Question

Given a rational number, a, and an irrational number, b, prove that the sum of a and b is irrational.

Answer 1

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. ... This is our contradiction, so it must be the case that the sum of a rational and an irrational number is irrational. And that's our proof!

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Answer 2

Step-by-step explanation:

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 = c/d and a + b = m/n

Plugging a = c/d into a + b = m/n gives the following:

c/d + b = m/n

Now, let's subtract c/d from both sides of the equation.

b = m/n - c/d, or

b = m/n + (-c/d)

Since the rational numbers are closed under addition, b = m/n + (-c/d) 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 irrational.

And that's our proof!