Number line

On a number line, the number farther to the right is greater. \(-13\) lies to the right of \(-20\) and is also closer to zero, so \(-13 > -20\). “Both are equal” is incorrect because the two numbers are different. Exam tip: among negative numbers, the number closer to zero is greater.

The successor of an integer is found by adding 1. Thus, \( -25 + 1 = -24 \), so \( -24 \) is correct. \( -26 \) is the predecessor because it lies to the left of \( -25 \) on the number line. Exam tip: for negative integers, numbers increase as you move to the right.

The midpoint of two numbers on a number line is their average: \(\frac{-8+12}{2}=\frac{4}{2}=2\). Hence, \(2\) is correct. Although \(0\) lies between the two numbers, it is not at equal distance from both. Exam tip: Add the two numbers and divide by 2 to find their midpoint.

The distance between two numbers on a number line is the absolute value of their difference: \(|9-(-2)|=|11|=11\). Therefore, the correct answer is 11. Choosing 10 usually results from an error while subtracting a negative number. Exam tip: Always take the absolute value of the difference, so distance is never negative.

Since \(2^2=4\) and \(3^2=9\), we have \(4<5<9\). Therefore, \(2<\sqrt{5}<3\), so option A is correct. Exam tip: compare the number with nearby perfect squares to locate a square root.

On a number line, the number farther to the right is greater.
-6 is closer to zero and lies to the right of -6.5, so -6 is greater. Although -6.25 is greater than -6.5, it is still less than -6. Exam tip: among negative numbers, the number closer to zero is greater.

For negative numbers, the number closer to zero is greater. Since \(-17=-17.0\) and \(-17.0>-17.8\), \(-17\) is greater. Although \(-17.08\) is close to \(-17\), it is still smaller than \(-17\). Exam tip: Write negative decimals to equal decimal places before comparing them.

-0.125 is smaller than -0.12 because, among negative numbers, the number farther to the left on the number line is smaller. Write -0.12 as -0.120: -0.125 < -0.120. Hence, option B is correct. -0.12 is larger because it is closer to zero. Exam tip: Add zeros at the end of decimals, when needed, to make the number of decimal places equal before comparing.

The interval between any two distinct real numbers contains infinitely many rational as well as irrational numbers, so B is correct. Exam tip: remember this as the density property of number sets.

For rational \(p<q\), \(p+\frac{q-p}{\sqrt{2}}\) is irrational and lies between them, since \(0<\frac{1}{\sqrt{2}}<1\). Thus A is correct. Exam tip: remember that both rational and irrational numbers are dense.

The midpoint of two numbers on a number line is their average: \(\frac{-9+4}{2}=\frac{-5}{2}=-2.5\). Hence, \( -2.5 \) is correct. Although \( -3 \) is close, it is not at an equal distance from both numbers. Exam tip: add the two numbers first, then divide the sum by 2.

On a number line, the distance between two numbers is the absolute value of their difference: \(|7-(-13)|=|20|=20\). Therefore, the correct answer is 20. A value such as 18 can result from handling the negative sign incorrectly. Exam tip: always take the absolute value of the difference when finding distance.

The correct answer is -3.05. We can write -3.5 as -3.50, and -3.05 is greater than -3.50. On a number line, among negative numbers, the number closer to zero is greater. Therefore, -3.05 > -3.5. Exam tip: Add trailing zeros when needed to make the decimal places equal before comparing decimals.

\(0\) is greater than \(-0.001\) because \(-0.001\) is a negative number. On a number line, every negative number lies to the left of zero, so zero is greater. The option ‘Both numbers are equal’ is incorrect because one number is zero and the other is negative. Exam tip: On a number line, the number farther to the right is always greater.

The distance between two numbers on a number line is their absolute difference: \(|9-(-4)|=|13|=13\). Therefore, the correct distance is 13 units. The value 5 can result from an incorrect calculation such as \(|-4|+1\), which does not represent the distance here. Exam tip: When subtracting a negative number, carefully account for its sign.

C is correct. With \(n(b-a)>1\), a rational \(m/n\) can lie between \(a\) and \(b\); infinitely many irrationals lie there too. Thus A, B and D fail. Exam tip: recall the density property.

Since \(3.5^2=12.25\) and \(4^2=16\), we get \(12.25<13<16\). Hence \(\sqrt{13}\) lies between 3.5 and 4, not between 3 and 3.5. Exam tip: compare nearby squares.

The midpoint of two numbers on a number line is their average: \(\frac{-8+(-2)}{2}=\frac{-10}{2}=-5\). Therefore, \( -5 \) is correct. Although \( -4 \) lies between \( -8 \) and \( -2 \), it is not equally distant from both numbers. Exam tip: include the signs of negative numbers while adding them for a midpoint.

On a number line, the distance between two numbers is the absolute value of their difference: \(\lvert -15-(-5)\rvert=\lvert -10\rvert=10\). Therefore, the correct answer is 10. The value 5 is only the difference of the magnitudes, not the distance between the actual positions. Exam tip: a distance is always non-negative.

\( -6.7 = -6.70 \) because adding a zero at the end of the decimal part does not change the value of a number. Therefore, both numbers represent the same point on the number line, so neither is greater. It is incorrect to treat \( -6.70 \) as smaller merely because it has more decimal places. Exam tip: Remove trailing zeros before comparing decimals.