This question tests understanding that irrational numbers cannot be expressed as fractions a/b and have non-terminating, non-repeating decimals, like √2 and π, while rational numbers can be expressed as fractions and have terminating or repeating decimals. Rational numbers include integers like 5 (which is 5/1), fractions like 3/4, terminating decimals like 0.5 (which is 1/2), and repeating decimals like 0.333... (which is 1/3 with the '3' repeating); irrational numbers cannot be written as fractions, and their decimals go on forever without a repeating pattern, such as √2 ≈ 1.41421356... (non-repeating) and π ≈ 3.14159265... (non-repeating), and remember that every number has a decimal expansion where rational ones eventually repeat, including terminating ones as repeating zeros like 0.5 = 0.5000.... Examples include √9 = 3, which is rational because it's a perfect square, √2 ≈ 1.414... which is irrational because it's not a perfect square, 0.666... which is rational because it repeats '6' and equals 2/3, and π ≈ 3.14159... which is irrational with no repeating pattern. The number √3 is irrational because 3 is not a perfect square, so its decimal is non-terminating and non-repeating, while the others are rational: √4 = 2 (integer), 0.125 = 1/8 (terminating), and 0.777... = 7/9 (repeating). A common error is thinking √4 is irrational, but it's wrong because it's a perfect square equal to 2, which is rational; another is assuming 0.777... is irrational, but it repeats so it's rational, or claiming all decimals are irrational, or believing π = 22/7 exactly, but 22/7 approximates π and π is truly irrational. For classification: (1) check if it can be a fraction—integers and simple fractions are clearly rational; (2) check the decimal— if it terminates, it's rational; if it repeats, it's rational; if neither, it's irrational; (3) for square roots, if it's a perfect square like √4, √9, or √16, it's rational, but if non-perfect like √2, √3, or √5, it's irrational. Converting a repeating decimal like 0.666...: let x = 0.666..., then 10x = 6.666..., subtract to get 9x = 6, so x = 2/3, showing repeating decimals are rational as they can be expressed as fractions; common mistakes include thinking all square roots are irrational (missing perfect squares), claiming repeating decimals are irrational (but they're rational), or believing π = 22/7 exactly (it's an approximation, π is irrational).

This question tests understanding that irrational numbers cannot be expressed as fractions $a/b$ and have non-terminating, non-repeating decimals, like $√2$ and $π$, while rational numbers can be expressed as fractions and have terminating or repeating decimals. Rational numbers include integers like 5 (which is $5/1$), fractions like $3/4$, terminating decimals like 0.5 (which is $1/2$), and repeating decimals like 0.333... (which is $1/3$ with the '3' repeating); irrational numbers cannot be written as fractions, and their decimals go on forever without a repeating pattern, such as $√2$ ≈ 1.41421356... (non-repeating) and $π$ ≈ 3.14159265... (non-repeating), and remember that every number has a decimal expansion where rational ones eventually repeat, including terminating ones as repeating zeros like 0.5 = 0.5000.... Examples include $√9$ = 3, which is rational because it's a perfect square, $√2$ ≈ 1.414... which is irrational because it's not a perfect square, 0.666... which is rational because it repeats '6' and equals $2/3$, and $π$ ≈ 3.14159... which is irrational with no repeating pattern. The repeating decimal 0.272727... converts to the fraction 27/99 = $3/11$ in simplest form, as shown by letting x = $0.272727\ldots$, multiplying by 100 to get 100x = $27.272727\ldots$, subtracting to yield 99x = 27, so x = $27/99$ = $3/11$. A common error is thinking $√4$ is irrational, but it's wrong because it's a perfect square equal to 2, which is rational; another is assuming all decimals are irrational, or claiming 0.333... is irrational, but it repeats so it's rational, or believing $π$ = 22/7 exactly, but 22/7 approximates $π$ and $π$ is truly irrational. For classification: (1) check if it can be a fraction—integers and simple fractions are clearly rational; (2) check the decimal— if it terminates, it's rational; if it repeats, it's rational; if neither, it's irrational; (3) for square roots, if it's a perfect square like $√4$, $√9$, or $√16$, it's rational, but if non-perfect like $√2$, $√3$, or $√5$, it's irrational. Common mistakes include thinking all square roots are irrational (missing perfect squares), claiming repeating decimals are irrational (but they're rational), or believing $π$ = 22/7 exactly (it's an approximation, $π$ is irrational).

