Question 1208936: True or false? The sum of rational and irrational number is rational
Answer by math_tutor2020(3817)   (Show Source):
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Answer: False
Explanation
x = rational number
y = irrational number
x = a/b for some integers a,b where b is nonzero
y cannot be written as a fraction of integers because it is set up to be irrational.
Assume x+y was rational.
We'll do a proof by contradiction to show that x+y is instead irrational.
x+y = rational
x+y = p/q
(a/b) + y = p/q
y = (p/q) - (a/b)
y = (bp)/(bq) - (aq)/(bq)
y = (bp-aq)/(bq)
y = (some integer)/(some integer)
y = rational number
But wait, we made y irrational and now it's rational.
This is clearly a contradiction.
A number cannot be both rational and irrational at the same time.
The term "irrational" literally means "not rational".
This contradiction means the assumption x+y = rational is false, so x+y = irrational must be the case.
A template to write down in your notes would be rational + irrational = irrational
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