Approximating Irrational Numbers

Irrational numbers are real numbers that cannot be expressed as a fraction of two integers. Their decimal expansions are non-repeating and non-terminating. A common example is $\pi$, which is approximately $3.14$.

Key Concepts

• Irrational Numbers: Numbers that cannot be expressed as a fraction.
• Approximation: Finding a close value for an irrational number.

Steps to Approximate an Irrational Number

1. Identify the irrational number you want to approximate (e.g., $\sqrt{2}$).
2. Determine a simple fraction or decimal that is close to the number.
3. Use a calculator for greater precision if necessary.

Example: Approximating $\sqrt{2}$

The value of $\sqrt{2}$ is approximately $1.41421356...$ To approximate:

• Start with a simple fraction. We can try $\frac{3}{2} = 1.5$.
• Check smaller fractions. $\frac{14}{10} = 1.4$ is closer.
• Refine further if needed (e.g., $1.41$).

Thus, a good approximation for $\sqrt{2}$ could be $1.41$.

Key Questions

• What is an irrational number?
• How can we approximate $\pi$?
• Why are some numbers irrational?

Conclusion

Approximating irrational numbers helps us use them in practical situations, such as measurements and calculations.