End of Chapter Exercises

Convert the following rational numbers in the form of a terminating decimal or non-terminating and repeating decimal, whichever the case may be, by the process of long division: \(\frac{3}{50}\)

Convert the following rational numbers in the form of a terminating decimal or non-terminating and repeating decimal, whichever the case may be, by the process of long division: \(\frac{2}{9}\)

Prove that √5 is an irrational number.

Convert the following decimal numbers in the form of \(\frac{p}{q}\): 12.6

Convert the following decimal numbers in the form of \(\frac{p}{q}\): 0.0120

Convert the following decimal numbers in the form of \(\frac{p}{q}\): \( 3.0\overline{52} \)

Convert the following decimal numbers in the form of \(\frac{p}{q}\): \( 1.2\overline{35} \)

Convert the following decimal numbers in the form of \(\frac{p}{q}\): \( 2.\overline{1625} \)

Convert the following decimal numbers in the form of \(\frac{p}{q}\): \( 0.\overline{23} \)

Convert the following decimal numbers in the form of \(\frac{p}{q}\): \( 2.0\overline{5} \)

Convert the following decimal numbers in the form of \(\frac{p}{q}\): \( 2.12\overline{5} \)

Convert the following decimal numbers in the form of \(\frac{p}{q}\): \( 3.12\overline{5} \)

Locate the following rational numbers on the number line. \(0.532\)

Locate the following rational numbers on the number line. \( 1.1\overline{5} \)

Find 6 rational numbers between 3 and 4.

Find 5 rational numbers between \(\frac{2}{5}\) and \(\frac{3}{5}\).

Find 5 rational numbers between \(\frac{1}{6}\) and \(\frac{2}{5}\).

If \(\frac{x}{3} + \frac{x}{5} = \frac{16}{15}\), find the rational number x.

Let a and b be two non-zero rational numbers such that \(a + \frac{1}{b} = 0\). Without assigning any numerical values, determine whether ab is positive or negative. Justify your answer.

A rational number has a terminating decimal expansion whose last non-zero digit occurs in the 4th decimal place. Show that such a number can be written in the form \(\frac{p}{10^4}\), where p is an integer not divisible by 10. Is it necessary that the denominator of this rational number, when written in the lowest form, is divisible by \(2^4\) or \(5^4\)? Give reasons.

Without performing division, determine whether the decimal expansion of \(\frac{18}{125}\) is terminating or non-terminating. If it terminates, state the number of decimal places.

A rational number in its lowest form has denominator \(2^3 × 5\) . How many decimal places will its decimal expansion have? Explain your answer.

Let \(a = \frac{7}{12}\), \(b = \frac{5}{6}\). Express both \(a\) and \(b\) in the form \(\frac{k_1}{m}\) and \(\frac{k_2}{m}\) where \(k_1, k_2\) and \(m\) are integers and \(k_2 − k_1 = 6\). Using the same denominator m, write exactly five distinct rational numbers lying between \(a\) and \(b\) keeping an integer numerator. Explain why the condition \(k_2 − k_1 > n + 1\) is necessary to find \(n\) such rational numbers between the two rational numbers \(a\) and \(b\) using this method.

Three rational numbers \(x, y, z\) satisfy \(x + y + z = 0\) and \(xy + yz + zx = 0\). Show that all the rational numbers \(x, y, z\) must be simultaneously zero.

Show that the rational number \(\frac{a+b}{2}\) lies between the rational numbers \(a\) and \(b\).

Find the lengths of the hypotenuses of all the right triangles in the figure, which is referred to as the square root spiral.