ThmDex – An index of mathematical definitions, results, and conjectures.

Formulation 0

Let $x, y \in \mathbb{R}$ each be a D993: Real number such that

(i)	\begin{equation} \sqrt{x^2 + y^2} = 1 \end{equation}

Then

(1)	\begin{equation} x^2 + x y \leq 1 \end{equation}
(2)	\begin{equation} x^2 + x y = 1 \quad \iff \quad x = 1, \; y = 0 \end{equation}
(3)	\begin{equation} x^2 + x y \geq 0 \end{equation}
(4)	\begin{equation} x^2 + x y = 0 \quad \iff \quad x = y = 0 \end{equation}

Proofs

Proof 0

Let $x, y \in \mathbb{R}$ each be a D993: Real number such that

(i)	\begin{equation} \sqrt{x^2 + y^2} = 1 \end{equation}

This result is a special case of R5613: Tight upper and lower bounds for the value of a real polynomial of two variables given that the arguments lie on the unit circle when applied to the matrix \begin{equation} A = \begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix} \end{equation} $\square$