Number line

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.

\(\sqrt{5}\) is irrational because \(4<5<9\) and 5 is not a perfect square. Its decimal expansion is non-terminating and non-repeating. \(0.125\) terminates, so it is rational. Exam tip: the square root of a non-perfect-square natural number is irrational.

For rational \(r<s\), \(r+(s-r)\frac{\sqrt2}{n}\) with \(n\ge2\) lies between them and is irrational. Different values of \(n\) give infinitely many such numbers. Exam tip: every interval contains both rational and irrational numbers.

On a number line, the distance between two numbers is the absolute value of their difference: \(\lvert 8-(-1)\rvert=\lvert 9\rvert=9\). Hence, the correct answer is \(9\). Choosing \(7\) usually results from an error while subtracting a negative number. Exam tip: a distance is always non-negative.

Since \(2^2<5<3^2\), \(\sqrt{5}\) lies between 2 and 3. Also, \(\left(\frac{11}{5}\right)^2=\frac{121}{25}<5\), so \(\frac{11}{5}<\sqrt{5}\). Thus option B reverses their order. Exam tip: for positive numbers, comparing squares is often quicker.

On the number line, the integers between -2 and 5 must be greater than -2 and less than 5: -1, 0, 1, 2, 3, and 4. Therefore, 1 is correct. -3 lies to the left of -2, while 5 and 6 are not less than 5. Exam tip: “Between” usually excludes the two endpoints.

On a number line, the distance between two numbers is the absolute value of their difference: \(|10-(-10)|=|20|=20\). Therefore, the correct answer is 20. The value 10 is the distance of either number from 0, not the distance between the two numbers. Exam tip: Always take the absolute value of the difference when finding distance.

( -3.001 ) is more negative than ( -3 ). On a number line, the more negative number lies farther to the left and is therefore smaller. Hence, ( -3.001 ) is correct; the two numbers are not equal because their decimal values differ. Exam tip: Among negative numbers, the number with the greater magnitude is the smaller number.

The absolute value of a number is its distance from zero on the number line, so it is never negative. Since \(-12\) is 12 units from zero, \(|-12|=12\). \(-12\) is the original number, not its absolute value. Exam tip: for a negative number, remove the minus sign to find its absolute value.

Write 1.2 as 1.20 so that both decimals have the same number of decimal places. Comparing 1.20 and 1.02, the tenths digit 2 is greater than 0, so 1.2 is greater. The 2 in the hundredths place of 1.02 does not make it equal to 1.2. Exam tip: add zeros to the right of decimals before comparing place values.

The midpoint of two numbers on a number line is their average: \(\frac{-5+5}{2}=\frac{0}{2}=0\). Therefore, \(0\) is at an equal distance from \(-5\) and \(5\). The numbers \(5\) and \(-5\) are the given endpoints, not the midpoint. Exam tip: The midpoint of opposite numbers \(a\) and \(-a\) is always \(0\).

The distance between two numbers on a number line is their absolute difference: \(\lvert -9-(-1)\rvert=\lvert -8\rvert=8\). Therefore, the correct answer is 8. The number 9 is only the absolute value of \(-9\), not the distance between the two points. Exam tip: a distance is always non-negative.