Representing real numbers on the number line

The governing idea is repeated equal movement along a number line. Starting at 2, each step has length 1/2. After four steps, the total movement is 4×1/2=2 units. Adding this to the starting position gives 2+2=4, so option C is correct. Equivalently, the successive points are 2.5 after one step, 3 after two, 3.5 after three, and 4 after four. Option A stops after two steps, option B after three steps, and option D moves five steps. The wording “fourth point” is interpreted as four equal intervals from the starting point, consistent with the preceding division setup.

\(\sqrt{\frac{1}{2}}=\frac{1}{\sqrt{2}}=\frac{\sqrt{2}}{2}\approx0.7071\), so its closest decimal estimate is \(0.707\). The square of 0.5 is only 0.25, whereas \(0.707^2\approx0.5\), confirming that 0.707 is correct. An exam tip is to square the options to verify a square-root estimate.

(-0.1) is less than zero, so it is not between (0) and (1). Compare with both bounds when checking between values.

(\frac{2}{5}=\frac{4}{10}), so the distance is (\frac{7}{10}-\frac{4}{10}=\frac{3}{10}). Use a common denominator before subtracting.

Given \(\sqrt{a}=4.5\), squaring both sides gives \(a=(4.5)^2=20.25\). Therefore, option C is correct. Option A is only twice \(4.5\), but removing a square root requires squaring the number. Exam tip: whenever \(\sqrt{a}=x\), write \(a=x^2\).

First, \(\sqrt{0.04}=0.2\). Moving \(0.2\) units to the right on the number line means adding \(0.2\) to \(-0.4\): \(P=-0.4+0.2=-0.2\). Option A results from subtracting \(0.2\), which represents moving left instead. Exam tip: moving right increases the number, while moving left decreases it.

(\sqrt{6}\approx2.45), so (-\sqrt{6}\approx-2.45), which is left of (-2). The smaller number lies to the left.

For positive numbers, (1<2<3) gives (\sqrt{1}<\sqrt{2}<\sqrt{3}). Square root preserves order.

(-\sqrt{4}=-2) and (\sqrt{16}=4), so the midpoint is (\frac{-2+4}{2}=1). Simplify roots first.

(\frac{13}{4}=3+\frac{1}{4}), so it is the first quarter point after (3). Think of improper fractions as mixed numbers.

(\sqrt{5}\approx2.24), which is closest to (2) among the options. Compare each distance for closeness.

(-\sqrt{25}=-5) and (-\sqrt{36}=-6), so (-5) is to the right. Among negatives, the one with smaller magnitude is greater.

(\sqrt{0.81}=0.9), so the distance of (-\sqrt{0.81}) from (0) is (0.9). Distance equals magnitude.

Since (4^2<24<5^2), (\sqrt{24}) is between (4) and (5) and is not equal to (24). Do not treat a square root as the original number.

For the symmetric point (3) is the midpoint so the other point is (2\times3-4.6=1.4). In symmetry the midpoint is equally distant from both endpoints.

Since (6^2<37<7^2), (6<\sqrt{37}<7) and (0<\sqrt{37}-6<1). Subtract the same number from a root interval to locate the value.

Since (2^2=4) and (3^2=9), (\sqrt{7}) lies between (2) and (3). In exams, compare squares first.

Because (1.4^2=1.96) and (1.5^2=2.25), (\sqrt{2}) lies between them. For hard questions, use squares of decimals.

Since (2^2<5<3^2), (\sqrt{5}) lies between (2) and (3), so (-\sqrt{5}) lies between (-3) and (-2). On the negative side, order reverses.

The midpoint of (0) and (1) is (\frac{0+1}{2}=\frac{1}{2}). To find a midpoint on a number line, take the average.