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Answer:
An irrational number is a number that can not be written as the quotient of two integer numbers.
Then if we have:
A = a rational number
B = a irrational number.
Then we can write:
A = x/y
Then the product of A and B can be written as:
A*B = (x/y)*B
Now, let's assume that this product is a rational number, then the product can be written as the quotient between two integer numbers.
(x/y)*B = (m/n)
If we isolate B, we get:
B = (m/n)*(y/x)
We can rewrite this as:
B = (m*y)/(n*x)
Where m, n, y, and x are integer numbers, then:
m*y is an integer
n*x is an integer.
Then B can be written as the quotient of two integer numbers, but this contradicts the initial hypothesis where we assumed that B was an irrational number.
Then the product of an irrational number and a rational number different than zero is always an irrational number.
We need to add the fact that the rational number is different than zero because if:
B is an irrational number
And we multiply it by zero, we get:
B*0 = 0
Then the product of an irrational number and zero is zero, which is a rational number.