Rational vs. Irrational Numbers

Objectives

• Understand the difference between rational and irrational numbers.
• Classify numbers as rational or irrational.

Definitions

• Rational Numbers: Numbers that can be expressed as the quotient of two integers, where the denominator is not zero. For example, $\frac{3}{4}$, $0.5$, and $-2$ are rational numbers.
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction. Their decimal expansions are non-repeating and non-terminating. Examples include $\sqrt{2}$ and $\pi$.

Key Properties

• A rational number can always be written in the form $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.
• An irrational number cannot be expressed as $\frac{a}{b}$.

Example

Determine whether the following numbers are rational or irrational:

1. $\sqrt{16}$
   • $\sqrt{16} = 4$, which is rational.
2. $\sqrt{3}$
   • Cannot be expressed as a fraction, so it is irrational.
3. $\frac{7}{0}$
   • Undefined, thus not a rational number.

Practice

Classify the following numbers:

• $5.75$
• $\pi$
• $\frac{-3}{8}$
• $\sqrt{5}$

Key Questions

• What makes a number rational or irrational?
• Can you give examples of rational and irrational numbers?
• How can you identify irrational numbers?

Conclusion

Understanding the difference between rational and irrational numbers is fundamental in mathematics. Knowing how to identify and classify them helps in various math applications.