Manin’s trick: Writes any symbol {α, β} as a linear combination of generating symbols of the form r_{i}{0, ∞}.

The trick implies that the symbols r_{i}{0, ∞} generate

The trick: Suffices to consider {0, b/a}. Expand b/a as a continued fraction and consider the successive convergents in lowest terms:

\frac{b}{a} = \frac{b_{n}}{a_{n}}, \frac{b_{n−1}}{a_{n−1}}, …, \frac{b_{0}}{a_{0}} = \frac{b_{0}}{1}, \frac{b_{−1}}{a_{−1}} = \frac{1}{0}, \frac{b_{−2}}{a_{−2}} = \frac{0}{1}

(the last two are added formally).

Then

b_{k}a_{k−1} − b_{k−1}a_{k} = (−1)^{k−1},

so that

g_{k} = (\begin{matrix}b_{k} & (−1)^{k−1}b_{k−1} \\ a_{k} & (−1)^{k−1}a_{k−1} \end{matrix} ) ∈ SL_{2}(Z).

Hence

\{\frac{b_{k−1}}{a_{k−1}}, \frac{b_{k}}{a_{k}}\} = g_{k}\{0, ∞\} = r_{i}\{0, ∞\},

for some i, is of the required special form.

Example: Let N = 11, and consider {0, 4/7}. We have

\frac{4}{7} = 0 + \frac{1}{1 + _{\frac{1}{1+\frac{1}{3}}}} ,

so partial convergents are

\frac{b_{−2}}{a_{−2}} = \frac{0}{1}, \frac{b_{−1}}{a_{−1}} = \frac{1}{0}, \frac{b_{0}}{a_{0}} = \frac{0}{1}, \frac{b_{1}}{a_{1}} = \frac{1}{1}, \frac{b_{2}}{a_{2}} = \frac{1}{2}, \frac{b_{3}}{a_{3}} = \frac{4}{7}.

Thus

{0, 4/7} = {0, ∞} + {∞, 0} + {0, 1} + {1, 1/2} + {1/2, 4/7}
= (\begin{matrix}1 & −1 \\ 2 & −1 \end{matrix}) \{0, ∞\} + (\begin{matrix}4 & 1 \\ 7 & 2 \end{matrix}) \{0, ∞\}
= 2 · [(\begin{matrix}1 & 4 \\ 1 & 5 \end{matrix}) \{0, ∞\}]