Problems and Theorems in Analysis · 1

If $r$ and $s$ are rational numbers and $r+s\sqrt{-3}$ is an integer, then $2r$ and $2s$ must be rational integers and $2r\equiv 2s \pmod2$; $r$ and $s$ need not necessarily be rational integers.

If both $\vartheta /\pi$ and $\cos \vartheta$ are rational, then $\cos \vartheta$ must have one of the following five values:

Consider the interior angle included between two contiguous faces of a regular solid and call it

according as the solid is a tetra-, hexa-, octa-, dodeca-, or icosahedron (a hexahedron is a cube). Show that the ratios

are irrational.

The numbers $\alpha_1,\alpha_2,\ldots ,\alpha_m$ are called linearly dependent if there exist rational integers $a_1,a_2,\ldots ,a_m$ not all zero such that $a_1\alpha_1+a_2\alpha_2+\cdots +a_m\alpha_m=0$. If no relation of this kind exists then $\alpha_1,\alpha_2,\ldots ,\alpha_m$ are called linearly independent. Thus, if $\beta \neq 0$, the numbers $\alpha$ and $\beta$ are linearly dependent or independent according as $\alpha /\beta$ is rational or irrational. The number $\alpha$ is algebraic if there exists an $n$ (a positive rational integer) such that $1,\alpha ,\alpha^2,\ldots ,\alpha^n$ are linearly dependent.

The numbers (angles) $T,H$ and $O$ are linearly dependent; in fact

(see 197.2; it can be also proved by elementary geometry). Yet $T$ and $O$ are linearly independent.

(continued). Show that $T,H,D$ and $I$ are linearly independent. Also $H,O,D$ and $I$ are linearly independent, and any linear homogeneous relation with rational coefficients between $T,H,O,D$ and $I$ can differ only trivially (that is, only by a numerical factor) from the one given in 197.3.

If both $\vartheta /\pi$ and $\tan^2\vartheta$ are rational then $\tan^2\vartheta$ must have one of the following five values (regarding $\infty =1/0$ as ``rational''):

With the exception of the four cases $n=1,2,4$, and 8, $\tan 2\pi /n$ is irrational for all positive rational integers $n$.