product of rational number and irrational number is
irrational
an irrational number.
When we multiply a rational number (a number that can be expressed as a ratio of two integers) and an irrational number (a number that cannot be expressed as a ratio of two integers), the result will always be an irrational number.
To understand why this is the case, consider the following example:
Let’s say we have a rational number, like 1/2, and an irrational number, like √2. If we multiply these two numbers together, we get:
1/2 x √2 = (√2)/2
The result (√2)/2 is irrational, because √2 is irrational and cannot be expressed as a ratio of two integers. Therefore, any product of a rational number and an irrational number will always be irrational.
This can also be proven using the definition of irrational numbers. An irrational number is defined as a number that cannot be expressed exactly as a ratio of two integers. Since a rational number can always be expressed as a ratio of two integers, multiplying a rational number and an irrational number can never result in a number that can be expressed exactly as a ratio of two integers, making it irrational.
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