Decimal Representation Of Rational Numbers

Decimal Representation Of Rational Numbers

Example 1:    Express \(\frac { 7 }{ 8 }\) in the decimal form by long division method.
Solution:    We have,
Decimal Representation Of Rational Numbers 1
∴ \(\frac { 7 }{ 8 }\) = 0.875

Example 2:    Convert \(\frac { 35 }{ 16 }\) into decimal form by long division  method.
Solution:    We have,  
Decimal Representation Of Rational Numbers 2

Example 3:    Express \(\frac { 2157 }{ 625 }\) in the decimal form.
Solution:    We have,
Decimal Representation Of Rational Numbers 3

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,
Decimal Representation Of Rational Numbers 4

Example 5:    Find the decimal representation of \(\frac { 8 }{ 3 }\) .
Solution:    By long division, we have
Decimal Representation Of Rational Numbers 5

Example 6:    Express \(\frac { 2 }{ 11 }\) as a decimal fraction.
Solution:    By long division, we have
Decimal Representation Of Rational Numbers 6

Example 7:    Find the decimal representation of \(\frac { -16 }{ 45 }\)
Solution:    By long division, we have
Decimal Representation Of Rational Numbers 7

Example 8:    Find the decimal representation of \(\frac { 22 }{ 7 }\)
Solution:    By long division, we have
Decimal Representation Of Rational Numbers 8
Decimal Representation Of Rational Numbers 9
So division of rational number gives decimal expansion. This expansion represents two types
(A) Terminating (remainder = 0)
Decimal Representation Of Rational Numbers 10
So these are terminating and non repeating (recurring)
(B) Non terminating recurring (repeating)
(remainder ≠ 0, but equal to devidend)
Decimal Representation Of Rational Numbers 11
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).

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