Blaise Pascal · Mathematics

The Arithmetical Triangle and Proof by Induction

Pascal&rsquo;s <em>Treatise on the Arithmetical Triangle</em>: how one simple table of numbers unifies the binomial coefficients, combinations, and figurate numbers - and how, to prove one of its properties for infinitely many rows at once, Pascal gave one of the first fully explicit proofs by <em>mathematical induction</em>.

What you'll be able to recall

You learned that Pascal&rsquo;s arithmetical triangle collects the binomial coefficients (the numbers of combinations) in one table, and that in proving its &lsquo;Twelfth Consequence&rsquo; he laid out proof by induction: a base case plus a step that carries the property from each row to the next. Explain the triangl…

Leads to Gottfried Wilhelm Leibniz.

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