Number line

Between any \(a<b\), there are infinitely many rational as well as irrational numbers, so C is correct. In exams, remember that the real number line has no gaps between two real numbers.

Distance equals the absolute difference of the two coordinates. In exams, treat the negative sign as direction, not distance.

Since (7^2=49) and (8^2=64), (\sqrt{52}) lies between them. In exams, locate square roots by comparing squares.

The coordinate of the midpoint is the average of the coordinates of the two endpoints. Thus, \(\frac{-1.25+2.75}{2}=\frac{1.50}{2}=0.75\). Therefore, option B is correct. Option A, 0.50, results from an arithmetic error in adding or averaging the coordinates. Exam tip: Add the endpoint coordinates and divide the sum by 2 to find the midpoint.

Since (\sqrt{10}) lies between (3) and (4), (-\sqrt{10}) lies between (-4) and (-3). In exams, place negative surds to the left.

Distance means (x=3\pm5), so the values are (-2) and (8). In exams, always consider both directions.

This decimal is non-recurring and non-terminating, so it is irrational. In exams, identify non-recurring infinite decimals as irrational.

(\frac{11}{4}=2.75), so it lies between (2) and (3). In exams, convert fractions to decimals or mixed numbers for placement.

(-\frac{5}{2}=-2.5), and (-2.45) is greater than (-2.5) but less than (-2.4). In exams, compare negative numbers carefully.

Moving left means subtracting, so (1.6-0.9=0.7). In exams, add to move right and subtract to move left.

(-\frac{4}{5}=-0.8), which is the smallest among the given values. In exams, the farthest-left number is the least number.

From (x^2=\frac{1}{9}), (x=\pm\frac{1}{3}), and the given interval needs the negative value. In exams, use conditions to choose the correct sign.

By Pythagoras theorem, the hypotenuse is (\sqrt{2}). In exams, transfer the hypotenuse length to the number line for surd construction.

The middle number is the average, so (\frac{\frac{2}{7}+\frac{3}{7}}{2}=\frac{5}{14}). In exams, find the mean for the midway value.

The distance between two points on a number line is the absolute value of the difference of their coordinates. Thus, \(MN=|-1.8-(-6.2)|=|-1.8+6.2|=4.4\). Therefore, 4.4 is correct. A value such as 3.4 can result from handling the negative signs incorrectly. Exam tip: distance or length is always positive.

(\sqrt{18}\approx4.24), so the nearest integer is (4). In exams, first locate it between consecutive squares.

The number line represents real numbers, and each real number has a unique position. In exams, connect the number line with real numbers.

The equidistant point is the midpoint, so (\frac{-3+4}{2}=\frac{1}{2}). In exams, use the average for equidistant points.

(\sqrt{\frac{1}{2}}) lies between (0) and (1) and cannot be simplified to a rational form. In exams, identify square roots of non-perfect squares as irrational.

A positive number is greater than all negative numbers, so (0.003) is the greatest. In exams, the number to the right is greater.