Understanding irrational numbers means recognizing that they cannot be expressed as fractions a/b where a and b are integers and b ≠ 0, and their decimal expansions are non-terminating and non-repeating, like √2 or π, while rational numbers can be expressed as such 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, however, cannot be written as fractions, and their decimals go on forever without a repeating pattern, such as √2 ≈ 1.41421356... or π ≈ 3.14159265.... Examples include √9 = 3, which is rational as it's a perfect square, while √2 is irrational since it's not a perfect square; 0.666... is rational as it equals 2/3, but π is irrational with no repeating pattern. The correct choice is 1.41421356... because it represents √2, which has a non-terminating, non-repeating decimal, making it irrational, while 0.125 = 1/8 terminates, 0.3¯3 = 1/3 repeats, and 2.7 = 27/10 terminates. A common error is assuming all long decimals are irrational, but repeating ones like 0.333... are rational, or thinking terminating decimals like 0.5 are irrational when they can be fractions; another is claiming π = 22/7 exactly, but it's an approximation since π is irrational. To classify, check if it can be a fraction: for decimals, terminating or repeating means rational, neither means irrational; you can convert repeating like let x = 0.666..., 10x = 6.666..., 9x = 6, x = 2/3; avoid mistakes like thinking all non-terminating decimals are irrational if they actually repeat.

Understanding irrational numbers means recognizing that they cannot be expressed as fractions a/b where a and b are integers and b ≠ 0, and their decimal expansions are non-terminating and non-repeating, like √2 or π, while rational numbers can be expressed as such 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, however, cannot be written as fractions, and their decimals go on forever without a repeating pattern, such as √2 ≈ 1.41421356... or π ≈ 3.14159265.... Examples include √9 = 3, which is rational as it's a perfect square, while √2 is irrational since it's not a perfect square; 0.666... is rational as it equals 2/3, but π is irrational with no repeating pattern. The correct statement is that √16 is rational because √16 = 4, which is 4/1, while claiming all square roots are irrational ignores perfect squares, its decimal is 4.0 (terminating), and it's not neither. A common error is thinking √4 is irrational, but it's 2 and rational; another is assuming 0.333... is irrational when it repeats and equals 1/3, or that π = 22/7 exactly, but π is irrational. To classify square roots: if perfect square like √16 = 4, rational; non-perfect like √2, irrational; convert repeating decimals like x = 0.666..., 10x = 6.666..., 9x = 6, x = 2/3 to show rational; avoid thinking all roots are irrational or repeating decimals irrational.

This question tests understanding that irrational numbers cannot be expressed as fractions a/b where a and b are integers with b ≠ 0, and they have non-terminating, non-repeating decimal expansions, like √2 or π, while rational numbers terminate or repeat. Rational numbers include integers like 5 (which is 5/1), fractions like 3/4, terminating decimals like 0.5 (equal to 1/2), and repeating decimals like 0.333... (equal to 1/3, with the pattern repeating). Irrational numbers cannot be written as such fractions, and their decimals continue forever without a repeating pattern, such as √2 ≈ 1.41421356... (non-repeating) or π ≈ 3.14159265... (non-repeating); every number has a decimal expansion where rational ones eventually repeat (including terminating decimals as repeating zeros, like 0.5 = 0.5000...), but irrational ones never settle into a repeating pattern. For example, √9 = 3 is rational because it's a perfect square, √2 ≈ 1.414... is irrational as it's not a perfect square, 0.666... is rational because it repeats '6' and equals 2/3, and π ≈ 3.14159... is irrational with no repeating pattern. The counterexample is D: 0.75, which is a terminating decimal equal to 3/4, proving that not all decimals are irrational since this one is rational. 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 that 0.333... is irrational, but it repeats and is rational, or believing π = 22/7 exactly, but 22/7 is just an approximation and π is truly irrational. To classify, first check if it can be expressed as a fraction: integers and simple fractions are clearly rational; second, examine the decimal— if it terminates or repeats, it's rational, if neither, it's irrational; third, for square roots, if it's a perfect square like √4, √9, or √16, it's rational, but non-perfect like √2, √3, or √5 are irrational. Converting a repeating decimal like x = 0.666..., multiply by 10 to get 10x = 6.666..., subtract to find 9x = 6, so x = 2/3, showing repeating decimals are rational; common mistakes include thinking all square roots are irrational (missing perfect squares), claiming repeating decimals are irrational (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 where a and b are integers with b ≠ 0, and they have non-terminating, non-repeating decimal expansions, like √2 or π, while rational numbers terminate or repeat. Rational numbers include integers like 5 (which is 5/1), fractions like 3/4, terminating decimals like 0.5 (equal to 1/2), and repeating decimals like 0.333... (equal to 1/3, with the pattern repeating). Irrational numbers cannot be written as such fractions, and their decimals continue forever without a repeating pattern, such as √2 ≈ 1.41421356... (non-repeating) or π ≈ 3.14159265... (non-repeating); every number has a decimal expansion where rational ones eventually repeat (including terminating decimals as repeating zeros, like 0.5 = 0.5000...), but irrational ones never settle into a repeating pattern. For example, √9 = 3 is rational because it's a perfect square, √2 ≈ 1.414... is irrational as it's not a perfect square, 0.666... is rational because it repeats '6' and equals 2/3, and π ≈ 3.14159... is irrational with no repeating pattern. The true statement is D: 22/7 = 3.142857142857… is rational because its decimal repeats with the block 142857, correctly identifying repeating decimals as rational. 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 that 0.333... is irrational, but it repeats and is rational, or believing π = 22/7 exactly, but 22/7 is just an approximation and π is truly irrational. To classify, first check if it can be expressed as a fraction: integers and simple fractions are clearly rational; second, examine the decimal— if it terminates or repeats, it's rational, if neither, it's irrational; third, for square roots, if it's a perfect square like √4, √9, or √16, it's rational, but non-perfect like √2, √3, or √5 are irrational. Converting a repeating decimal like x = 0.666..., multiply by 10 to get 10x = 6.666..., subtract to find 9x = 6, so x = 2/3, showing repeating decimals are rational; common mistakes include thinking all square roots are irrational (missing perfect squares), claiming repeating decimals are irrational (they're rational), or believing π = 22/7 exactly (it's an approximation, π is irrational).

