Understanding Rational and Irrational Numbers
What are Rational and Irrational Numbers?
In mathematics, numbers are often classified as rational or irrational:
- Rational Numbers: Numbers that can be expressed as a ratio of two integers, where the denominator is not zero.
- Irrational Numbers: Numbers that cannot be expressed as a ratio of two integers.
Mathematical Definition
For a number \(x\):
- \(x\) is rational if it can be written as \(x = \frac{a}{b}\), where \(a\) and \(b\) are integers and \(b \neq 0\).
- \(x\) is irrational if it cannot be written in this form.
Calculation Method
To determine if a number \(x\) is rational or irrational:
- Check if \(x\) can be written as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers.
- If not, check if \(x\) is an integer.
- If not, check if \(x\) has a finite decimal representation.
- If none of the above conditions are met, \(x\) is irrational.
Example Calculation
Let's determine if \(\frac{22}{7}\) is rational or irrational:
- \(\frac{22}{7}\) is already in the form \(\frac{a}{b}\) where \(a = 22\) and \(b = 7\) are integers.
- Therefore, \(\frac{22}{7}\) is a rational number.
Visual Representation
This diagram represents \(\frac{22}{7}\) as a rational number, visualized as a fraction.