Question

Convert the following number into decimal form $\dfrac{-16}{45}$. \[\]

Answer 1

Hint: We recall the definitions of rational number, decimal number terminating decimal and non-terminating decimal. We divide the numerator $-16$ of the given rational number by the denominator 45 by decimal division to convert into decimal form.

Complete step-by-step answer:
We know that a rational number is in the form $\dfrac{p}{q}$ where $p,q$are integers are and $q$ is not zero. Here $p$ is called the numerator of the rational number and $q$ is called denominator. \[\]

The decimal representation of numbers has two parts: integral part and fractional part which are separated by decimal point, for example in the decimal number 1.23 the integral part is 1 and decimal part is 23. \[\]
A decimal number is called a terminating decimal if it has a terminating digit in the fractional part, for example in 10.12 the digit 2 is the terminating digit and hence 10.12 is a terminating decimal. If the decimal number does not have a terminating digit; for example in 10.122222... it is called non-terminating decimal because 2 repeats itself infinite times.\[\]
We convert the rational numbers into decimal by dividing the numerator by denominator. Here the numerator is $-16$ and denominator is 45. We first removed the negative sign on 16 and divided it by 45.
\[45\overset{{}}{\overline{\left){16}\right.}}\]
Since 45 is greater than 16 , we put a decimal point after 0 and add a after the one’s place of the dividend 45.
\[\begin{align}
  & 45\overset{0\cdot 3}{\overline{\left){\begin{align}
  & 160 \\
 & \underline{135} \\
\end{align}}\right.}} \\
 & \hspace{1.2 cm} 25 \\
\end{align}\]
We add 0 after the one’s place of remainder 25 and continue dividing

\[\begin{align}
  & 45\overset{0\cdot 3555}{\overline{\left){\begin{align}
  & 160 \\
 & \underline{135} \\
\end{align}}\right.}} \\
 & \hspace{1 cm}250 \\
 & \hspace{1 cm} \underline{225} \\
 & \hspace{1 cm} 250 \\
 & \hspace{1 cm} \underline{225} \\
 & \hspace{1 cm}250 \\
 & \hspace{1 cm} \underline{225} \\
 & \hspace{1 cm}25 \\
\end{align}\]
We see that in the decimal part of the quotient 5 repeat itself infinite times. So the quotient when we divide 16 by 45 is $0.35555...$. We put back to negative sign represent the given rational number in decimal as
\[\dfrac{-16}{45}=-0.35555..\]
We know that we can represent the repeating digits with a bar on them. So we have;
\[\dfrac{-16}{45}=-0.35555..=-0.3\overline{5}\]

Note: We note that when we added negative sign on $0.35555...$ as $-0.35555...=-0.3\overline{5}$ , the negative sign does not represent the negative on 0 but it represents the negative on the fractional part $0.35555...$. We note that when the numerator is less than the denominator the integral part in the decimal part is always zero.