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Rational and Irrational Numbers

Explained with examples and non examples

Rational Numbers

Can be expressed as the quotient of two integers (ie a fraction) with a denominator that is not zero. Many people are surprised to know that a repeating decimal is a rational number. The Venn diagram below shows which numbers the various sets and subsets of rational numbers.
Set of Real Numbers 6enn Diagram set of real numbers venn diagram
Examples of Rational Numbers
5 You can express 5 as $$ \frac{5}{1} $$ which is the quotient of the integer 5 and 1
2 You can express 2 as $$ \frac{2}{1} $$ which is the quotient of the integer 2 and 1
$$ \sqrt{9} $$ is rational because you can simplify the square root to 3 which is the quotient of the integer 3 and 1
$$ .\overline{11} $$ All repeating decimals are rational. It's a little bit tricker to show why so I will do that elsewhere
$$ .9 $$ is rational because it can be expressed as $$ \frac{9}{10} $$ ( All terminating decimals are also rational numbers)
$$ .73 $$ is rational because it can be expressed as $$ \frac{73}{100} $$
$$ 1.5 $$ is rational because it can be expressed as $$ \frac{3}{2} $$

Irrational Numbers

Can not be expressed as the quotient of two integers (ie a fraction) such that the denominator is not zero
Examples of Irrational Numbers
$$ \sqrt{7} $$ Unlike $$ \sqrt{9} $$, you cannot simplify $$ \sqrt{7} $$ .
$$ \frac{5}{0} $$ If a fraction, has a dominator of zero, then it's irrational
$$ \sqrt{5} $$ Unlike $$ \sqrt{9} $$, you cannot simplify $$ \sqrt{5} $$ .
$$ \pi $$ $$ \pi $$ is probably the most famous irrational number out there!
pi
$$ \frac{ \sqrt{2}}{3} $$ Although this number can be expressed as a fraction, we need more than to be a rational number. The fraction's numerator and denominator must both be integers, and $$\sqrt{2} $$ cannot be expressed as an integer.
$$.2020020002 ...$$ This non terminating decimal does not repeat but , like $$ \pi $$ constantly changes and so can not be represented as a quotient of two integers d
Practice Problems


problem 1) Is the number $$ \fbox{ -12 } $$ rational or irrational ?
Answer


problem 2) Is the number $$ \boxed{ \sqrt{ 25} } $$ rational or irrational ?
Answer


problem 3) Is the number $$ \boxed{ 0.09009000900009...} $$ rational or irrational ?
Answer


problem 4) Is the number $$ \boxed{ 0.\overline{201}} $$ rational or irrational ?
Answer


problem 5) Is the number $$ \boxed { \frac{ \sqrt{3}}{4} } $$ rational or irrational ?
Answer


problem 6) Is the number $$ \boxed { \frac{ \sqrt{9}}{25} } $$ rational or irrational ?
Answer


problem 7) Is the number $$ \boxed { \frac{ \pi}{\pi} } $$ rational or irrational ?
Answer


problem 8) Is the number $$ \boxed { \frac{ \sqrt{2}}{ \sqrt{2} } }$$ rational or irrational ?
Answer

Proof that repeating decimals are rational numbers

$$ \text{1) let }x = .\overline{1} \\ \fbox{2) multiply both sides by 10} 10 \cdot x = 10 \cdot .\overline{1} \\ 10x = 1.\overline{1} \\ \fbox{3) subtract equation 1 from 2} 10x - 1x = 1.\overline{1} - .\overline{1} \\ 9x = 1 \\ x = \frac{1}{9} $$
. Yes, the repeating decimal $$ .\overline{1} $$ is equivalent to the fraction $$ \frac{1}{9} $$

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Rational Number Game
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