# Rational Plus Irrational?

Explain why the sum of a rational number and irrational number MUST BE irrational?

Relevance

Let's assume that the sum of the rational number a/b (where a and b are integers) and irrational number x is also a rational number c/d (where c and d are some integer), i.e.

(a/b) + x = c/d

Then x = (c/d) - (a/b)

RHS is clearly rational while LHS is irrational. This is a contradiction.

Hence the sum must be irrational

• If u (rational) + β (irrational) = v (rational),

then u - v (rational) = -β (irrational);

So u + β = γ (irrational),

• Suppose it is not true. In that case, let there be a rational number r, and an irrational number x. Suppose their sum is rational, so this relation must be true:

r + x = a/b, for some certain integers a and b

Since r is rational, r = n/m, for some integers m and n.

r + x = a/b

m/n + x = a/b

x = a/b - m/n

x = (an - bm)/(bn)

But a, b, m, and n are all integers. Therefore, so are (an - bm) and bn. That means x is the ratio of two integers, which cannot be so for an irrational x. By contradiction, we must reject the proposition that r + x can be rational. Since the sum cannot be rational, it must be irrational.

• Because the irrational number can't be represented by a fraction, so for instance, 4 + pi will be irrational, something like 7.14159(continue on into infinity).