Question

The sum of a rational number and an irrational number is irrational.

Answer 1

Answer: Always

Proof:

Let

• A = some rational number
• B = some irrational number
• C = some rational number (that isn't necessarily equal to A)

We'll show that a contradiction happens if we tried to say A+B = C.

In other words, we'll show a contradiction happens for the form rational+irrational = rational.

Because A and C are rational, we can say

A = p/q

C = r/s

where p,q,r,s are integers. The q and s in the denominators cannot be zero.

So,

A+B = C

B = C - A

B = (r/s) - (p/q)

B = (qr/qs) - (ps/qs)

B = (qr-ps)/(qs)

B = (some integer)/(some other integer)

B = some rational number

But wait, we stated that B was irrational. The term "irrational" literally means "not rational". Irrational numbers cannot be written into fraction form of two integers. This is a contradiction and shows that A+B = C is never possible if A,C are rational while B is irrational.

This must lead to the conclusion that A+B must always be irrational if A is rational and B is irrational.

The template is this

rational + irrational = irrational

rational + rational = rational

Answer 2

ALWAYSS
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