Real Numbers
Real numbers encompass all numbers that can be plotted on a number line. This includes both rational and irrational numbers.
Irrational Numbers
Irrational numbers are a subset of real numbers. They cannot be expressed as a simple fraction, meaning they cannot be written as a ratio of two integers. They have decimal representations that are non-terminating and non-repeating.
Key Differences:
- Rational Numbers: Can be expressed as a fraction of two integers (e.g., 1/2, 3/4, -5/7). Their decimal representations either terminate (e.g., 0.5) or repeat (e.g., 0.333...).
- Irrational Numbers: Cannot be expressed as a fraction of two integers. Their decimal representations are non-terminating and non-repeating (e.g., pi (π) = 3.14159265..., √2 = 1.41421356...).
Examples:
- Rational: 2, -3, 0.75, 1/3
- Irrational: π, √2, √3, e
In Summary:
All irrational numbers are real numbers, but not all real numbers are irrational. Irrational numbers have decimal representations that are non-terminating and non-repeating, while rational numbers can be expressed as a fraction of two integers.
