Euler's formula from the Taylor series

About this app

Proof of Euler's formula in eight steps: the Taylor series of eˣ only needs plus and times and therefore also works for complex numbers. With z = iφ every term turns another 90°, and the arrows form a spiral towards the unit circle. Sorted into terms without and with i, they become the series of cosine and sine, shown on a helix with cos and sin shadows. De Moivre's theorem follows at the end.

Subject: Complex numbers

More from Complex numbers

• Complex addition
• Complex division
• Complex multiplication
• Complex numbers workshop
• n-th roots
• Powers of complex numbers
• Number system ladder: from ℕ to ℂ

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