Number line

The midpoint of two numbers on a number line is their average: \(\frac{4+(-12)}{2}=\frac{-8}{2}=-4\). Therefore, \( -4 \) is correct. \( -8 \) is the sum of the two numbers, not their midpoint. Exam tip: add the two numbers and divide by 2 to find the midpoint.

The distance between two numbers on a number line is the absolute value of their difference: \(|20-(-60)|=|80|=80\). Therefore, the correct answer is 80. A value such as 79 can result from handling the negative sign incorrectly. Exam tip: always take the absolute value of the difference when finding distance.

Since \(1^2=1\) and \(2^2=4\), \(\sqrt{2}\) lies between 1 and 2. Also, \(1.4^2=1.96\) and \(1.5^2=2.25\), so \(\sqrt{2}\approx1.414\), which is closer to 1 than to 2. Therefore the stated option should identify it as nearer 1; check midpoint \(1.5\) in exams.

-0.03 can be written as -0.0300. On comparison, -0.0300 is greater than -0.0302 because, among negative numbers, the number closer to zero is greater. Therefore, ( -0.03 ) is the correct answer. Exam tip: Add trailing zeros when needed to make the number of decimal places equal before comparing decimals.

Real numbers include both rational and irrational numbers; for example, \(\sqrt{2}\) has a definite point on the number line. A single point cannot represent two real numbers. Exam tip: remember the one-to-one correspondence between real numbers and points.

For negative numbers, the number closer to zero is greater.

\(-0.07\) is closer to zero than \(-0.7\), so it lies to the right of \(-0.7\) on the number line. Hence, \(-0.07\) is greater. Writing \(-0.7\) as \(-0.70\) makes the comparison clear: \(-0.07>-0.70\). Exam tip: While comparing negative decimals, first write them with the same number of decimal places.

The real number line is dense. If \(x<y\), there are infinitely many rational as well as infinitely many irrational numbers between \(x\) and \(y\). Exam tip: the phrase “between two real numbers” often tests density.

The midpoint of two numbers on a number line is their average: \(\frac{-3+8}{2}=\frac{5}{2}\). Hence, \(\frac{5}{2}\) is correct. The option 3 is not correct because the sum of the two numbers is 5, not 6. Exam tip: add the two endpoints first and then divide by 2.

On a number line, the distance between two numbers is the absolute value of their difference. Thus, \(\lvert -18-(-6)\rvert=\lvert -12\rvert=12\). Therefore, 12 is correct. A value such as 13 may result from an error while handling negative signs. Exam tip: distance is always non-negative.

The distance from the origin is \(|x|\). Here \(|x|=\sqrt{2}\), and the point lies in the positive direction, so \(x=\sqrt{2}\). \(-\sqrt{2}\) is equally far from the origin but lies to the left. Exam tip: use the direction to choose the sign.

For rational \(r<s\), \(r+\frac{s-r}{\sqrt2}\) is irrational and lies between them, so A is correct and B is false. Exam tip: rational and irrational numbers are both dense on the number line.

The midpoint of two numbers on a number line is their average: \(\frac{-4+15}{2}=\frac{11}{2}=5.5\). Therefore, \(5.5\) is correct. \(4.5\) is not half of the sum, so it cannot be the midpoint. Exam tip: while finding a midpoint, add the numbers with their signs before dividing by 2.

On a number line, the distance between two numbers is the absolute value of their difference. Thus, \(\lvert -9-(-21)\rvert=\lvert 12\rvert=12\). Therefore, the correct answer is 12. Choosing 11 may result from incorrectly counting the integers between the numbers rather than finding their distance. Exam tip: always take the absolute value of the difference when finding distance.

For negative numbers, the number closer to zero is greater. Here, -0.009 = -9/1000 and -0.01 = -10/1000; therefore, -9/1000 > -10/1000. Hence, -0.009 is greater. “Both are equal” is incorrect because the decimal values are different. Exam tip: Write negative decimals to the same number of decimal places before comparing them.

The midpoint coordinate is \(\frac{a+b}{2}\). For it to be the origin, \(\frac{a+b}{2}=0\), so \(a+b=0\). In contrast, \(a-b=0\) gives the same coordinate for both points. Exam tip: use the average of coordinates for a midpoint.

Since \(P^2=2\) and \(P>0\), \(P=\sqrt{2}\). Its decimal expansion \(1.414\ldots\) is non-terminating and non-repeating, so it is irrational. Every real number has a point on the number line. Exam tip: being between two integers does not make a number rational.

By Pythagoras’ theorem, hypotenuse² = \(3^2+2^2=13\), so the length obtained is \(\sqrt{13}\). For \(\sqrt{10}\), use perpendicular sides 3 and 1. Exam tip: always add the squares of the two legs first.

The midpoint of two numbers on a number line is their average: \(\frac{-6+10}{2}=\frac{4}{2}=2\). Therefore, the correct answer is 2. Option 3 is not the average of -6 and 10. Exam tip: Add the two numbers and divide by 2 to find their midpoint.

The correct answer is -0.405. Write -0.45 as -0.450 for comparison. Since -0.405 is closer to zero than -0.450, -0.405 is greater. -0.45 is smaller because among negative numbers, the number with the greater magnitude is smaller. Exam tip: Write negative decimals to the same number of decimal places before comparing them.

Between any two distinct real numbers, there is at least one rational number; in fact, infinitely many exist. An integer is not necessary: there is no integer between 0.2 and 0.3. Exam tip: the average of two numbers often gives a number between them.