Real analysis/Section 1 Exercises/Answers

Exercises

1. Show that $-1<0$
2. Show that $\mathrm{\forall }x,y\in \mathbb{Z},-(xy)=(-x)y=x(-y)$
3. Show that $\mathrm{\forall }z\in \mathbb{Z},z^2\ge 0$
4. Show that $\mathrm{\forall }z,\in \mathbb{Z},z\ge 0\iff -z\le 0$
5. Show that $\mathrm{\forall }x,y,z\in \mathbb{Z},(x<0)\cap (y<z)\iff (xy>xz)$
6. Let $p$ be any prime. Show that $\sqrt{p}$ is irrational.
7. Complete the proofs of the simple results given above.
8. Show that the complex numbers $ℂ$ cannot be made into an ordered field.
9. Complete the proof of the square roots theorem by giving details for the case $x<1$.

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6. With the view of getting a contradiction, assume $\sqrt{p}$ is rational. Then $\sqrt{p}=r/s$ for some integers $s$ and $r$ such that they have no common factor other than one, (that is, $s$ and $r$ are in lowest terms). Squaring both sides and rearranging terms gives $s^{2}p=r^2$. Since $p$ is prime, $r$ must be divisible by $p$, say $r=pt$ for some integer $t$. By substitution, $s^{2}p=p^2{t}^2$, so that $s^2=p{t}^2$, and thus $s$ must also be divisible by $p$, a contradiction.
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