Rational numbers:
Rational numbers are numbers that can be written in the form \(\frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\).
Irrational numbers:
A number \(x\) is said to be an irrational number if it cannot be expressed in the form of \(\frac{p}{q}\) where \(q \neq 0\).
Example:
\(\sqrt{2}\), \(\sqrt{5}\), \(\sqrt{7}\) are some examples of irrational numbers.
Properties of rational and irrational numbers:
- The sum of two rational numbers is always rational.
- The difference of two rational numbers is always rational.
- The product of two rational numbers is always rational.
- The division of two rational numbers is always rational (If divisor \(\neq 0\)) .
- All the decimal numbers (terminating or non-terminatin) are rational numbers.
- All the intergers are rational numbers.
- The sum of a rational and an irrational number is always irrational.
- The product of a rational and an irrational number is always irrational.
Working rule to prove an irrational number:
Let \(x\) be the given irrational number or expression.
Step - 1: Assume the given number \(x\) is rational.
Step - 2: Write in the form \(x = \frac{p}{q}\) where \(p\) and \(q\) are integers and \(q \neq 0\).
Step - 3: Rearrange and simplify the given expression algebraically.
Step - 4: Arrive at a contradiction that an irrational number is a rational number or an impossible condition like \(p\) and \(q\) having a common factor when assumed coprime.
Step - 5: Thus, arrive at a conclusion that the assumption is false and prove the given number or expression \(x\) is irrational.
Example:
Prove that \(8 - \sqrt{3}\) is an irrational number.
Proof: Let us assume that \(8 - \sqrt{3}\) is a rational number.
Therefore, it can be written as \(8 - \sqrt{3} = \frac{p}{q}\) where \(p, q\) are integers and \(q \neq 0\).
Here, \(p\) and \(q\) are coprime(no common factors between \(p\) and \(q\)).
Thus, \(8 - \sqrt{3} = \frac{p}{q}\)
\(8 - \frac{p}{q} = \sqrt{3}\)
\(\frac{8q - p}{q} = \sqrt{3}\)
Since \(8\), \(p\) and \(q\) are integers, then \(\frac{8q - p}{q}\) is rational. This implies that \(\sqrt{3}\) is a rational number.
But, we know that \(\sqrt{3}\) is an irrational number.
Therefore, our assumption is wrong.
Thus, \(8 - \sqrt{3}\) is an irrational number.
Hence, we proved.