Solution
Note: Sets of Rational Numbers
Therefore, the answer is
Note:
(1) Any number that can be written as a fraction is RATIONAL
(2) Any number that CANNOT be written as a fraction is not Rational
(3) Examples of rational numbers
[tex]\frac{4}{2},\frac{3}{5},\frac{1}{10},-\frac{5}{4},...[/tex]
(4) Example of Irrational Numbers
[tex]\sqrt{2},\sqrt{11},tan1,log5,e^2[/tex]
(5). How to differentiate decimal numbers that is rational from the Irrational numbers
Example 1: Which is rational and which is Irrational
[tex]\begin{gathered} 0.666666666666667 \\ and \\ 0.4771212547 \end{gathered}[/tex]
Answer: (a). If the decimal can be converted to fraction, then it is rational, but if it can't be converted to fraction, then it is irrational
(b). The decimal of a rational number (or fraction) most times repeat it's digit.
Now, for the above question
[tex]\begin{gathered} 0.666666666666667\text{ =}\frac{2}{3} \\ So\text{ it is rational} \\ Notice\text{ the repetition of digits} \end{gathered}[/tex]
and
[tex]\begin{gathered} 0.4771212547 \\ The\text{ digits does not follow any repetition pattern} \\ There\text{ is no fraction this decimals can represent} \\ So\text{ it is not Rational} \\ It\text{ is an Irrational number} \\ Indeed,\text{ } \\ 0.4771212547=log\mleft(3\mright) \end{gathered}[/tex]
