UPSKILL MATH PLUS
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Learn moreThe decimal expansion of a rational number is will terminating or non-terminating and recurring. Conversely, the decimal expansion of a number is terminating, or non-terminating recurring is a rational number.
Example:
1. Prove that \(0.77777... = 0.\) is a rational number. That is, show that \(0.\) can be expressed in \(p/q\), where \(p\) and \(q\) are integers with \(q\)\(0\).
Solution:
Let us take the provided number as \(x\).
That is \(x = 0.77777...\)
Note that the number \(x\). Only the one-digit \(7\) repeats here.
Here we have to make multiplies of \(x\) in such a way that the repeated decimals will be the same.
Let us multiply \(x\) by \(10\).
\(10x = 7.77777...\)
Now subtract \(x\) from \(10x\),
\(10x - x =7.77777... - 0.777777...\)
\(9x = 7\)
\(x = 7/9\)
Therefore, the fractional form of the rational number \(0.\) is \(7/9\).
2. Prove that \(0 .2363636... = 0.2\) is a rational number. That is, show that \(0.2\) can be expressed in \(p/q\), where \(p\) and \(q\) are integers with \(q\)\(0\).
Solution:
Let us take the provided number as \(x\).
That is \(x = 0.2363636...\)
Note the number \(x\) - two of digits\(36\) repeats here.
Here we have to make multiplies of \(x\) in such a way that the repeated decimals will be the same.
\(10x = 2.363636...\)
Multiply \(x\) by \(1000\).
\(1000x = 236.363636...\)
Subtract \(10x\) from \(1000x\),
\(1000x - 10x = 236.363636... - 2.363636...\)
\(990x = 234\)
\(x = 234/990\)
Therefore, the fractional form of the rational number \(0.2\) is \(234/990\).
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