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.

Write -2.12 as -2.120. Comparing -2.125 and -2.120, -2.125 is more negative, so it lies further left on the number line and is smaller. “Both are equal” is incorrect because their decimal parts are different. Exam tip: among negative decimals, the number with the greater magnitude is the smaller number.

Between any two distinct real numbers, there are infinitely many rational as well as infinitely many irrational numbers. Hence A is correct; choices claiming a finite number are false. Exam tip: remember this as the density property of rational and irrational numbers.

Since \(9<10<16\), \(3<\sqrt{10}<4\). Negating reverses order, so \(-4<-\sqrt{10}<-3\); A is correct. The next interval is too far right. Tip: reverse inequalities when negating.

The midpoint of two numbers on a number line is their average: \(\frac{-13+5}{2}=\frac{-8}{2}=-4\). Therefore, \(-4\) is correct. \(-3.5\) is not at an equal distance from both numbers. Exam tip: add the two numbers first, then divide the sum by 2.

Every real number, whether rational or irrational, corresponds to exactly one unique point on the number line. For example, \(\sqrt{2}\) has a fixed position. Option A is wrong because irrational numbers are real. Exam tip: all real numbers can be located on the number line.

The distance between two numbers on a number line is the absolute value of their difference: \(\lvert -27-(-9)\rvert=\lvert -18\rvert=18\). Therefore, the correct answer is 18. \(-18\) can be the difference, but a distance can never be negative. Exam tip: always take the absolute value of the difference when finding distance.

Option A is correct. \(\frac{a+b}{2}\) is rational and lies between \(a\) and \(b\); repeated midpoints give infinitely many rationals. Every open interval also contains infinitely many irrationals. Exam tip: both sets are dense.

The correct answer is ( -4.705 ). Among negative numbers, the number closer to zero is greater. Since \( -4.705 > -4.750 \), ( -4.705 ) is greater. Writing ( -4.75 ) as ( -4.750 ) makes the comparison easier. Exam tip: While comparing negative decimals, first make the number of decimal places equal.

The midpoint of two numbers on a number line is their average: \(\frac{-6+9}{2}=\frac{3}{2}=1.5\). Therefore, \(1.5\) is correct. \(3\) is the sum of the two numbers, not the midpoint. Exam tip: add the two numbers and divide by 2 to find their midpoint.

The successor of an integer is 1 greater than that integer. Hence, \(-41+1=-40\), so \(-40\) is correct. \(-42\) is 1 less than \(-41\), so it is the predecessor, not the successor. Exam tip: To find a successor, add 1 even when the number is negative.

The predecessor of a whole number is exactly 1 less than that number. Therefore, \(32-1=31\), so 31 is the predecessor of 32. In contrast, 33 is the successor of 32, while 30 is 2 less than 32. Exam tip: subtract 1 to find a predecessor.

Since \(2^2=4<7<9=3^2\), \(\sqrt{7}\) lies between 2 and 3. As 7 is not a perfect square, its square root is irrational. Option C has the correct interval but the wrong type. Exam tip: use adjacent perfect squares to locate roots.

\(2.6^2=6.76\) and \(2.7^2=7.29\). Since \(6.76<7<7.29\), \(x=\sqrt{7}\) lies between \(2.6\) and \(2.7\). In exams, compare nearby squares to locate an irrational number quickly.

For distinct rational numbers a and b, their midpoint \(\frac{a+b}{2}\) is also rational and lies between them. Repeating this process gives infinitely many rational numbers. Exam tip: use the midpoint method whenever numbers between two rationals are asked.

For negative numbers, the number farther to the left on the number line is smaller. Since -0.0002 is more negative than -0.00012, ( -0.0002 ) is the smaller number. They are not equal because their decimal values differ. Exam tip: when comparing negative decimals, the number with the greater magnitude is smaller.

Infinitely many rational and irrational numbers lie between any two distinct real numbers, so A is correct. B and D ignore this density property. Exam tip: every open interval contains both types of numbers.

Reema squared 2.7 incorrectly. Since \(2.7^2=7.29\), which is greater than 7, the positive square root \(\sqrt{7}\) must be less than 2.7. Exam tip: compare squares of nearby positive decimals to locate a square root.

The distance is \(|\sqrt{3}-(-\sqrt{3})|=|2\sqrt{3}|=2\sqrt{3}\). Equal magnitudes do not mean the same point; these points lie on opposite sides of zero. Exam tip: always use the absolute value of the difference for distance.

The distance between two numbers on a number line is their absolute difference: \(|25-(-70)|=|25+70|=95\). Therefore, 95 is correct. Choosing 94 results from an addition error of 1. Exam tip: when subtracting a negative number, change it to addition, and keep distance non-negative.

The correct answer is -3.141. Among negative numbers, the number with the greater magnitude is smaller. Since -3.141 is more negative than -3.140, we have -3.141 < -3.14. Writing -3.14 as -3.140 makes the comparison clearer. Exam tip: Add zeros at the end of decimals when needed to compare the same number of decimal places.

The midpoint of two numbers on a number line is their average: \(\frac{-18+6}{2}=\frac{-12}{2}=-6\). Therefore, \( -6 \) is correct. Although \(0\) lies between the two numbers, it is not equally distant from both. Exam tip: add the two numbers and divide by 2 to find their midpoint.