The sum or product of a non-zero rational number and an irrational number is always irrational.
To understand why this is the case, let’s first define our terms. A rational number is any number that can be expressed as the quotient of two integers (where the denominator is not zero), such as 1/2 or -3. An irrational number, on the other hand, cannot be expressed as such a fraction. Examples of irrational numbers include π (pi) and √2.
Now, when we add a rational number to an irrational number, the result cannot be rational. This is because if we assume their sum is rational, we could rearrange the equation to show that the irrational number must be rational as well, which leads to a contradiction.
Similarly, when we multiply a non-zero rational number by an irrational number, the product is also irrational. Again, if the product were rational, then by dividing the product by the non-zero rational number, we would find that the irrational number is also rational, which is impossible.
In summary, both the sum and the product of a non-zero rational number and an irrational number result in an irrational number. This property highlights the inherent differences between rational and irrational numbers.