Why Multiplying A Rational Number With An Irrational Number Always Results In An Irrational Number: A Simple Proof By Contradiction

product of rational number and irrational number is

irrational

an irrational number.

When we multiply a rational number (a number that can be expressed in the form p/q, where p and q are integers and q is not equal to zero) by an irrational number (a number that cannot be expressed in the form p/q), the result is always an irrational number.

This can be demonstrated through a simple proof by contradiction. Suppose the product of a rational number and an irrational number is a rational number. Let x be a rational number, and y be an irrational number. Then we can write:

x * y = z, where z is a rational number

If we square both sides, we get:

(x * y)^2 = z^2

Expanding the left-hand side, we get:

x^2 * y^2 = z^2

Rearranging the terms, we get:

y^2 = (z^2) / (x^2)

Since z^2 and x^2 are both integers, y^2 must be a rational number. But this contradicts the fact that y is an irrational number. Therefore, our assumption that the product of a rational number and an irrational number is a rational number is false, and the product must be an irrational number.

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