This question tests understanding that irrational numbers cannot be expressed as fractions $a/b$ and have non-terminating, non-repeating decimals, like $\sqrt{2}$ and $\pi$, while rational numbers can be expressed as fractions and have terminating or repeating decimals. Rational numbers include integers like 5 (which is $5/1$), fractions like $3/4$, terminating decimals like 0.5 (which is $1/2$), and repeating decimals like $0.333\ldots$ (which is $1/3$ with the '3' repeating); irrational numbers cannot be written as fractions, and their decimals go on forever without a repeating pattern, such as $\sqrt{2} \approx 1.41421356\ldots$ (non-repeating) and $\pi \approx 3.14159265\ldots$ (non-repeating), and remember that every number has a decimal expansion where rational ones eventually repeat, including terminating ones as repeating zeros like 0.5 = $0.5000\ldots$. Examples include $\sqrt{9} = 3$, which is rational because it's a perfect square, $\sqrt{2} \approx 1.414\ldots$ which is irrational because it's not a perfect square, $0.666\ldots$ which is rational because it repeats '6' and equals $2/3$, and $\pi \approx 3.14159\ldots$ which is irrational with no repeating pattern. The correct list includes only the irrational numbers $\sqrt{2}$ and $\pi$, as $\sqrt{2}$ is not a perfect square so its decimal is non-repeating and non-terminating, and $\pi$ is a known irrational with a non-repeating decimal, while the others are rational: $0.75 = 3/4$ (terminating), $\sqrt{9} = 3$ (integer), and $0.333\ldots = 1/3$ (repeating). A common error is thinking $\sqrt{9}$ is irrational like $\sqrt{2}$, but it's wrong because $\sqrt{9}$ is a perfect square equal to 3, which is rational; another mistake is assuming all non-terminating decimals are irrational, but $0.333\ldots$ is rational because it repeats, or believing $\pi$ can be exactly $22/7$, but $22/7$ is just an approximation and $\pi$ is truly irrational. For classification: (1) check if it can be a fraction—integers and simple fractions are clearly rational; (2) check the decimal— if it terminates, it's rational; if it repeats, it's rational; if neither, it's irrational; (3) for square roots, if it's a perfect square like $\sqrt{4}$, $\sqrt{9}$, or $\sqrt{16}$, it's rational, but if non-perfect like $\sqrt{2}$, $\sqrt{3}$, or $\sqrt{5}$, it's irrational. Converting a repeating decimal like $0.666\ldots$: let $x = 0.666\ldots$, then $10x = 6.666\ldots$, subtract to get $9x = 6$, so $x = 2/3$, showing repeating decimals are rational as they can be expressed as fractions; common mistakes include thinking all square roots are irrational (missing perfect squares), claiming repeating decimals are irrational (but they're rational), or believing $\pi = 22/7$ exactly (it's an approximation, $\pi$ is irrational).

This question tests understanding that irrational numbers cannot be expressed as fractions a/b and have non-terminating, non-repeating decimals, like √2 and π, while rational numbers can be expressed as fractions and have terminating or repeating decimals. Rational numbers include integers like 5 (which is 5/1), fractions like 3/4, terminating decimals like 0.5 (which is 1/2), and repeating decimals like 0.333... (which is 1/3 with the '3' repeating); irrational numbers cannot be written as fractions, and their decimals go on forever without a repeating pattern, such as √2 ≈ 1.41421356... (non-repeating) and π ≈ 3.14159265... (non-repeating), and remember that every number has a decimal expansion where rational ones eventually repeat, including terminating ones as repeating zeros like 0.5 = 0.5000.... Examples include √9 = 3, which is rational because it's a perfect square, √2 ≈ 1.414... which is irrational because it's not a perfect square, 0.666... which is rational because it repeats '6' and equals 2/3, and π ≈ 3.14159... which is irrational with no repeating pattern. The true statement is that π is irrational, and 22/7 is only a rational approximation, because π cannot be exactly expressed as a fraction and its decimal is non-repeating, while 22/7 is a close but not exact rational value. A common error is claiming π is rational because 22/7 equals π exactly, but it's wrong as 22/7 is an approximation; another is saying π is rational because its decimal repeats, but it doesn't, or thinking π is irrational only if rounded, but it's inherently irrational, or assuming all decimals are irrational, but many are rational. For classification: (1) check if it can be a fraction—integers and simple fractions are clearly rational; (2) check the decimal— if it terminates, it's rational; if it repeats, it's rational; if neither, it's irrational; (3) for square roots, if it's a perfect square like √4, √9, or √16, it's rational, but if non-perfect like √2, √3, or √5, it's irrational. Converting a repeating decimal like 0.666...: let x = 0.666..., then 10x = 6.666..., subtract to get 9x = 6, so x = 2/3, showing repeating decimals are rational as they can be expressed as fractions; common mistakes include thinking all square roots are irrational (missing perfect squares), claiming repeating decimals are irrational (but they're rational), or believing π = 22/7 exactly (it's an approximation, π is irrational).

