A formula for finding any power of a binomial without multiplying at length.
- At the academy Gauss independently discovered Bode's law, the binomial theorem and the arithmetic - geometric mean, as well as the law of quadratic reciprocity and the prime number theorem.
- It appears that this formula may have led Gregory to the binomial theorem and more generally to the first discovery of the Taylor series expansion about forty years before Taylor.
- The discovery of the binomial theorem for integer exponents by al-Karaji was a major factor in the development of numerical analysis based on the decimal system.
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