Representing real numbers on the number line

Correct answer: B. 1.75

Moving right means adding 2.25, while moving left means subtracting 0.5. Thus, the final point is \(0+2.25-0.5=1.75\). Option A can result from an incorrect subtraction; remember to add for rightward movement and subtract for leftward movement.

Correct answer: B. (1.7)

(\sqrt{3}) is about (1.732), so (1.7) is closest. For estimation place (\sqrt{3}) between (1) and (2).

Correct answer: A. (2.5) and (-2.5)

Both (2.5) and (-2.5) are (2.5) units from (0). Opposite numbers are equally distant from the origin.

Correct answer: B. (\frac{1}{3})

(0.333\ldots=\frac{1}{3}), so both are at the same point. Connect recurring decimals with fractions.

Correct answer: B. (4)

(\sqrt{20}) is about (4.47), so it is closer to (4). Place (\sqrt{20}) between (4) and (5) and compare distances.

Correct answer: B. (\frac{1}{4})

The total length is (2) and there are (8) parts, so each part is (\frac{2}{8}=\frac{1}{4}). Divide total length by the number of parts.

Correct answer: C. 1.5

The distance is the absolute difference between the numbers: |0.3 − (−1.2)| = |1.5| = 1.5. Because the numbers lie on opposite sides of zero, their distances from zero, 1.2 and 0.3, add to 1.5.

Correct answer: B. (\sqrt{3}) is to the right

Since (3>2), (\sqrt{3}>\sqrt{2}). For positive square roots, the root of the larger number is larger.

Correct answer: B. (0.95)

(\frac{9}{10}=0.9), whose distance from (1) is (0.1), while (0.95) has distance (0.05). The smaller distance is closer.

Correct answer: B. 4

Since \(\sqrt{a^2}=|a|\), substituting \(a=-4\) gives \(\sqrt{(-4)^2}=\sqrt{16}=4\). Therefore, the represented point is 4. Option A is incorrect because the principal square root is never negative, while 16 is only the value of \(a^2\). Exam tip: rewrite \(\sqrt{a^2}\) as \(|a|\) before substituting the value of \(a\).

Correct answer: B. (-0.7,-\frac{3}{5},-0.2)

(-\frac{3}{5}=-0.6), so the order is (-0.7<-0.6<-0.2). Among negatives, the farther left number is smaller.

Correct answer: C. It will be between (3) and (4)

Since (3^2<12<4^2), (\sqrt{12}) lies between (3) and (4). A square root can be much smaller than the number.

Correct answer: C. 1.005

Writing the decimals with the same number of places gives 1.050, 1.500, and 1.005. Since 1.005 is the smallest value, it lies farthest to the left on the number line. Remember that adding a zero at the end of a decimal does not change its value, so 1.05 = 1.050.

Correct answer: C. \(\sqrt{8}<3\)

Since \(3^2=9\) and \(8<9\), we get \(\sqrt{8}<\sqrt{9}=3\). Therefore, \(\sqrt{8}\) lies to the left of 3 on the number line. Option B is incorrect because \(\sqrt{8}\) is not equal to \(\sqrt{9}\). Exam tip: For positive numbers, compare their squares to determine their order.

Correct answer: C. Many points like (\frac{1}{2},\frac{1}{3},\sqrt{\frac{1}{4}})

Between (0) and (1), there are infinitely many rational and irrational numbers. Between any two real numbers, more numbers can be found.

Correct answer: B. (0.2)

The total distance is (0.8-0.2=0.6) and (\frac{0.6}{3}=0.2). Divide total distance by the number of parts for equal sections.

Correct answer: B. (\frac{7}{8})

(\frac{5}{6}\approx0.833) and (\frac{7}{8}=0.875), so (\frac{7}{8}) is to the right. Use decimals or cross multiplication to compare.

Correct answer: C. (A) and (C)

The direct answer is option C: the pair A and C. Distance between two number-line points is the absolute difference of their coordinates. For A and B, \(|0-(-0.5)|=0.5\). For B and C, \(|0.75-0|=0.75\). For A and C, \(|0.75-(-0.5)|=|1.25|=1.25\). Since 1.25 is greatest, AC is the required pair. Option A is smaller at 0.5, and option B is smaller at 0.75. Option D is wrong because the three distances are not equal. The farthest points are the two outermost points on the number line. Compare coordinates by absolute subtraction, not merely by looking at signs.

Correct answer: B. (\frac{3}{5})

The midpoint is (\frac{\frac{2}{5}+\frac{4}{5}}{2}=\frac{3}{5}). To find the exact middle point, take the average of the two points.

Correct answer: B. (3) and (4)

Since (3^2=9) and (4^2=16), (\sqrt{10}) lies between (3) and (4). In exams, bracket square roots between perfect squares.

Correct answer: C. (-5) and (-4)

Since (\sqrt{18}) lies between (4) and (5), (-\sqrt{18}) lies between (-5) and (-4). In exams, the direction reverses for negative square roots.

Correct answer: B. (3) and (4)

(\sqrt{3}) lies between (1) and (2), so (2+\sqrt{3}) lies between (3) and (4). In exams, adding a number shifts the whole interval forward.

Correct answer: B. (3) and (4)

(\sqrt{2}) lies between (1) and (2), so (5-\sqrt{2}) lies between (3) and (4). In exams, change limits carefully while subtracting.

Correct answer: A. (3+\frac{1}{4})

(\frac{13}{4}=3+\frac{1}{4}), so it lies one-fourth after (3). In exams, convert an improper fraction into a mixed number.

Correct answer: B. (-3-\frac{2}{5})

(-\frac{17}{5}=-3-\frac{2}{5}), so it lies between (-4) and (-3). In exams, keep the sign of a negative mixed number correct.