This question tests approximating irrational numbers using rational bounds, successive refinement, number line location, and comparisons via decimal truncation. Irrational numbers like √2, π have non-repeating decimals; π≈3.14, so π²≈3.14²=9.8596≈9.86. Specific example: √5 approximation showing √4=2<√5<√9=3, refine 2.2²=4.84<5<5.29=2.3², so 2.2<√5<2.3, estimate √5≈2.24. The closest estimate is about 9.86, calculated directly from the approximation. Error like bad approximation (about 6.28 which is 2π, not π²) or miscalculation (12.56=4π). Process: (1) use given approximation π≈3.14, (2) square it: 3.14×3.14=9.8596, (3) round to nearest. Mistakes: confusing π² with other multiples or arithmetic errors in multiplication.

This question tests approximating irrational numbers using rational bounds, successive refinement, number line location, and comparisons via decimal truncation. Irrational numbers like √2, π have non-repeating decimals; for √5, bound between 2 and 3 since 4<5<9, refine to 2.2<√5<2.3 as 4.84<5<5.29, and on a number line from 2 to 3 with tenths, place at about 2.24, between 2.2 and 2.3, closer to 2.2. Specific example: ordering √2≈1.41, 1.5, 1.7, √3≈1.73 giving 1.5<√2<1.7<√3? Wait, actually √2<1.5<1.7<√3. The correct placement is at about 2.24, between 2.2 and 2.3, closer to 2.2 since √5≈2.236 is 0.036 from 2.2 and 0.064 from 2.3. Errors include bad approximations like at 2.50 (halfway, but 2.5²=6.25>5) or at 2.90 (too close to 3, since 2.9²=8.41>5 much higher). Process: (1) bound with perfect squares (4 and 9), (2) refine to tenths, (3) place proportionally on number line (5 is 1 from 4 to 9, but square root scales differently, better use refined bounds). Number line: mark 2.0 to 3.0, place √5 using approximation √5≈2.236 near 2.2; mistakes: misplaced due to linear thinking instead of squaring check.

This question tests approximating irrational numbers using rational bounds, successive refinement, number line location, and comparisons via decimal truncation. Irrational numbers like √2, π have non-repeating decimals; for √50, bound between 7 and 8 since 49<50<64, and ≈7.07 as it's close to √49=7, or 5√2≈5×1.414=7.07. Specific example: ordering √2≈1.41, 1.5, 1.7, √3≈1.73 giving √2<1.5<1.7<√3. The number closest to √50 is 7.1, as 7.1²=50.41 close to 50, while 7²=49 and 7.07 is nearer to 7.1 than others. Error like bad approximation (8.5 too high since 8²=64>50 far). Process: (1) identify nearby perfect squares (49 and 64), (2) estimate closer to 7, (3) compare distances. Mistakes: wrong perfect squares or misjudging proximity.

This question tests approximating irrational numbers using rational bounds, successive refinement, number line location, and comparisons via decimal truncation. Irrational numbers like √2 or π have non-repeating decimals, such as π ≈ 3.14159265... continuing without pattern, approximated by truncation like to 3.14. Truncating π to 3.14 and squaring gives 3.14² = 9.8596 ≈ 9.86, which is closest to choice C. This is accurate as it matches the calculation directly. An error might be confusing with 2π ≈ 6.28 or π + π ≈ 6.28, leading to wrong choices like A. The process is: (1) truncate to the given digits; (2) square the approximation; (3) round to match options. This provides a reasonable estimate, avoiding mistakes like using rounding instead of truncation.

This question tests approximating irrational numbers using rational bounds, successive refinement, number line location, and comparisons via decimal truncation. Irrational numbers like √2 or π have non-repeating decimals, such as √2 ≈ 1.41421356... continuing without pattern, placed on a number line between bounds like 1.4 and 1.5. For √5 ≈ 2.236 on a number line with P at 2.2 and Q at 2.3, since 2.2 < 2.236 < 2.3, it belongs between P and Q, confirming choice C. This is determined by refining bounds: 2.2² = 4.84 < 5 < 5.29 = 2.3². A mistake would be placing it left of 2.2 if underestimating, like confusing with √4 = 2. The process is: (1) bound with decimals; (2) compare to points; (3) locate accordingly. This visualizes the position accurately, preventing misplaced approximations.

