Understanding Irrational Numbers
Irrational numbers are numbers that cannot be expressed as a fraction of two integers. They have non-repeating, non-terminating decimal expansions. An example of an irrational number is the square root of 2 (√2).
Product of an Irrational and a Rational Number
When an irrational number is multiplied by a rational number (a number that can be expressed as a fraction), the product will still be irrational.
Example of an Irrational Number with a Rational Product
- Consider the irrational number √2.
- Let’s multiply √2 by a rational number, such as 2.
Calculation
- The product is 2 * √2.
Explanation
- The number 2 is rational because it can be expressed as 2/1.
- The product, 2√2, remains irrational because if it were rational, it could be expressed as a fraction of two integers, which contradicts the definition of √2.
Conclusion
- Therefore, the product of an irrational number (√2) and a rational number (2) is 2√2, which is irrational.
Key Takeaways
- Irrational numbers cannot be written as a simple fraction.
- The product of an irrational number and a non-zero rational number is always irrational.
- Example: 2 * √2 = 2√2 (irrational).
This understanding is crucial in recognizing the nature of numbers in mathematics.