The World of Numbers | EOT

Question 1

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) $\dfrac{2}{9}$

Solution

Understand the Question

To convert a rational number $\dfrac{p}{q}$ into decimal form, divide the numerator $p$ by the denominator $q$ using long division.
Terminating Decimal: The remainder becomes zero after a finite number of steps.
Non-Terminating Repeating Decimal: The remainder never becomes zero and starts repeating in a cycle.

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

Step 1 · Divide 3 by 50

Perform long division of $3$ by $50$ :

\begin{aligned} 3 \div 50 &= 0.06 \end{aligned}

$3 < 50$ , so write $0$ and place a decimal point: $3.0$
$30 < 50$ , add another zero: $3.00$
$50 \times 6 = 300$
Remainder $= 300 - 300 = 0$

$\dfrac{3}{50} = 0.06$

Since the remainder is $0$ , the decimal terminates.

Answer

(i) $0.06$ (Terminating decimal)

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

Step 1 · Divide 2 by 9

Perform long division of $2$ by $9$ :

$2 < 9$ , so write $0$ and place a decimal point: $2.0$
$9 \times 2 = 18$ , remainder $= 20 - 18 = 2$
Bringing down $0$ gives $20$ , $9 \times 2 = 18$ , remainder $= 2$
The remainder $2$ repeats continuously.

\dfrac{2}{9} = 0.222\dots = 0.\bar{2}

Since the remainder never becomes zero and repeats indefinitely, the decimal is non-terminating and repeating.

Answer

(ii) $0.\bar{2}$ (Non-terminating and repeating decimal)

Common Mistakes

Decimal Place Error: Writing $\dfrac{3}{50} = 0.6$ instead of $0.06$ . Always insert a zero in the tenths place when $30 < 50$ .
Missing Bar Notation: Writing $0.2$ or rounding to $0.22$ instead of indicating repetition with bar notation $0.\bar{2}$ for non-terminating decimals.

More questions in EOT

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) $2.\overline{1625}$

Locate the following rational numbers on the number line.

(i) $0.532$

(ii) $1.1\bar{5}$

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.

Q10

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.

Q11

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.

Q12

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.

Q13

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.

Q14

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.

Q15

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

Q16

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