Representing real numbers on the number line

Correct answer: A. 2

\(\sqrt{4}=2\) 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 \(\sqrt{9}=3\).

Correct answer: A. 3

The principal square root of 9 is 3 because \(3^2=9\). Therefore, \(\sqrt{9}\) is located at the point 3 on the number line.

Correct answer: A. 2 and 3

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.

Correct answer: A. (3) and (4)

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

Correct answer: A. left of (0)

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

Correct answer: A. 2.1

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.

Correct answer: A. −0.5

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.

Correct answer: A. Neither positive nor negative

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.

Correct answer: A. 3 units

Distance is the absolute difference between the two numbers: |3 − 0| = 3 units. Therefore, the distance is 3 units.

Correct answer: A. 5 units

Distance is the absolute difference between the coordinates: |1 − (−4)| = |5| = 5 units. Therefore, the distance is 5 units.

Correct answer: A. 3

The opposite of a number on the number line is its additive inverse. An additive inverse is the number that produces zero when added to the original number. For −3, we need a number y such that −3 + y = 0; solving gives y = 3. Geometrically, −3 and 3 are equally distant from zero, each three units away, but they lie on opposite sides of the origin. Therefore option A is correct. Option B is the reciprocal-related value −1/3, not the additive inverse. Option C is the midpoint or origin, not the point opposite to −3, and option D is −6, which lies farther in the same negative direction rather than on the opposite side. The essential rule is that the opposite of a is −a, so the opposite of −3 is −(−3) = 3.

Correct answer: A. they are equally far from (0)

The direct answer is option A: the two numbers are equally far from zero. On a number line, positive numbers lie to the right and negative numbers lie to the left. Distance from zero is measured by absolute value: \(|2.4|=2.4\) and \(|-2.4|=2.4\). Hence both points are 2.4 units from zero, although they are on opposite sides. Option A is correct. Option B is wrong because \(-2.4\) is left of zero. Option C is wrong because \(2.4\) is right of zero. Option D is wrong because neither point is zero, so their distances from zero are not zero. The sign shows direction; the absolute value shows distance.

Correct answer: A. first mark among three equal parts

For (\frac{1}{3}), divide (0) to (1) into (3) equal parts and take the first mark. The denominator tells the number of parts.

Correct answer: A. (1)

(\frac{7}{5}=1.4), so it comes after (1) and before (2). Think of an improper fraction in mixed form.

Correct answer: A. between (-1) and (0)

(-\frac{1}{4}=-0.25), so it lies between (-1) and (0). Small negative fractions are close to (0) on the left.

Correct answer: A. (3.5)

The direct answer is A: \(3\frac12=3.5\). A mixed number contains a whole-number part and a fractional part. Here the whole part is 3 and the fraction is \(\frac12\). Since \(\frac12=0.5\), add them: \(3+0.5=3.5\). Therefore the number has the same position on the number line as 3.5. Option A is correct. Option B, 3.2, would mean adding only 0.2 to 3, not one-half. Option C, 2.5, is smaller by 1 and would correspond to \(2\frac12\). Option D, 4.5, is larger by 1 and would correspond to \(4\frac12\). Another method is to convert the mixed number to an improper fraction: \(3\frac12=\frac{3\times2+1}{2}=\frac72=3.5\). Memory cue: one-half always equals 0.5.

Correct answer: A. Both will be at the same point

4.0 = 4 because a zero written at the end of the decimal part does not change the value of a number. Therefore, both are represented by the same point on the number line. They would lie to the left or right of each other only if their values were different. Exam tip: Adding or removing trailing zeros after the decimal does not change a number, so 4.0 = 4.00 = 4.

Correct answer: A. (0.25)

The smallest number is (0.25), so it is the leftmost. Smaller numbers lie to the left on the number line.

Correct answer: A. (-1.2)

(-1.2) is the greatest number, so it is rightmost. Among negative numbers, the one closer to (0) is greater.

Correct answer: A. real numbers

Every point on the number line represents a real number. Between (1) and (2), both rational and irrational numbers can occur.

Correct answer: A. Both are located at the same point

Since 16 is a perfect square, \(\sqrt{16}=4\). Therefore, \(\sqrt{16}\) and 4 represent the same point on the number line. Options B and C are incorrect because the two numbers are equal, and option D is incorrect because 4 is positive. Exam tip: Recognize the square roots of perfect squares directly, such as \(\sqrt{9}=3\) and \(\sqrt{16}=4\).

Correct answer: A. (\frac{9}{4})

(2+\frac{1}{4}=\frac{8}{4}+\frac{1}{4}=\frac{9}{4}). Convert the integer to a fraction with the same denominator.

Correct answer: A. (-\frac{3}{2})

(-2+\frac{1}{2}=-\frac{4}{2}+\frac{1}{2}=-\frac{3}{2}). Be careful with direction when adding a fraction to a negative integer.

Correct answer: A. 0

The midpoint of two numbers \(a\) and \(b\) is found using \(\frac{a+b}{2}\). Here, \(\frac{2+(-2)}{2}=\frac{0}{2}=0\), so the correct answer is 0. The numbers 2 and -2 are the given endpoints, not the midpoint. Exam tip: the midpoint of a number and its opposite is always 0.

Correct answer: A. 3 units

The distance between two numbers on a number line is the absolute value of their difference: \(|8-5|=3\) units. Therefore, one must move 3 steps to the right from 5 to reach 8. The value 13 is their sum, not the distance. In an exam, subtract the smaller number from the larger number to find the distance.