Number line

For negative numbers, the number closer to zero is greater. \( -18 \) is closer to zero than \( -18.25 \); on a number line, \( -18 \) lies to the right of \( -18.25 \). Therefore, \( -18 \) is greater. \( -18.5 \) is smaller because it is more negative. Exam tip: when comparing negative numbers, the less negative number is greater.

Rational numbers are dense on the number line, so a rational number lies between any two distinct real numbers, including irrationals. Since \(2<\frac{9}{4}<3\), \(\sqrt{2}<\frac{3}{2}<\sqrt{3}\). Exam tip: test “always” statements using a counterexample.

Both rational and irrational numbers lie between any two distinct real numbers, so C is correct. There is no immediate next rational number after a rational number. Exam tip: remember that the real number line is dense.

The midpoint of two numbers on a number line is their average: \(\frac{-9+11}{2}=\frac{2}{2}=1\). Therefore, the correct answer is \(1\). The value \(-1\) may result from an incorrect calculation of the average. Exam tip: always use \(\frac{x+y}{2}\) to find the midpoint of two numbers.

The distance between two numbers on a number line is the absolute value of their difference: \(|5-(-13)|=|18|=18\). Therefore, the correct answer is 18. The value 16 can result from incorrectly subtracting instead of adding the negative number. Exam tip: always use the absolute value of the difference for distance.

The correct answer is
( -0.04 ). Write the decimals to the same number of places:
-0.04 = -0.040 and
-0.004 = -0.004. On a number line, the more negative number lies farther to the left, so
-0.040 < -0.004. Since
-0.004 is closer to zero, it is greater. Exam tip: for negative decimals, the number farther from zero is smaller.

For \(p<q\), \((p+q)/2\) is rational and lies between them. Repeatedly taking midpoints gives infinitely many rationals, and every such interval also contains infinitely many irrationals. Exam tip: check words such as “always” carefully.

Adding or removing zeros to the right of a decimal does not change the value of a number. Therefore, \( -6 = -6.00 \), and both represent the same point on the number line. Neither \( -6 \) nor \( -6.00 \) is greater than the other. Exam tip: Zeros written at the end of a decimal do not change its value.

We can write \(-2.75\) as \(-2.750\). Comparing \(-2.750\) and \(-2.705\), \(-2.750\) is more negative, so it is smaller. \(-2.705\) is closer to zero and is therefore greater. Exam tip: Among negative decimals, the number farther from zero is the smaller number.

On a number line, larger real numbers are located to the right. Since \(b-a>0\), the distance from P to Q is positive, so Q must be to the right of P. Exam tip: use the order relation, not merely the signs of the numbers.

The midpoint of two numbers on a number line is their average: \(\frac{-7+3}{2}=\frac{-4}{2}=-2\). Therefore, \( -2 \) is correct. \( -1 \) is not correct because it is not equally distant from \(-7\) and \(3\). 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 positive value of their difference: \(\lvert -12-(-3)\rvert=\lvert -9\rvert=9\). Therefore, the correct answer is \(9\). \(15\) is obtained by adding the magnitudes of the numbers, but that is not the distance when both numbers are negative. Exam tip: always use \(\lvert a-b\rvert\) to find distance on a number line.

A zero added at the end of a decimal does not change its value. Hence, \(-0.90=-0.9\), so both numbers are equal. It is incorrect to treat \(-0.90\) as more negative; the extra zero only changes the way the decimal is written. Exam tip: Remove trailing zeros before comparing decimals.

The correct answer is -1.2. Writing both numbers to the same number of decimal places gives -1.2 = -1.20. On a number line, the more negative number lies farther to the left, so -1.20 is smaller than -1.02. The ‘both are equal’ option is incorrect because -1.20 and -1.02 have different values. Exam tip: When comparing negative decimals, first write them with equal decimal places.

\(2\) is greater than \(-4\) and less than \(9\); that is, \(-4 < 2 < 9\). Hence, \(2\) is the integer between the two numbers. \(-5\) and \(-6\) are less than \(-4\), while \(10\) is greater than \(9\). Exam tip: For “between” questions, compare the number with both endpoints.

The midpoint of two numbers is half of their sum: \(\frac{-15+1}{2}=\frac{-14}{2}=-7\). Therefore, \( -7 \) is correct. At \( -6 \), the distances from the two given numbers are not equal. Exam tip: Keep the negative sign carefully while adding integers.

On a number line, the distance between two numbers is the absolute value of their difference: \(\lvert -20-(-10)\rvert=\lvert -10\rvert=10\). Therefore, the correct answer is 10. The value 20 is the sum of the magnitudes of the two numbers, not their distance. Exam tip: distance is always non-negative.

Take \(x=\sqrt{2}\). It is irrational, but \(x^2=(\sqrt{2})^2=2\), which is rational. In contrast, \(-x\), \(x+1\), and \(1/x\) remain irrational. Exam tip: test “can be” statements using a suitable example.

Every rational and irrational number has one fixed point; \( \sqrt{2} \) is irrational yet locatable. Hence, the claim that only rationals occur is false. Exam tip: recall the one-to-one correspondence.

The correct answer is ( -8 ). On a number line, the number farther to the right is greater. ( -8 ) lies to the right of ( -8.1 ) because it is closer to zero. “Both are equal” is incorrect because the two decimal values are different. Exam tip: Among negative numbers, the number closer to zero is greater.

The midpoint of two numbers on a number line is their average: \(\frac{-6+12}{2}=\frac{6}{2}=3\). Therefore, 3 is correct. \(6\) is half of the sum before dividing it by 2, so it is not the midpoint. Exam tip: use \(\frac{x+y}{2}\) for the midpoint of two numbers.

On a number line, the distance between two numbers is the absolute value of their difference. Thus, \(\left|-14-(-6)\right|=\left|-8\right|=8\). Therefore, 8 is the correct answer. Values such as 6 or 7 can result from skipping a point while counting. Exam tip: a distance is always non-negative.

For comparison, write -0.33 as -0.330. Since -0.333 is more negative than -0.330, it lies further to the left on the number line and is smaller. The option ‘Both are equal’ is incorrect because -0.330 and -0.333 have different decimal values. Exam tip: Among negative decimals, the number with the greater magnitude is smaller.

The midpoint of two numbers is their average: \(\frac{-10+(-2)}{2}=\frac{-12}{2}=-6\). Therefore, -6 is the correct answer. -4 is not the average of -10 and -2. Exam tip: To find a midpoint on a number line, add the two numbers and divide by 2.

The distance between two numbers on a number line is the absolute value of their difference: \(|9-(-11)|=|20|=20\). Therefore, the correct answer is 20. The value 18 can result from handling the negative sign incorrectly. Exam tip: Always use the absolute value of the difference when finding distance.