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Question

Prove that the sum of a rational number and an irrational number is always irrational.

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Solution

Assume that a is rational, b is irrational, and a+b is rational. Since a and a+b are rational, we can write them as fractions.
Let a=cd and a+b=mn
Substituting a=cd in a+b=mn gives the following:
cd+b=mn
Now, let's subtract cd from both sides of the equation.
b=mncd, or

b=mn+(cd)
Since the rational numbers are closed under addition, b=mn+(cd) is a rational number. However, the assumptions said that b is irrational, and b cannot be both rational and irrational. This is our contradiction, so it must be the case that the sum of a rational and an irrational number is always irrational.

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