4. Irrational Numbers

• Irrational numbers cannot be expressed as a simple fraction. They have non-repeating, non-terminating decimal expansions.
• Examples include $\sqrt{2}$ and $\pi$.

Example

Let's classify the number 4.5:

• Whole Number? No, because it has a decimal.
• Integer? No, because it is not a whole number.
• Rational? Yes, because it can be expressed as $\frac{9}{2}$.
• Irrational? No, because it can be expressed as a fraction.

Key Questions

• What is the difference between whole numbers and integers?
• Can all rational numbers be represented as decimals? Why or why not?
• Give an example of an irrational number.

Summary

• Whole Numbers: Non-negative integers (0, 1, 2, ...)
• Integers: Whole numbers and their negatives (..., -2, -1, 0, 1, 2, ...)
• Rational Numbers: Can be expressed as $\frac{a}{b}$ where $b = 0$
• Irrational Numbers: Cannot be expressed as a fraction, non-terminating decimals.

Understanding these number classifications helps us in various math concepts and applications.