97.7k views
4 votes
The sum of a rational number and an irrational number is irrational.

ALWAYS
NEVER
SOMETIMES

2 Answers

3 votes
ALWAYSS
HAVE A GOOD DAY
User Discomurray
by
8.5k points
6 votes

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

User Emin Mastizada
by
8.6k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories