

Flashcard 11: Estimate $pi^2$ using $pi \approx 3.14$.

Answer: $\pi^2 \approx 9.86$. Square the approximation: $3.14^2 = 9.8596 approx 9.86$.

Flashcard 12: What is the definition of an irrational number in terms of its decimal form?

Answer: A number with a nonterminating, nonrepeating decimal. Irrational decimals go on forever without repeating patterns.

Flashcard 13: Which statement is always true about squaring positive numbers: if $a<b$, then what about $a^2$ and $b^2$?

Answer: If $0<a<b$, then $a^2<b^2$. Squaring preserves inequality for positive numbers.

Flashcard 14: Identify the best first integer bounds for $sqrt{2}$: which inequality is correct?

Answer: $1<\sqrt{2}<2$. Since $1^2=1$ and $2^2=4$, and $1<2<4$.

Flashcard 15: What inequality shows $sqrt{2}$ is between $1.4$ and $1.5$ using squares?

Answer: $1.4^2<\sqrt{2}^2<1.5^2$. Since $1.4^2=1.96$ and $1.5^2=2.25$, and $1.96<2<2.25$.

Flashcard 16: Which is closer to $sqrt{2}$: $1.41$ or $1.42$?

Answer: $1.41$. Since $1.41^2=1.9881$ is closer to $2$ than $1.42^2=2.0164$.

Flashcard 17: Which is closer to $sqrt{5}$: $2.23$ or $2.24$?

Answer: $2.24$. Since $2.24^2=5.0176$ is closer to $5$ than $2.23^2=4.9729$.

Flashcard 18: What is a common rational approximation for $pi$ to the nearest hundredth?

Answer: $3.14$. Standard approximation of $pi$ rounded to hundredths.

Flashcard 19: What is the definition of a rational number in terms of fractions?

Answer: 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.

Flashcard 20: Which inequality is true: $6<\sqrt{40}<7$ or $7<\sqrt{40}<8$?

Answer: $6<\sqrt{40}<7$. Since $6^2=36$ and $7^2=49$, and $40$ is between them.

Flashcard 21: Which inequality is true: $1.41<\sqrt{2}<1.42$ or $1.42<\sqrt{2}<1.43$?

Answer: $1.41<\sqrt{2}<1.42$. Since $1.41^2=1.9881$ and $1.42^2=2.0164$, and $2$ is between them.

Flashcard 22: What is an irrational number (in terms of its decimal form)?

Answer: A number with a decimal that is nonterminating and nonrepeating. Cannot be expressed as a fraction of integers.

Flashcard 23: What is a rational approximation of an irrational number?

Answer: A nearby rational number, often a truncated or rounded decimal. Used to estimate and compare irrational values.

Flashcard 24: Which inequality is true: $1<\sqrt{2}<2$ or $2<\sqrt{2}<3$?

Answer: $1<\sqrt{2}<2$. Since $1^2=1$ and $2^2=4$, and $2$ is between them.

Flashcard 25: Which inequality is true: $1.4<\sqrt{2}<1.5$ or $1.5<\sqrt{2}<1.6$?

Answer: $1.4<\sqrt{2}<1.5$. Since $1.4^2=1.96$ and $1.5^2=2.25$, and $2$ is between them.

Flashcard 26: What is $\sqrt{2}$ rounded to the nearest tenth?

Answer: $1.4$. $\sqrt{2} \approx 1.414$, which rounds to $1.4$.

Flashcard 27: What is $\sqrt{2}$ rounded to the nearest hundredth?

Answer: $1.41$. $\sqrt{2} \approx 1.414$, which rounds to $1.41$.

Flashcard 28: Which is larger: $\sqrt{5}$ or $2.2$?

Answer: $\sqrt{5}$. $\sqrt{5} \approx 2.236$, which is greater than $2.2$.

Flashcard 29: Which is larger: $\sqrt{10}$ or $3.1$?

Answer: $\sqrt{10}$. $\sqrt{10} \approx 3.162$, which is greater than $3.1$.

Flashcard 30: Which is larger: $\pi$ or $3.14$?

Answer: $\pi$. $\pi \approx 3.14159$, which is greater than $3.14$.