משפט ליוביל (קירוב דיופנטי) – הבדלי גרסאות

: <math>\left\vert \alpha -\frac{p/}{q}\right\vert = \frac{\left\vert f(\alpha)- f(\frac{p/}{q})\right\vert / }{\left\vert f'(x_0x_{0}) \right\vert }= \left\vert \frac{f(p/q) / }{f'(x_0x_{0}) }\right\vert \,</math>

: <math>\left\vert f(\frac{p/}{q}) \right\vert = \left\vert \sum_{i=0}^n c_i p^i q^{-i} \right\vert = \frac{\left\vert \sum_{i=0}^n c_i p^i q^{n-i} \right\vert}{q^n} \ge \frac {1}{q^n} </math>

:<math>\ \left\vert \alpha - \frac{p/}{q }\right\vert = \left\vert \frac{f(\frac{p/}{q}) / }{f'(x_0x_{0}) }\right\vert \ge \frac{1/(}{q^{n}\left|f'(x_0x_{0})\right|) }\ge \frac{1/(Mq}{q^{n) > }M}\geq\frac{A/}{q^{n }}\ge \left\vert \alpha - \frac{p/}{q }\right\vert </math>