Representing real numbers on the number line

The midpoint of two numbers is found using \(\frac{a+b}{2}\). Here, \(\frac{0+2}{2}=1\), so the correct answer is 1. \(\frac{1}{2}\) is the midpoint of 0 and 1, not of 0 and 2. Exam tip: calculate the average of the two endpoints to find their midpoint.

The midpoint of two numbers is found using
\(\frac{a+b}{2}\)
. Here,
\(\frac{-1+1}{2}=0\)
, so the correct answer is 0. The numbers −1 and 1 are symmetric about 0, while −1 and 1 themselves are the endpoints. Exam tip: Two numbers with equal magnitudes and opposite signs always have 0 as their midpoint.

A decimal can be changed into a fraction by writing it over a power of 10. The number 2.25 has two digits after the decimal point, so 2.25=225/100. Now reduce this fraction by dividing numerator and denominator by their common factor 25: 225÷25=9 and 100÷25=4. Thus 2.25=9/4, making choice A correct.

The result can also be checked by dividing 9 by 4: 4 goes into 9 twice with remainder 1, and the remainder gives 0.25, so the quotient is 2.25. The fraction 4/9 is less than 1, 5/4 equals 1.25, and 11/4 equals 2.75, so none of those represents 2.25. Therefore the point on the number line has value 9/4.

The middle number between (0) and (1) is (\frac{0+1}{2}=\frac{1}{2}). In exams, use the average for the midpoint.

(-\frac{3}{2}=-1.5), so it lies between (-2) and (-1). For negative numbers, values increase to the right.

The denominator of (\frac{3}{4}) is (4), so divide (0) to (1) into (4) equal parts. The numerator tells you to move to the third mark.

(-\frac{7}{3}\approx -2.33), so it lies between (-3) and (-2). Check the position of negative decimals carefully.

A terminating decimal can be converted into a fraction by writing it over the appropriate power of ten. Since 1.25 has two digits after the decimal point, 1.25 = 125/100. Dividing the numerator and denominator by their common factor 25 gives 125/100 = 5/4. Therefore the point representing 1.25 on the number line is the point 5/4 units to the right of zero. Option A is correct. The value 4/5 equals 0.8, so option B lies to the left of 1; 3/4 equals 0.75, so option C is also smaller; and 7/4 equals 1.75, so option D lies farther to the right. The governing idea is equivalence between decimal and fractional representations, not merely matching the digits 1 and 25.

Converting 0.6 into a fraction gives \(0.6=\frac{6}{10}=\frac{3}{5}\). Therefore, 0.6 and \(\frac{3}{5}\) represent the same point on the number line. Option B, \(\frac{5}{3}\), is greater than 1, whereas 0.6 is less than 1. Exam tip: write the decimal over 10, 100, or 1000 according to its place value, then simplify.

The integer part of (2.75) is (2), so it lies between (2) and (3). The integer part gives the first clue.

\(\sqrt{4}=2\) because the square of 2 is 4. The principal square root is always non-negative, so −2 is not the value of \(\sqrt{4}\), although \((-2)^2\) is also 4. Exam tip: remember the square roots of perfect squares, such as \(\sqrt{1}=1\), \(\sqrt{4}=2\), and \(\sqrt{9}=3\).

\(\sqrt{9}=3\) because the square of 3 is 9. Therefore, \(\sqrt{9}\) is represented by the point 3 on the number line. Option 9 is the radicand, not its square root, and -3 is not the principal square root. Exam tip: The principal square root is always non-negative.

To locate √5 between consecutive integers, compare 5 with nearby perfect squares. We know that 2² = 4 and 3² = 9. Since 4 < 5 < 9, taking positive square roots gives 2 < √5 < 3. Thus √5 lies between the integers 2 and 3, and option A is correct. In fact, √5 is approximately 2.236, which confirms the same placement, although the perfect-square comparison is the exact and more useful method. Option B would contain numbers whose squares range only from 1 to 4, so it cannot contain √5. Options C and D are also impossible because their intervals are respectively much larger and much smaller than the value determined by the inequalities. The governing concept is ordering irrational numbers through neighboring perfect squares.

Since (3^2=9) and (4^2=16), (\sqrt{10}) lies between (3) and (4). Use perfect squares to locate square roots.

(-\sqrt{3}) is negative, so it lies to the left of (0). Negative numbers are placed on the left side of the number line.

Since 2.1 is greater than 1.8, it lies to the right of 1.8 on the number line. The closest distractor is 1.8, but being the smaller number, it lies to the left of 2.1. Exam tip: On a number line, numbers increase as we move from left to right.

On a number line, the smaller number lies to the left. Since −0.5 < −0.2, −0.5 is located to the left. Although 0.5 has a larger magnitude, it is positive and lies to the right of both numbers. For exams, compare negative decimals carefully: the negative number with the greater magnitude is smaller.

0 is neither positive nor negative; it represents the origin on the number line. Therefore, option A is correct. Options B and C are incorrect because positive numbers are greater than 0, while negative numbers are less than 0. Exam tip: 0 is an integer and a rational number, but it is neither positive nor negative.

The distance between two numbers is the absolute value of their difference: |3−0|=3 units. Therefore, the correct answer is 3 units. −3 may represent directed displacement, but it cannot represent distance because distance is always non-negative. Exam tip: On a number line, subtract the smaller number from the larger number to find the distance.

The distance between two points is the absolute value of the difference of their coordinates: \(|1-(-4)|=|5|=5\) units. Therefore, option A is correct. Option B results from an incorrect subtraction, and a distance cannot be negative, so -5 is also incorrect. In exams, always take the absolute value when finding distance.