Decimal Representation Of Rational Numbers
Decimal Representation of Rational Numbers
We can get the decimal representation of the rational number \(\frac {p}
{q}\) by long division method.
When we divide p and q using long division method either the remainder becomes zero or the remainder never becomes zero.
Case (i) (Remainder = 0)
Ex : \(
\frac{7}
{{16}} = 0.4375\)
Were we observe that the remainder becomes zero after a few steps i.e., decimal expansion of \(\frac {7}
{16}\) terminates
Ex : \(
\frac{1}
{2} = 0.5, - \frac{8}
{{25}} = - 0.32\) etc
Terminating Decimal :
When the decimal expansion of \(\frac {p}
{q}\),\(q\ne0\) terminates (i.e., comes to an end), the decimal expansion is called terminating.
Case 2 : (Remainder \(\ne0\))
Ex : \(
\frac{5}
{{11}} = 0.4545......,\frac{7}
{6} = 1.1666.......\)
Hence, we observe that the remainder never becomes zero.
In dividing, we note that the remainders repeat after some steps. So, we have a repeating (recurring) block of digits in the quotient.
Non-terminating and recurring decimal:
In the decimal expansion of \(
\frac{p}
{q},q \ne 0\) when the remainder never becomes zero, we have a repeating (recurring) block of digits in the quotient. This decimal expansion is called non
terminating and recurring.
To simplify the notation, we place a bar over the first block of the repeating (recurring) part & neglect remaining.
\(
\therefore \frac{5}
{{11}} = 0.\overline {45} ,\frac{7}
{6} = 1.1\overline 6 \)
Thus, a rational number can be expressed by either a terminating or a non terminating and recurring (repeating) decimal expansion.
The converse of this statement is also true.