No, multiplying an irrational number by a rational number always results in an irrational number, not a rational one.
In general, the product of an irrational number and a rational number is irrational, meaning that it cannot be expressed as a simple fraction of two integers.
An irrational number is defined as a number that cannot be represented as the quotient of two integers, and examples include the square root of 2 or the mathematical constant π.
When you multiply an irrational number by a rational number, the result typically retains the irrational nature of the original number.
The reasoning lies in the fact that multiplying an irrational number by any non-zero rational number does not eliminate the non-repeating, non-terminating decimals inherent in irrational numbers.
For instance, if you consider the irrational number √2 and multiply it by a rational number like 2/3, the result (√2) * (2/3) remains irrational.
The product is a combination of the irrationality of √2 and the rationality of 2/3, resulting in an irrational number.
Therefore, it is a fundamental mathematical property that the multiplication of an irrational number by a rational number typically yields an irrational result.
Question
Can you multiply an irrational number by a rational number and the answer is rational?