0

Any idea?

I've started with $a= p/q$, meaning $q$ has to be rational.

Then $1/a= q/p$ But surely this could be irrational if, say $p=9$, $q=4$?

mrtaurho
  • 16,103
AnoUser1
  • 169
  • 2
  • 9
  • 3
    I think $4/9$ is a rational number. – Angina Seng Nov 18 '18 at 17:31
  • A rational number is the ratio of integers. Suppose $a = p/q$ is rational and nonzero. Then $p,q$ are integers, with $p,q\neq 0$. Then $1/a = q/p$. Since $p,q$ are still integers, notice that $1/a$ is rational. I think what you're thinking of are improper fractions as opposed to irrational numbers. (Well ... maybe. I'm not sure.) – PrincessEev Nov 18 '18 at 17:32
  • 2
    A number is rational if it has the form $\frac{n}{m}$ for $n,m\in\mathbb{Z}$ and $m\not = 0$. So both $\frac{p}{q}$ and $\frac{q}{p}$ are rationals. – Yanko Nov 18 '18 at 17:32
  • Yeah my bad I'm being thick today lol – AnoUser1 Nov 18 '18 at 17:36
  • Q is generally defined as being the field extension of the integers. According to that definition, the fact that the multiplicative inverse is in Q simply follows from the definition of field. So the first questions to ask with this sort of problem are "What definition of a rational number are we working with?" and "What is the definition of 1/a?" – Acccumulation Nov 18 '18 at 17:45

1 Answers1

1

By definition, a real number $x$ is irrational if and only if there are integers $m$ and $n\not=0$ such that $x=m/n$. If $x\not=0$, then $m\not= 0$, so, as you pointed out $1/x=n/m$. This number is by definition rational!

ervx
  • 12,208