Understanding irrational numbers means recognizing that they cannot be expressed as fractions a/b where a and b are integers and b ≠ 0, and their decimal expansions are non-terminating and non-repeating, like √2 or π, while rational numbers can be expressed as such 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, however, cannot be written as fractions, and their decimals go on forever without a repeating pattern, such as √2 ≈ 1.41421356... or π ≈ 3.14159265.... Examples include √9 = 3, which is rational as it's a perfect square, while √2 is irrational since it's not a perfect square; 0.666... is rational as it equals 2/3, but π is irrational with no repeating pattern. The always true statement is that a number is rational if its decimal terminates or repeats, while the others are false: non-terminating non-repeating is irrational, terminating is rational, and a/b form is rational. A common error is thinking all non-terminating decimals are irrational, but if they repeat, they're rational like 0.333...; another is claiming π = 22/7 exactly, but it's an approximation since π is irrational. To classify: check decimal type or fraction form; convert repeating like x = 0.666..., 10x = 6.666..., 9x = 6, x = 2/3; for square roots, perfect are rational; avoid reversing definitions.

This question tests understanding that irrational numbers cannot be expressed as fractions a/b where a and b are integers with b ≠ 0, and they have non-terminating, non-repeating decimal expansions, like √2 or π, while rational numbers terminate or repeat. Rational numbers include integers like 5 (which is 5/1), fractions like 3/4, terminating decimals like 0.5 (equal to 1/2), and repeating decimals like 0.333... (equal to 1/3, with the pattern repeating). Irrational numbers cannot be written as such fractions, and their decimals continue forever without a repeating pattern, such as √2 ≈ 1.41421356... (non-repeating) or π ≈ 3.14159265... (non-repeating); every number has a decimal expansion where rational ones eventually repeat (including terminating decimals as repeating zeros, like 0.5 = 0.5000...), but irrational ones never settle into a repeating pattern. For example, √9 = 3 is rational because it's a perfect square, √2 ≈ 1.414... is irrational as it's not a perfect square, 0.666... is rational because it repeats '6' and equals 2/3, and π ≈ 3.14159... is irrational with no repeating pattern. The counterexample is D: 0.75, which is a terminating decimal equal to 3/4, proving that not all decimals are irrational since this one is rational. 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 that 0.333... is irrational, but it repeats and is rational, or believing π = 22/7 exactly, but 22/7 is just an approximation and π is truly irrational. To classify, first check if it can be expressed as a fraction: integers and simple fractions are clearly rational; second, examine the decimal— if it terminates or repeats, it's rational, if neither, it's irrational; third, for square roots, if it's a perfect square like √4, √9, or √16, it's rational, but non-perfect like √2, √3, or √5 are irrational. Converting a repeating decimal like x = 0.666..., multiply by 10 to get 10x = 6.666..., subtract to find 9x = 6, so x = 2/3, showing repeating decimals are rational; common mistakes include thinking all square roots are irrational (missing perfect squares), claiming repeating decimals are irrational (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).

Understanding irrational numbers means recognizing that they cannot be expressed as fractions a/b where a and b are integers and b ≠ 0, and their decimal expansions are non-terminating and non-repeating, like √2 or π, while rational numbers can be expressed as such 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, however, cannot be written as fractions, and their decimals go on forever without a repeating pattern, such as √2 ≈ 1.41421356... or π ≈ 3.14159265.... Examples include √9 = 3, which is rational as it's a perfect square, while √2 is irrational since it's not a perfect square; 0.666... is rational as it equals 2/3, but π is irrational with no repeating pattern. The correct fraction is 3/11 because letting x = 0.272727..., 100x = 27.272727..., subtracting gives 99x = 27, x = 27/99 = 3/11 in simplest form, while 27/100 = 0.27 (terminating), 27/99 = 3/11 (not simplest listed), and 11/3 ≈ 3.666... don't match. A common error is thinking repeating decimals are irrational, but they convert to fractions and are rational; another is assuming π = 22/7 exactly, but π is irrational. To classify, check if fraction possible; for repeating decimals, use algebra to convert; avoid mistakes like confusing terminating with repeating or thinking all long decimals are 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 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).