Question

Write irrational numbers between 0.1 and 0.2.

Answer 1

Hint: We recall the definitions of rational numbers and irrational numbers. We convert the given decimal numbers into fractions and square them. We find four factions in the form of $\dfrac{p}{q}$ in between squared fractions of the given decimals. We convert back the fractions to decimals and take square root of them,

Complete step by step answer:
We know that a rational number can be expressed in the form $\dfrac{p}{q}$ such that $p,q$ are integers and $q$ is not zero. An irrational numbers is a number that cannot be expressed in the form $\dfrac{p}{q}$. We also know the inequality of square roots;
\[a < b\Rightarrow \sqrt{a} < \sqrt{b}\]

We are asked in the question to find 4 irrational numbers between 0.1 and 0.2. We know how to convert decimals into fractions. We write the number without decimal point as numerator and put same number of zeros as the number of digits after the decimal point. So the fraction form the given numbers are $0.1=\dfrac{1}{10}$ and $0.2=\dfrac{2}{10}$.
We take the square of obtained factions . So we have;
\[\begin{align}
  & \Rightarrow {{\left( 0.1 \right)}^{2}}={{\left( \dfrac{1}{10} \right)}^{2}}=\dfrac{1}{10\times 10}=\dfrac{1}{100} \\
 & \Rightarrow {{\left( 0.2 \right)}^{2}}={{\left( \dfrac{2}{10} \right)}^{2}}=\dfrac{2\times 2}{10\times 10}=\dfrac{4}{100} \\
\end{align}\]
We multiply 10 and get equivalent fractions.
\[\begin{align}
  & \Rightarrow {{\left( \dfrac{1}{10} \right)}^{2}}=\dfrac{1}{100}=\dfrac{1\times 10}{1000}=\dfrac{10}{1000} \\
 & \Rightarrow {{\left( \dfrac{2}{10} \right)}^{2}}=\dfrac{4}{100}=\dfrac{4\times 10}{100\times 10}=\dfrac{40}{1000} \\
\end{align}\]
We can now easily find 4 numbers in between ${{\left( 0.1 \right)}^{2}}$ and ${{\left( 0.2 \right)}^{2}}$in fraction form by taking a numbers in between 10 and 40 as numerator and 1000 as denominator. We take 11,23,28,37 as numerators and denominator as 1000. We have the fractions as
\[\dfrac{11}{1000},\dfrac{23}{1000},\dfrac{28}{1000},\dfrac{37}{1000}\]
So we have;
\[\Rightarrow \dfrac{10}{100} < \dfrac{11}{100} < \dfrac{23}{100} < \dfrac{28}{100} < \dfrac{37}{100} < \dfrac{40}{100}\]
We take square root and use the inequality of square roots and have;
\[\Rightarrow \sqrt{\dfrac{10}{1000}} < \sqrt{\dfrac{11}{1000}}<\sqrt{\dfrac{23}{1000}} < \sqrt{\dfrac{28}{1000}}<\sqrt{\dfrac{37}{1000}} < \sqrt{\dfrac{40}{1000}}\]
We convert these factions into decimals and a
\[\begin{align}
  & \Rightarrow \sqrt{{{\left( 0.1 \right)}^{2}}} < \sqrt{0.011} < \sqrt{0.023} < \sqrt{0.028} < \sqrt{0.037}< \sqrt{{{\left( 0.2 \right)}^{2}}} \\
 & \Rightarrow 0.1 < \sqrt{0.011} < \sqrt{0.023} < \sqrt{0.028}< \sqrt{0.037} < 0.2 \\
\end{align}\]

So four irrational numbers between 0.1 and 0.2 are $\sqrt{0.011},\sqrt{0.23},\sqrt{0.028},\sqrt{0.037}$.\[\]

Note: We have converted the denominator to 1000 for easier conversion to decimal otherwise we can find any fraction in between $\dfrac{1}{100},\dfrac{4}{100}$ and then take their square root to get an irrational number. We can alternatively solve using the fact that irrational numbers have non-terminating and non-repeating decimal representation, for example 0.121221222... is an irrational number in between 0.1 and 0.2.