This question tests understanding that irrational numbers cannot be expressed as fractions a/b and have non-terminating, non-repeating decimals, like √2 and π, while rational numbers can be expressed as fractions and have terminating or repeating decimals. Rational numbers include integers like 5 (which is 5/1), fractions like 3/4, terminating decimals like 0.5 (which is 1/2), and repeating decimals like 0.333... (which is 1/3 with the '3' repeating); irrational numbers cannot be written as fractions, and their decimals go on forever without a repeating pattern, such as √2 ≈ 1.41421356... (non-repeating) and π ≈ 3.14159265... (non-repeating), and remember that every number has a decimal expansion where rational ones eventually repeat, including terminating ones as repeating zeros like 0.5 = 0.5000.... Examples include √9 = 3, which is rational because it's a perfect square, √2 ≈ 1.414... which is irrational because it's not a perfect square, 0.666... which is rational because it repeats '6' and equals 2/3, and π ≈ 3.14159... which is irrational with no repeating pattern. The correct statement is that √16 = 4 is rational, and √17 is irrational because 17 is not a perfect square, meaning √16 simplifies to an integer while √17 has a non-terminating, non-repeating decimal. A common error is thinking √16 is irrational just because it's a square root, but it's rational as it equals 4; another is claiming all square roots are irrational, or saying √17 is rational because it's close to 4, but proximity doesn't make it rational, or believing π = 22/7 exactly, but 22/7 approximates π and π is truly irrational. For classification: (1) check if it can be a fraction—integers and simple fractions are clearly rational; (2) check the decimal— if it terminates, it's rational; if it repeats, it's rational; if neither, it's irrational; (3) for square roots, if it's a perfect square like √4, √9, or √16, it's rational, but if non-perfect like √2, √3, or √5, it's irrational. Converting a repeating decimal like 0.666...: let x = 0.666..., then 10x = 6.666..., subtract to get 9x = 6, so x = 2/3, showing repeating decimals are rational as they can be expressed as fractions; common mistakes include thinking all square roots are irrational (missing perfect squares), claiming repeating decimals are irrational (but they're rational), or believing π = 22/7 exactly (it's an approximation, π is irrational).

This question tests understanding that irrational numbers cannot be expressed as fractions a/b and have non-terminating, non-repeating decimals, like √2 and π, while rational numbers can be expressed as fractions and have terminating or repeating decimals. Rational numbers include integers like 5 (which is 5/1), fractions like 3/4, terminating decimals like 0.5 (which is 1/2), and repeating decimals like 0.333... (which is 1/3 with the '3' repeating); irrational numbers cannot be written as fractions, and their decimals go on forever without a repeating pattern, such as √2 ≈ 1.41421356... (non-repeating) and π ≈ 3.14159265... (non-repeating), and remember that every number has a decimal expansion where rational ones eventually repeat, including terminating ones as repeating zeros like 0.5 = 0.5000.... Examples include √9 = 3, which is rational because it's a perfect square, √2 ≈ 1.414... which is irrational because it's not a perfect square, 0.666... which is rational because it repeats '6' and equals 2/3, and π ≈ 3.14159... which is irrational with no repeating pattern. The statement that it does not terminate and does not repeat, so it is irrational is true, because the decimal of √2 continues indefinitely without settling into a repeating pattern, which is the hallmark of irrational numbers. A common error is claiming it terminates so it's rational, but it doesn't terminate; another is saying it repeats a pattern so it's irrational, but it doesn't repeat, or thinking non-terminating means rational, but non-terminating repeating decimals are rational while non-repeating are irrational, or believing π = 22/7 exactly, but 22/7 is just an approximation and π is truly irrational. For classification: (1) check if it can be a fraction—integers and simple fractions are clearly rational; (2) check the decimal— if it terminates, it's rational; if it repeats, it's rational; if neither, it's irrational; (3) for square roots, if it's a perfect square like √4, √9, or √16, it's rational, but if non-perfect like √2, √3, or √5, it's irrational. Converting a repeating decimal like 0.666...: let x = 0.666..., then 10x = 6.666..., subtract to get 9x = 6, so x = 2/3, showing repeating decimals are rational as they can be expressed as fractions; common mistakes include thinking all square roots are irrational (missing perfect squares), claiming repeating decimals are irrational (but they're rational), or believing π = 22/7 exactly (it's an approximation, π is irrational).

