Number line

Writing \(-0.13\) as \(-0.130\) makes the comparison clear. On a number line, the more negative number lies farther to the left and is smaller. Since \(-0.130 < -0.103\), \(-0.13\) is the correct answer. \(-0.103\) is closer to zero, so it is greater. Exam tip: When comparing negative decimals, add trailing zeros if needed to make the decimal places equal.

Option C is correct. Between any two distinct real numbers, there are infinitely many rational numbers and infinitely many irrational numbers. Hence both types lie between \(\sqrt{2}\) and \(\sqrt{3}\). Exam tip: remember this as the density property of real numbers.

The absolute value of a number is its distance from 0 on the number line. Since \(-25\) is 25 units away from 0, its absolute value is \(25\). \(-25\) is the original number, not its absolute value. Exam tip: The absolute value of a negative number is always positive.

The midpoint of two numbers on a number line is their average: \(\frac{-2+14}{2}=\frac{12}{2}=6\). Therefore, 6 is correct. For example, 7 is not the midpoint because it is not equally distant from -2 and 14. Exam tip: add the two numbers and divide the sum by 2.

Since \(2.2^2=4.84<5\), we have \(2.2<\sqrt{5}\), so Q lies to the right of P. Distance is larger coordinate minus smaller coordinate: \(\sqrt{5}-2.2\). Exam tip: check the order before subtracting.

On a number line, the distance between two numbers is the absolute value of their difference: \(|7-(-1)|=|8|=8\). Therefore, the correct answer is 8. The number 7 is only the larger coordinate, not the distance between the two points. Exam tip: while finding distance, subtracting a negative becomes addition, and the final distance is always non-negative.

In increasing order, numbers are written from smallest to greatest. Among negative decimals, the number with the greater magnitude is smaller, so -0.2 is less than -0.02. This is followed by 0 and then 0.2. Hence, the correct order is ( -0.2, -0.02, 0, 0.2 ). In option C, -0.02 and -0.2 are placed in the wrong order. Exam tip: Visualise decimals from left to right on the number line when comparing them.

If \(x<y\), choose \(n\) such that \(1/n<y-x\); then some \(m/n\) lies between them. Hence A is correct. D is false because irrationals also lie between them. Exam tip: both sets are dense.

On the number line, the integers to the right of \( -12 \) and to the left of \( -6 \) are \( -11, -10, -9, -8, -7 \). Therefore, \( -10 \) lies between \( -12 \) and \( -6 \). \( -13 \) and \( -15 \) are less than \( -12 \), while \( -5 \) is greater than \( -6 \). Exam tip: For negative numbers, values increase as you move to the right on the number line.

The midpoint of two numbers on a number line is their average: \(\frac{-4+8}{2}=\frac{4}{2}=2\). Therefore, \(2\) is correct. \(4\) is the sum of the two numbers before dividing by \(2\), so it is not the midpoint. Exam tip: add the two numbers and divide the result by \(2\) to find their midpoint.

On a number line, the distance between two numbers is the positive value of their difference. Thus, \(\lvert -15-0\rvert=\lvert-15\rvert=15\). Therefore, 15 is correct. The value 14 is merely one less and is not the distance. Exam tip: always use the absolute value of the difference, since distance cannot be negative.

On the number line, among negative numbers the value closer to zero is greater. The decimal -0.0003 is closer to zero than -0.003. Equivalently, -0.003=-30/10000 and -0.0003=-3/10000; since -3 is greater than -30, -0.0003 is greater. They are not equal, so option B is correct.

The midpoint of two numbers on a number line is their average: \(\frac{-27+13}{2}=\frac{-14}{2}=-7\). Therefore, the correct answer is \((-7)\). \(6\) is incorrect because it is not equally distant from both numbers. Exam tip: add the two numbers first, then divide by 2 to find the midpoint.

The distance between two numbers on a number line is the absolute value of their difference: \(\lvert -18-(-9)\rvert=\lvert -9\rvert=9\). Therefore, the correct answer is 9. Although 8 may seem close, distance must be found using the absolute value of the difference. Exam tip: distance is never negative.

Between any two distinct real numbers, there are infinitely many rational and irrational numbers because both sets are dense. An integer need not lie between them. Exam tip: for “always” questions, recall density.

The successor of an integer is obtained by adding 1. Thus, \( -45+1=-44 \), so -44 is correct. -46 is one less than -45, so it is the predecessor. Exam tip: Even for negative numbers, the successor lies to the right on the number line.

For negative numbers, the number closer to zero is greater. The magnitude of ( -0.00008 ) is 0.00008, which is smaller than 0.0008; therefore, ( -0.00008 ) is closer to zero and is greater. ( -0.0008 ) lies further to the left on the number line. Exam tip: When comparing negative decimals, the number with the smaller magnitude is greater.

In right triangle OAB, OA = 3 and AB = 1. Thus, OB² = 3² + 1² = 10, so OB = \(\sqrt{10}\). The arc places P at a distance \(\sqrt{10}\) from O. In such constructions, apply Pythagoras’ theorem first.

The midpoint of two numbers on a number line is their average: \(\frac{-35+18}{2}=\frac{-17}{2}=-8.5\). Therefore, \(-8.5\) is correct. Although \(-8\) 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.

The distance between two numbers on a number line is the absolute value of their difference: \(\lvert -19-(-7)\rvert=\lvert -12\rvert=12\). Therefore, 12 is correct. Choosing 11 can result from a counting error, but distance is always the positive value of the difference. Exam tip: Use brackets carefully when subtracting a negative number.