Question 4

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:

(i) $\dfrac{3}{50}$

(ii) $\dfrac{2}{9}$

Prove that $\sqrt{5}$ is an irrational number.

Convert the following decimal numbers in the form of $\dfrac{p}{q}$.

(i) $12.6$

(ii) $0.0120$

(iii) $3.05\overline{2}$

(iv) $1.2\overline{35}$

(v) $0.\overline{23}$

(vi) $2.0\overline{5}$

(vii) $2.12\overline{5}$

(viii) $3.12\overline{5}$

(ix) $2.\overline{1625}$

Find 6 rational numbers between $3$ and $4$.

Find 5 rational numbers between $\dfrac{2}{5}$ and $\dfrac{3}{5}$.

Find 5 rational numbers between $\dfrac{1}{6}$ and $\dfrac{2}{5}$.

If $\dfrac{x}{3} + \dfrac{x}{5} = \dfrac{16}{15}$, find the rational number $x$.

Let $a$ and $b$ be two non-zero rational numbers such that $a + \dfrac{1}{b} = 0$. 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 $\dfrac{p}{10^4}$, 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 $\dfrac{18}{125}$ is terminating or non-terminating. If it terminates, state the number of decimal places.

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

Let $a = \dfrac{7}{12}$ and $b = \dfrac{5}{6}$. Express both $a$ and $b$ in the form $\dfrac{k_1}{m}$ and $\dfrac{k_2}{m}$ 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 $\dfrac{(a+b)}{2}$ lies between the rational numbers $a$ and $b$.

Find the lengths of the hypotenuses of all the right triangles in Fig. 3.14 which is referred to as the square root spiral.