Representing real numbers on the number line

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.

(2.4) and (-2.4) are opposite numbers, so they are equally far from (0). The sign tells the direction.

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

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

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

(3\frac{1}{2}=3+\frac{1}{2}=3.5). Converting a mixed number to a decimal makes its position clear.

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.

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

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

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

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 \(\sqrt{16}=4\).

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

(-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.

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.

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.

The distance between the two numbers is
|(-2)-(-6)|=|-2+6|=4
units. On the number line, one moves 4 steps to the right from
-6
to
-2
. Distance is always positive, so
-4
is not correct. Exam tip: Find the distance between two numbers by taking the absolute value of their difference.

Moving to the left on a number line means subtracting. Therefore, the required point is \(0-2.5=-2.5\). Option B represents moving 2.5 units to the right instead. Exam tip: Numbers to the left of 0 on the number line are negative.

Moving to the right on a number line increases the number. Therefore, the point reached is -1 + 3 = 2. The value -4 would result from moving 3 units to the left from -1. Exam tip: represent rightward movement by addition and leftward movement by subtraction.

To compare a decimal with a fraction, convert one representation into the other. The fraction 3/2 means three divided by two, and 3 ÷ 2 = 1.5. Equivalently, writing 1.5 as a fraction gives 15/10, which reduces by dividing numerator and denominator by 5 to 3/2. Since the two values are exactly equal, they correspond to the same point on the number line, not merely to nearby points. Therefore option A is correct. Option B and option C would require one value to be smaller, but equality has already been established. Option D is also false because opposite numbers have equal magnitudes with opposite signs, such as 1.5 and −1.5; both numbers here are positive and equal. The governing concept is equivalence of decimal and fractional forms.

(0.333\ldots=\frac{1}{3}) is a recurring decimal. Recurring decimals also represent fixed points on the number line.