Number line

In (\frac{5}{8}), the denominator is (8), so divide the unit into (8) equal parts and take the (5)th. In exams, the denominator gives the number of equal parts.

Movement to the left on a number line means subtracting the stated distance from the starting coordinate. First convert 7/2 to a decimal: 7/2 = 3.5. Therefore the coordinate of S is 3.5 − 2.25 = 1.25. Converting 1.25 to a fraction gives 1.25 = 125/100 = 5/4. Hence option A is correct. The value 23/4 would result from adding the distance instead of moving left, while 9/4 and 3/4 come from incorrect subtraction or conversion. The governing number-line rule is that rightward movement increases a coordinate and leftward movement decreases it.

(\sqrt{80}\approx8.94), so it lies slightly left of (9). In exams, connect approximation with direction near the nearest integer.

The approximate values are (-2.25), (-2.2), (-2.236), and (-2.3); from the right, (-2.2) comes first and (-\sqrt{5}) second. In exams, order negative decimals using approximation.

On the number line, the greater number lies to the right, so (b) is to the right of (a). In exams, connect inequalities with direction.

The governing concept is the geometric construction of an irrational length using the Pythagorean theorem. Construct a right triangle whose two perpendicular legs each have length 1. Its hypotenuse is √(1² + 1²) = √(1 + 1) = √2. This length can then be transferred to the number line with a compass, giving the point representing √2. Therefore, option B is correct. Legs 1 and 2 produce √5, legs 2 and 2 produce √8, and legs 3 and 1 produce √10. Although those lengths are also valid hypotenuses of right triangles, they do not construct √2. The equal-unit-leg triangle is the required starting construction.

The reliable governing method for locating a square root is to bracket its radicand between consecutive perfect squares. Find a whole number a such that a² < n < (a + 1)². Taking positive square roots gives a < √n < a + 1, so the correct interval on the number line is known even when √n has a non-terminating decimal. The spiral or compass can then construct and mark the exact length. Therefore option B is correct. Option A may be unnecessary and can introduce rounding errors; option C confuses the radicand with its square root; and option D is true only for special values such as roots between 1 and 2, not for every n.

\(\sqrt{7}\) is irrational because 7 is not a perfect square. Its decimal expansion is non-terminating and non-repeating, but it is a real number, so it has a unique point on the number line. Exam tip: the square root of a non-perfect square is irrational.

Since \(3^2=9\) and \(4^2=16\), and \(9<10<16\), we get \(3<\sqrt{10}<4\). Hence, \(\sqrt{10}\) is the correct option. \(\sqrt{8}<3\), while both \(\sqrt{17}\) and \(\sqrt{20}\) are greater than 4. Exam tip: locate a square root by comparing the number inside it with nearby perfect squares.