This question tests understanding that irrational numbers cannot be expressed as fractions a/b and have non-terminating, non-repeating decimals, like √2 and π, while rational numbers can be expressed as fractions and have terminating or repeating decimals. Rational numbers include integers like 5 (which is 5/1), fractions like 3/4, terminating decimals like 0.5 (which is 1/2), and repeating decimals like 0.333... (which is 1/3 with the '3' repeating); irrational numbers cannot be written as fractions, and their decimals go on forever without a repeating pattern, such as √2 ≈ 1.41421356... (non-repeating) and π ≈ 3.14159265... (non-repeating), and remember that every number has a decimal expansion where rational ones eventually repeat, including terminating ones as repeating zeros like 0.5 = 0.5000.... Examples include √9 = 3, which is rational because it's a perfect square, √2 ≈ 1.414... which is irrational because it's not a perfect square, 0.666... which is rational because it repeats '6' and equals 2/3, and π ≈ 3.14159... which is irrational with no repeating pattern. The repeating decimal 0.666... converts to the fraction 2/3 in simplest form, as shown by letting x = 0.666..., multiplying by 10 to get 10x = 6.666..., subtracting to yield 9x = 6, so x = 6/9 = 2/3. A common error is thinking √4 is irrational, but it's wrong because it's a perfect square equal to 2, which is rational; another is assuming all decimals are irrational, or claiming 0.333... is irrational, but it repeats so it's rational, or believing π = 22/7 exactly, but 22/7 approximates π and π is truly irrational. For classification: (1) check if it can be a fraction—integers and simple fractions are clearly rational; (2) check the decimal— if it terminates, it's rational; if it repeats, it's rational; if neither, it's irrational; (3) for square roots, if it's a perfect square like √4, √9, or √16, it's rational, but if non-perfect like √2, √3, or √5, it's irrational. Common mistakes include thinking all square roots are irrational (missing perfect squares), claiming repeating decimals are irrational (but they're rational), or believing π = 22/7 exactly (it's an approximation, π is irrational).

This question tests understanding that irrational numbers cannot be expressed as fractions a/b and have non-terminating, non-repeating decimals, like √2 and π, while rational numbers can be expressed as fractions and have terminating or repeating decimals. Rational numbers include integers like 5 (which is 5/1), fractions like 3/4, terminating decimals like 0.5 (which is 1/2), and repeating decimals like 0.333... (which is 1/3 with the '3' repeating); irrational numbers cannot be written as fractions, and their decimals go on forever without a repeating pattern, such as √2 ≈ 1.41421356... (non-repeating) and π ≈ 3.14159265... (non-repeating), and remember that every number has a decimal expansion where rational ones eventually repeat, including terminating ones as repeating zeros like 0.5 = 0.5000.... Examples include √9 = 3, which is rational because it's a perfect square, √2 ≈ 1.414... which is irrational because it's not a perfect square, 0.666... which is rational because it repeats '6' and equals 2/3, and π ≈ 3.14159... which is irrational with no repeating pattern. The correct classification is that A is rational because it repeats (it's the repeating decimal for 22/7, a fraction), and B is irrational because it does not repeat (it's π with its non-repeating expansion). A common error is thinking both are irrational because they don't terminate, but repeating non-terminating decimals like A are rational; another is claiming both are rational because they're decimals, but decimals can be irrational if non-repeating, or saying A is irrational and B rational, which reverses the truth, or believing π = 22/7 exactly, but 22/7 approximates π and π is truly irrational. For classification: (1) check if it can be a fraction—integers and simple fractions are clearly rational; (2) check the decimal— if it terminates, it's rational; if it repeats, it's rational; if neither, it's irrational; (3) for square roots, if it's a perfect square like √4, √9, or √16, it's rational, but if non-perfect like √2, √3, or √5, it's irrational. Converting a repeating decimal like 0.666...: let x = 0.666..., then 10x = 6.666..., subtract to get 9x = 6, so x = 2/3, showing repeating decimals are rational as they can be expressed as fractions; common mistakes include thinking all square roots are irrational (missing perfect squares), claiming repeating decimals are irrational (but they're rational), or believing π = 22/7 exactly (it's an approximation, π is irrational).