This question tests approximating irrational numbers using rational bounds, successive refinement, number line location, and comparisons via decimal truncation. Irrational numbers like √2, π have non-repeating decimals; for √30, bound with decimals like 5.4²=29.16<30<30.25=5.5², so 5.4<√30<5.5. Specific example: √5 approximation showing √4=2<√5<√9=3, refine 2.2²=4.84<5<5.29=2.3², so 2.2<√5<2.3, estimate √5≈2.24. The true comparison is √30<5.5, since 5.5²=30.25>30. Error like comparison reversed (√30>5.5 when actually <5.5) or wrong (√30<5.4 but 5.4²=29.16<30 so >5.4). Process: (1) test squares like 5.5²>30, (2) confirm direction. Comparison: use squaring to verify; mistakes: arithmetic errors or reversed direction.

This question tests approximating irrational numbers using rational bounds, successive refinement, number line location, and comparisons via decimal truncation. Irrational numbers like √2, π have non-repeating decimals (√2=1.41421356... continues without pattern). Approximate by bounding: √17 between integers (4²=16<17<25=5², so 4<√17<5), refine using decimals (4.1²=16.81<17<17.64=4.2², so 4.1<√17<4.2), continuing narrows to better approximation (√17≈4.123). For √5, between 2 and 3, closer to 2 since ≈2.24<2.5 midpoint. A common error is saying closer to 3, but 5 is nearer to 4 than 9. The process: (1) bound with perfect squares, (2) estimate distance, (3) compare to midpoint. Number line: place proportionally closer to 2.

This question tests approximating irrational numbers using rational bounds, successive refinement, number line location, and comparisons via decimal truncation. Irrational numbers like √2, π have non-repeating decimals (√2=1.41421356... continues without pattern). Approximate by bounding: √17 between integers (4²=16<17<25=5², so 4<√17<5), refine using decimals (4.1²=16.81<17<17.64=4.2², so 4.1<√17<4.2), continuing narrows to better approximation (√17≈4.123). For π² with π≈3.14, 3.14²=9.8596≈9.86. A common error is miscalculating as 6.28 (like 2π), but squaring gives about 9.86. The process: (1) use given approximation, (2) square it, (3) round to closest option. Comparison: verify by calculating manually.

This question tests approximating irrational numbers using rational bounds, successive refinement, number line location, and comparisons via decimal truncation. Irrational numbers like √2, π have non-repeating decimals; approximate by bounding: √17 between integers (4²=16<17<25=5², so 4<√17<5), refine (4.1²=16.81<17<17.64=4.2², so 4.1<√17<4.2). Specific example: √5 approximation showing √4=2<√5<√9=3, refine 2.2²=4.84<5<5.29=2.3², so 2.2<√5<2.3, estimate √5≈2.24. The correct statement is √17>4.1, since 4.1²=16.81<17 implies √17>4.1, and it's greater than 4. Error like comparison reversed (√17<4.1 when actually >4.1) or wrong bounds (√17<4 when >4). Process: (1) identify perfect squares near 17 (16 and 25), (2) bound (4<√17<5), (3) refine to compare with 4.1. Comparison: since 16.81<17, √17>4.1; mistakes: arithmetic errors squaring decimals or direction reversed.

This question tests approximating irrational numbers using rational bounds, successive refinement, number line location, and comparisons via decimal truncation. Irrational numbers like √2, π have non-repeating decimals (√2=1.41421356... continues without pattern). Approximate by bounding: √17 between integers (4²=16<17<25=5², so 4<√17<5), refine using decimals (4.1²=16.81<17<17.64=4.2², so 4.1<√17<4.2), continuing narrows to better approximation (√17≈4.123). The true statement is 4<√20<5 since 16<20<25. A common error is claiming equality like √20=4.5, but it's irrational. The process: (1) identify perfect squares around 20, (2) bound accordingly, (3) eliminate exact or wrong bounds. Comparison: verify by squaring the bounds.