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).