The Difference between Rational and Irrational Numbers
The main difference between rational and irrational numbers is that rational numbers are numbers that can be stated in the form of \(\frac{p}{q}\), where \(p\) and \(q\) are integers and \(q\neq 0\), whereas irrational numbers are numbers that cannot be expressed so (though both are real numbers).
When two numbers are divided if the digits in the quotient after the decimal point are non-terminating and non-repeating then the number is an irrational number. Whereas, if the digits in the quotient after the decimal point either terminates or are recurring or repeating then the number is a rational number.
Examples of Rational and Irrational Numbers
Rational Numbers | Irrational Numbers |
|---|---|
Examples: 525, 139, 16, 4, \(\frac{5}{2}\), \(\frac{-1}{3}\), 4.879, 2.\(\overline{34}\) | Examples: \(\sqrt{21}\), -\(\sqrt{5}\) , \(\sqrt{17}\), \(\sqrt{7}\), \(\sqrt{10}\), -1.34567... |
Quick Recall
Solved Rational and Irrational Examples
Example 1: Group the following set of numbers into rational and irrational numbers:
3.78, 21, \(\sqrt{31}\), 0, -2
Solution:
3.78 is a terminating decimal and that is why it is a rational number.
21 is a rational number, as it can be expressed in the form \(\frac{p}{q}\).
\(\sqrt{31}\) = 5.67764… The digits are non-recurring. Hence \(\sqrt{31}\) is an irrational number.
-2 is a rational number as it can be expressed in the form \(\frac{p}{q}\).
Example 2: Write three rational and irrational numbers between 4 and 5.
Solution:
Rational numbers between 4 and 5
4.5, 4.888…, 4.25
Irrational numbers between 4 and 5
4.23193…, \(\sqrt{17}\), 4.1356775…
Example 3: Write three rational numbers between \(\frac{2}{7}\) and \(\frac{3}{5}\)
Solution:
First let’s convert the given fractions to like fractions.
\(\frac{2}{7}=\frac{2\times 5}{7 \times 5}=\frac{10}{35}\)
\(\frac{3}{5}=\frac{3\times 7}{5 \times 7}=\frac{21}{35}\)
Hence, three rational numbers between \(\frac{10}{35}\) and \(\frac{21}{35}\) are:
\(\frac{11}{35}\), \(\frac{12}{35}\) and \(\frac{13}{35}\)
Half, \(\frac{1}{4}\), \(\frac{13}{2}\), 0.49, 1023, and so on are some examples of rational numbers.
Pi \((\pi)\) = 3.14159…., Euler’s Number (e) = (2.71828…), \(\sqrt{8}\), \(\sqrt{7}\) are irrational numbers.
Ratios of two integers (positive or negative) when the denominator is not equal to 0 can be used to represent rational numbers. Examples: -2, \(\frac{1}{4}\), 5, 0.222…
A number is rational if it is a terminating or a recurring decimal, such as \(\frac{1}{2}\) = 0.5, 0.99999… and so on. Irrational numbers, such as 0.31545673, are non-terminating and non-repeating decimals.