Number line

Real numbers are dense on the number line: between any two distinct real numbers, infinitely many rational and irrational numbers exist. Exam tip:
\(\sqrt{2}\) also corresponds to one definite point on the number line.

\( -6.6 = -6.60 \) because a zero added at the end of a decimal does not change its value. Hence, both numbers represent the same point on the number line. Choosing \( -6.6 \) as greater is incorrect; the extra zero only shows an additional decimal place. Exam tip: Remove trailing zeros before comparing decimals.

The midpoint of two numbers \(a\) and \(b\) is \(\frac{a+b}{2}\). Therefore, \(\frac{-20+(-4)}{2}=\frac{-24}{2}=-12\). Hence, \(-12\) is the correct answer. \(-10\) would be the midpoint of \(-20\) and \(0\), so it is not correct here. Exam tip: When adding negative numbers, add their magnitudes and keep the negative sign.

On a number line, the distance between two numbers is the absolute value of their difference: \(|9-(-3)|=|12|=12\). Therefore, the correct answer is 12. Although -12 can be a difference depending on the order of subtraction, distance is never negative. Exam tip: always take the absolute value of the difference in distance questions.

The integers strictly between ( -8 ) and ( -1 ) are ( -7 ), ( -6 ), ( -5 ), ( -4 ), ( -3 ), and ( -2 ). Therefore, ( -7 ) is correct. ( -9 ) lies to the left of ( -8 ) on the number line, while ( 0 ) and ( -1 ) are not inside the interval; ( -1 ) is an endpoint. Exam tip: “Between” usually excludes the two given endpoints.

0.005 is greater. In 0.005, the digit 5 is in the thousandths place, whereas in 0.0005 it is in the ten-thousandths place. Writing them as 0.0050 and 0.0005 shows that 50 ten-thousandths is greater than 5 ten-thousandths. Exam tip: Add zeros to the right of decimals to make the number of decimal places equal before comparing.

0 is greater than -11 because on the number line, 0 lies to the right of -11. A number to the right on a number line is always greater. “Both are equal” is incorrect because -11 and 0 are different numbers. Exam tip: Every negative number is less than 0.

The midpoint of two numbers on a number line is their average: \(\frac{5+(-11)}{2}=\frac{-6}{2}=-3\). Therefore, \(-3\) is correct. \(-2\) is not correct because it is not equally distant from 5 and \(-11\). Exam tip: add the two numbers and divide the sum by 2 to find the midpoint.

The distance between two numbers on a number line is the absolute value of their difference. Thus, the distance is \(|8-(-6)|=|14|=14\). Therefore, 14 is correct. A value such as 12 can result from handling the negative sign incorrectly. Exam tip: always use the absolute value for distance, so the answer cannot be negative.

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.

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.

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.