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39 votes
39 votes
The sum of a rational number and an irrational number equals:

A terminating decimal
A rational number
A fraction number
An irrational number

User Arman Hayots
by
2.9k points

1 Answer

29 votes
29 votes

Answer: D) An irrational number

rational + irrational = irrational

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Proof:

The claim is that A+B = C where,

  • A = some rational number
  • B = some irrational number
  • C = some other irrational number

Because A is rational, we can write it like A = p/q for some integers p,q. The value of q cannot be zero.

Let's for a moment consider the opposite scenario and let's assume that A+B was rational. We'll prove that a contradiction arises from this (hence this is a proof by contradiction).

If A+B = C was rational, then C = r/s for some integers r,s. The s cannot be zero.

From there we can have these steps to isolate B

A+B = C

B = C - A

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

B = (rq - ps)/(qs)

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

B = some rational number

But this directly contradicts B set up as an irrational number.

So if A is rational and B is irrational, then A+B cannot possibly be rational due to the proof by contradiction above. The only other possibility is that A+B must be irrational.

User Madamadam
by
3.2k points
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