Representing real numbers on the number line

The zero of a polynomial is the value of \(x\) for which the polynomial becomes zero. Thus, \(5x-10=0\), so \(5x=10\) and \(x=2\). Therefore, the correct point on the number line is 2. The value 5 is only the coefficient of \(x\), not the zero. In an exam, first set the polynomial equal to zero to find its zero.

An irrational number has a non-terminating, non-repeating decimal expansion, but it is still a real number and has one fixed point on the number line. For example, \(\sqrt{2}\) lies between 1 and 2. Exam tip: every real number corresponds to a point on the number line.

Moving (2.5) units to the right from (0) gives (2.5). In exams, moving right increases the number.

Moving (1.75) units to the left from (0) gives (-1.75). In exams, moving left decreases the number.

Since (8>2), (\sqrt{8}>\sqrt{2}). In exams, compare positive square roots using the numbers inside them.

Since (\sqrt{12}>\sqrt{3}), (-\sqrt{3}) is greater and lies to the right. In exams, negative signs reverse the comparison effect for square roots.

Since (3^2=9) and (4^2=16), (\sqrt{15}) lies between (3) and (4). In exams, use nearby perfect squares to locate square roots.

From (6x-18=0), we get (x=3). In exams, find the zero first and then place it correctly on the number line.

The governing result is the Pythagorean theorem: in a right triangle, the square of the hypotenuse equals the sum of the squares of the perpendicular sides. With both legs equal to 1, let the hypotenuse be h. Then h² = 1² + 1² = 1 + 1 = 2, so h = √2 because a length is positive. Therefore option B is correct. Choosing 1 ignores the second perpendicular side, choosing 2 adds lengths instead of their squares, and √3 does not follow from the Pythagorean calculation. This construction is used to transfer the irrational length √2 to the number line.

Since (4<5<9), (\sqrt{5}) lies between (2) and (3). Comparing squares helps locate square roots.

Since (1.375=1+0.375), move (0.375) unit from (1). Separate the integer part when locating decimals.

(-\frac{7}{4}=-1.75), so it lies between (-2) and (-1). Be careful with direction for negative fractions.

(\frac{5}{8}) means (5) parts out of (8) equal parts. Treat the denominator as divisions and the numerator as count.

Since (3^2<10<4^2), (\sqrt{10}) lies between (3) and (4). Use nearby perfect squares to locate roots.

By the Pythagorean theorem, hypotenuse² = (perpendicular side)² + (base)². Thus, hypotenuse = \(\sqrt{3^2+2^2}=\sqrt{9+4}=\sqrt{13}\), so option C is correct. Option B results from subtracting the squares, which is not applicable when finding the hypotenuse. Exam tip: in a right triangle, the hypotenuse is the longest side, and its squared length equals the sum of the squares of the other two sides.

First, \(\sqrt{4}=2\). The negative sign outside the square root makes the value negative, so \(-\sqrt{4}=-(\sqrt{4})=-2\). Therefore, the point is \(-2\) on the number line. Exam tip: do not confuse \(-\sqrt{4}\) with \(\sqrt{-4}\).

Write the numbers with equal decimal places: 2.6 = 2.60 and 2.7 = 2.70. Since 2.60 < 2.65 < 2.70, 2.65 lies between them. A useful exam tip is to add trailing zeros before comparing decimal numbers.

The distance between two points is \(|B-A|=|2-(-3)|=|5|=5\) units. Therefore, 5 is correct. The distractor 3 may result from incorrectly ignoring the negative sign, whereas subtracting a negative number gives \(2-(-3)=5\). Exam tip: Use \(|x_2-x_1|\) to find distance on a number line because distance is never negative.

On a number line, moving to the right means adding the stated distance, while moving to the left means subtracting it. Starting at −1 and moving 0.75 unit to the right gives −1 + 0.75. Write −1 as −1.00: −1.00 + 0.75 = −0.25. Therefore point P represents −0.25, so option B is correct. Option A would result from moving 0.75 unit left, option C incorrectly changes the sign of the starting point, and option D treats the starting value as positive. The calculation also places P between −1 and 0, exactly as the described rightward movement requires.

The governing concept is that values increase from left to right on a number line. To compare the mixed forms, convert the fraction: 1/2=0.5. The three values are therefore 0.4, 0.5, and 0.6, and their order is 0.4<0.5<0.6. Replacing 0.5 by its equivalent fraction gives the required left-to-right sequence 0.4, 1/2, 0.6, so option A is correct. Option B puts 1/2 before the smaller 0.4, option C reverses the overall order, and option D places 0.6 before 1/2. Exact conversion removes any ambiguity.