Decimal Representation Of Rational Numbers
Example 1: Express \(\frac { 7 }{ 8 }\) in the decimal form by long division method.
Solution: We have,
∴ \(\frac { 7 }{ 8 }\) = 0.875
Example 2: Convert \(\frac { 35 }{ 16 }\) into decimal form by long division method.
Solution: We have,
Example 3: Express \(\frac { 2157 }{ 625 }\) in the decimal form.
Solution: We have,
Example 4: Express \(\frac { -17 }{ 8 }\) in decimal form by long division method.
Solution: In order to convert \(\frac { -17 }{ 8 }\) in the decimal form, we first express \(\frac { 17 }{ 8 }\) in the decimal form and the decimal form of \(\frac { -17 }{ 8 }\) will be negative of the decimal form of \(\frac { 17 }{ 8 }\)
we have,
Example 5: Find the decimal representation of \(\frac { 8 }{ 3 }\) .
Solution: By long division, we have
Example 6: Express \(\frac { 2 }{ 11 }\) as a decimal fraction.
Solution: By long division, we have
Example 7: Find the decimal representation of \(\frac { -16 }{ 45 }\)
Solution: By long division, we have
Example 8: Find the decimal representation of \(\frac { 22 }{ 7 }\)
Solution: By long division, we have
So division of rational number gives decimal expansion. This expansion represents two types
(A) Terminating (remainder = 0)
So these are terminating and non repeating (recurring)
(B) Non terminating recurring (repeating)
(remainder ≠ 0, but equal to devidend)
These expansion are not finished but digits are continusely repeated so we use a line on those digits, called bar \((\bar{a})\).
So we can say that rational numbers are of the form either terminating, non repeating or non terminating repeating (recurring).