Representing real numbers on the number line

( \sqrt{2}\approx1.414) and ( \sqrt{3}\approx1.732), so the sum is about (3.146). Estimation is a safe method for such sums.

The distance from ( -\frac{5}{6}) to (-1) is ( \frac{1}{6}), and to (0) is ( \frac{5}{6}). Closeness depends on the smaller distance.

The governing concept is the midpoint formula for two points on a number line. If the endpoints are a and b, their midpoint is M=(a+b)/2, because this average is equally distant from both endpoints. Substituting a=-1.5 and b=4.5 gives M=(-1.5+4.5)/2=3/2=1.5. The result can also be checked geometrically: the distance from -1.5 to 1.5 is 3 units, and the distance from 1.5 to 4.5 is also 3 units. Therefore option A is correct. Option B is the total of the two values after an incorrect operation, option C is not their average, and option D does not divide the interval into two equal parts.

( \sqrt{8}\approx2.828), so (2.9) lies between it and (3). First estimate the irrational number.

( \sqrt{20}\approx4.47) and ( \sqrt{27}\approx5.19), so (5) lies between them. Use perfect squares to identify bounds.

From (8<\sqrt{n}<9), we get (64<n<81), so (70) is possible. Square positive bounds carefully.

( \frac{3}{7}\approx0.428), ( \frac{1}{2}=0.5), and ( \frac{4}{7}\approx0.571). Use decimals or cross multiplication to compare fractions.

( \sqrt{11}\approx3.316), so (p\approx-3.316), which is less than (-3). Be careful with direction while comparing negatives.

Moving \(\frac{7}{3}\) units to the right of \(-2\) gives \(-2+\frac{7}{3}=\frac{1}{3}\). Option B is also \(\frac{7}{3}\) units from \(-2\), but it lies to the left, so its direction is incorrect. The distances of options C and D from \(-2\) are \(\frac{5}{3}\) and \(\frac{13}{3}\), respectively. Exam tip: add the distance when moving right and subtract it when moving left.

Both (1.999) and (2.001) are at distance (0.001) from (2). Points at equal distance are equally close.

With perpendicular sides (1) and (1), the hypotenuse is ( \sqrt{2}). Understand the construction using Pythagoras theorem.

(0.4040040004\ldots) is non-terminating and non-repeating, so it is irrational. Observe the decimal pattern carefully.

( \sqrt{3}\approx1.732) and ( \frac{7}{4}=1.75), so ( \sqrt{3}) is slightly smaller. Use decimal comparison for close values.

Since \(7^2=49\) and \(8^2=64\), \(\sqrt{50}\) is just slightly greater than 7. More precisely, \(\sqrt{50}\approx7.071\), so \(a-7\approx0.071\), which is closest to 0.07. The value 0.7 is ten times larger than this difference. Exam tip: estimate the square root near the nearest perfect square before performing the required operation.

( -\sqrt{2}\approx-1.414), so (-1.35) lies between it and (-1.3). Read order carefully in negative intervals.

The midpoint is ( \frac{-\frac{9}{2}-\frac{1}{2}}{2}=-\frac{5}{2}). Add the fractions first, then divide by (2).

( \frac{22}{7}\approx3.142857) and ( \pi\approx3.14159), so ( \frac{22}{7}) is slightly larger. Do not treat common approximations as exactly equal.

(3.8^2=14.44) and (3.9^2=15.21), so ( \sqrt{15}) lies between them. Check squares for decimal bounds.

Moving \(\sqrt{13}\) units to the right from \(-4\) gives \(x=-4+\sqrt{13}\). Since \(3^2<13<4^2\), we have \(3<\sqrt{13}<4\). Adding \(-4\) throughout gives \(-1<x<0\), so \(x\) lies between \(-1\) and \(0\). Option B is incorrect because the value of \(x\) is still negative. Exam tip: locate a square root between two consecutive integers before adding or subtracting it.

Write the decimal as a fraction first: \(0.125=\frac{125}{1000}\). Dividing the numerator and denominator by \(125\) gives \(\frac{125}{1000}=\frac{1}{8}\), so option A is correct. The distractor \(\frac{1}{4}\) equals \(0.25\), not \(0.125\). Exam tip: for a decimal with three digits after the decimal point, initially use \(1000\) as the denominator and then simplify.