This question tests understanding that irrational numbers cannot be expressed as fractions a/b and have non-terminating, non-repeating decimals, like √2 and π, while rational numbers can be expressed as fractions and have terminating or repeating decimals. Rational numbers include integers like 5 (which is 5/1), fractions like 3/4, terminating decimals like 0.5 (which is 1/2), and repeating decimals like 0.333... (which is 1/3 with the '3' repeating); irrational numbers cannot be written as fractions, and their decimals go on forever without a repeating pattern, such as √2 ≈ 1.41421356... (non-repeating) and π ≈ 3.14159265... (non-repeating), and remember that every number has a decimal expansion where rational ones eventually repeat, including terminating ones as repeating zeros like 0.5 = 0.5000.... Examples include √9 = 3, which is rational because it's a perfect square, √2 ≈ 1.414... which is irrational because it's not a perfect square, 0.666... which is rational because it repeats '6' and equals 2/3, and π ≈ 3.14159... which is irrational with no repeating pattern. The best correction is that 0.333... is rational because it repeats, and all repeating decimals can be written as a fraction, specifically 0.333... = 1/3, so the non-terminating aspect doesn't make it irrational if it repeats. A common error is claiming 0.333... is irrational because it never ends, but it's rational due to the repeating pattern; another is saying it's irrational because it's a decimal, but many decimals are rational, or thinking it's rational only if it terminates, but repeating non-terminating are also rational, or believing π = 22/7 exactly, but 22/7 is just an approximation and π is truly irrational. For classification: (1) check if it can be a fraction—integers and simple fractions are clearly rational; (2) check the decimal— if it terminates, it's rational; if it repeats, it's rational; if neither, it's irrational; (3) for square roots, if it's a perfect square like √4, √9, or √16, it's rational, but if non-perfect like √2, √3, or √5, it's irrational. Converting a repeating decimal like 0.666...: let x = 0.666..., then 10x = 6.666..., subtract to get 9x = 6, so x = 2/3, showing repeating decimals are rational as they can be expressed as fractions; common mistakes include thinking all square roots are irrational (missing perfect squares), claiming repeating decimals are irrational (but they're rational), or believing π = 22/7 exactly (it's an approximation, π is irrational).

This question tests understanding that irrational numbers cannot be expressed as fractions a/b and have non-terminating, non-repeating decimals, like √2 and π, while rational numbers can be expressed as fractions and have terminating or repeating decimals. Rational numbers include integers like 5 (which is 5/1), fractions like 3/4, terminating decimals like 0.5 (which is 1/2), and repeating decimals like 0.333... (which is 1/3 with the '3' repeating); irrational numbers cannot be written as fractions, and their decimals go on forever without a repeating pattern, such as √2 ≈ 1.41421356... (non-repeating) and π ≈ 3.14159265... (non-repeating), and remember that every number has a decimal expansion where rational ones eventually repeat, including terminating ones as repeating zeros like 0.5 = 0.5000.... Examples include √9 = 3, which is rational because it's a perfect square, √2 ≈ 1.414... which is irrational because it's not a perfect square, 0.666... which is rational because it repeats '6' and equals 2/3, and π ≈ 3.14159... which is irrational with no repeating pattern. This number should be classified as irrational because it does not terminate and does not repeat a fixed block, as the increasing zeros prevent a periodic pattern, making it non-repeating like irrational decimals. A common error is thinking it's rational because it has a pattern, but the pattern isn't a fixed repeating block, so it's irrational; another is claiming it's rational because it doesn't terminate, but non-termination alone doesn't determine rationality, or saying all decimals are irrational, which is wrong, or believing π = 22/7 exactly, but 22/7 approximates π and π is truly irrational. For classification: (1) check if it can be a fraction—integers and simple fractions are clearly rational; (2) check the decimal— if it terminates, it's rational; if it repeats, it's rational; if neither, it's irrational; (3) for square roots, if it's a perfect square like √4, √9, or √16, it's rational, but if non-perfect like √2, √3, or √5, it's irrational. Converting a repeating decimal like 0.666...: let x = 0.666..., then 10x = 6.666..., subtract to get 9x = 6, so x = 2/3, showing repeating decimals are rational as they can be expressed as fractions; common mistakes include thinking all square roots are irrational (missing perfect squares), claiming repeating decimals are irrational (but they're rational), or believing π = 22/7 exactly (it's an approximation, π is irrational).