Representing real numbers 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.

(\frac{11}{5}=2.2), so it lies between (2) and (3). Convert an improper fraction to decimal or mixed form.

Since (2^2<7<3^2), (\sqrt{7}) lies between (2) and (3). Taking (\sqrt{7}) as (7) is a common mistake.

(-0.3) is greater than (-0.8), so it is to the right. Among negative numbers, the one closer to (0) is greater.

For (\sqrt{n}) to lie between (4) and (5), (16<n<25) is needed. Among the options (21) is correct.

The midpoint of two numbers is their average. Thus, \(\frac{1.2+1.3}{2}=\frac{2.5}{2}=1.25\). Therefore, the correct answer is 1.25. Although 1.21 is close to 1.2, it is not equidistant from 1.2 and 1.3. Exam tip: Add the two endpoint values and divide the sum by 2 to find their midpoint.

Dividing 9 by 5 gives 1.8, and the negative sign makes the value \(-1.8\). Thus, it is located 1.8 units to the left of zero on the number line. Option B has an incorrect decimal-place value, while option C omits the negative sign. Exam tip: After converting a fraction to a decimal, always check whether its sign has been retained.

(\sqrt{18}) is about (4.24), so it is closer to (4). For estimation, check nearby perfect squares (16) and (25).

The value of \(\sqrt{2}\) is approximately \(1.414\). Therefore, \(1+\sqrt{2}\approx 1+1.414=2.414\), which lies between 2 and 3. Hence, option B is correct. Exam tip: For such questions, estimate the irrational number first and then identify the consecutive integers containing the result.

Converting \(\frac{3}{4}\) into decimal form gives \(3 \div 4=0.75\). Therefore, \(\frac{3}{4}\) and 0.75 represent the same point on the number line. The other pairs are unequal; for example, \(\frac{2}{5}=0.4\), not 0.25. Exam tip: convert the fraction into decimal form and compare the values.

(-\sqrt{9}=-3) and (\sqrt{9}=3), so the distance is (6). The distance between opposite points is the sum of their magnitudes.

Write the decimals with equal place values: 0.09, 0.19, and 0.90. Comparing the hundredths and tenths places gives 9 < 19 < 90, so the increasing order is 0.09 < 0.19 < 0.9. Option D is incorrect because 0.9 is greater than 0.19, not smaller. Exam tip: Add trailing zeroes when needed so that decimals have the same number of places before comparing them.

We are given \(5<\sqrt{a}<6\). Since all quantities are positive, squaring all three parts preserves the inequality and gives \(25<a<36\). Option A incorrectly treats the range of \(\sqrt{a}\) as the range of \(a\). Exam tip: for positive bounds involving a square root, square both boundary values.

Since \(\sqrt{2}\approx 1.414\), we get \(2-\sqrt{2}\approx 2-1.414=0.586\). This value lies between \(0\) and \(1\). Option C is incorrect because \(2-\sqrt{2}\) is less than \(1\). Exam tip: estimate the irrational number first, perform the operation, and then identify the consecutive integers on either side of the result.

(\frac{-13}{6}\approx -2.17), so it lies between (-3) and (-2). Intervals with negative numbers can be tricky.