Section 1
Numbers
Prove that $\sqrt{5}$ is irrational and hence prove that $a+b \sqrt{5}$ is irrational for all rationals $a$ and $b, b \neq 0$. Deduce that the golden ratio $r$, defined by $r=1+1 / r, r>0$, is irrational.
Which of the following statements are true?(a) $x$ rational, $y$ irrational $\rightarrow x+y$ irrational.(b) $x$ rational, $y$ rational $\rightarrow x+y$ rational.(c) $x$ irrational, $y$ irrational $\Rightarrow x+y$ irrational.Prove the true ones and give a countercxample for each of the false ones.
Show that between any two distinct real numbers therc arc infinitely many rationals and infinitely many irrationals.
Prove that thcre is no rational number $x$ such that $10^{x}-2 .$ Deduce that $\log _{10} 2$ is irrational.
Let $x=\sqrt{3+2 \sqrt{2}}-\sqrt{3-2 \sqrt{2}}$ and calculatc $x^{2}$. Is $x$ irrational?