Answer by Joe for Rational + irrational = always irrational?
Lemma:If $a$ is rational, and $b$ is rational, then $a+b$ is rational.Proof:Let $a=p/q$, and $b=r/s$, where $p,q,r,s$ are integers. Then, $a+b=(ps+qr)/(qs)\blacksquare$Now, let $a$ be rational, so that...
View ArticleAnswer by Shaurya Gupta for Rational + irrational = always irrational?
Let x be rational and y be irrational. So let us assume that x + y is rational.(x+y) - x will also be a rational since rational - rational is always rational.therefore y is also rational. but that is a...
View ArticleAnswer by ncmathsadist for Rational + irrational = always irrational?
The rational numbers are closed under addition and subtraction. Let $w$ be any irrational number and $r$ be a rational number. Since$$ (r + w) - r = w$$and $w$ is irrational, one of the subtrahends...
View ArticleAnswer by user88377 for Rational + irrational = always irrational?
This follows essentially from the fact that as additive abelian groups $\mathbb Q$ is a normal subgroup of $\mathbb R$, which is true since both groups are abelian and the latter contains the former....
View ArticleAnswer by Fredrik Meyer for Rational + irrational = always irrational?
A proof in the style of "mathematics made difficult": Note that a number $r$ is rational if and only if $\mathbb Q(r) = \mathbb Q$. Now it is easy to see that $\mathbb Q(\gamma) = \mathbb Q(r+\gamma)$...
View ArticleAnswer by Asaf Karagila for Rational + irrational = always irrational?
Note that the sum of two rationals is always rational, and that if $n$ is rational then $-n$ is rational. Now suppose that $x$ is any number and $n$ is rational.Suppose $x+n$ is rational, then...
View ArticleAnswer by user87543 for Rational + irrational = always irrational?
Suppose $x$ is irrational and $x+\dfrac pq=\dfrac mn$ then, $x=\dfrac mn-\dfrac pq=\dfrac{mq-np}{nq}$ so, $x$ would then be rational. :)
View ArticleAnswer by DKal for Rational + irrational = always irrational?
In fact, for any rational number $ r $ it is true that the irrationality of $ x+r $ implies the irrationality of $ x $. This is due to the fact that the rationals are closed under addition. Assume that...
View ArticleAnswer by Ross Millikan for Rational + irrational = always irrational?
Yes, it is true that $x+1$ being irrational implies $x$ is irrational. Given that $x+1$ is irrational, assume $x=\frac ab$ with $a,b$ integers. Then $x+1=\frac {a+b}b$ would be rational as well.
View ArticleAnswer by Casteels for Rational + irrational = always irrational?
Look at the contrapositive: If $x$ is rational, then $x+n$ is rational. Clearly this is a true statement.
View ArticleRational + irrational = always irrational?
I had a little back and forth with my logic professor earlier today about proving a number is irrational. I proposed that 1 + an irrational number is always irrational, thus if I could prove that 1 +...
View ArticleAnswer by Michael Ejercito for Rational + irrational = always irrational?
Let $j$ be an irrational number, and $x$ a rational number, such that $x+1=j$.There are coprime integers $z_1$ and $z_2$ such that...
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