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Tuesday, September 10, 2013

Euler's Identity

Leonard Euler discovered one of the most stunning and beautiful equations in all of mathematics. It is called the Euler Identity. Here is is:



According to mathematician Keith Devlin: "Like a Shakespearean sonnet that captures the very essence of love, or a painting that brings out the beauty of the human form that is far more than just skin deep, Euler's equation reaches down into the very depths of existence"

The first step in understanding how the identity is derived is to study Maclauren and Taylor Series, as described in the following video.


The next step is to develop the Maclauren series for Cosine, as shown in the next video ...


... then the Maclauren series for Sine ...


The next step is to produce the series for e to the x ...


Now we can derive Euler's Identity ...



Additional explanatory videos.

The videos above assume some prior maths knowledge. Here are some videos and links filling in some of the assumed knowledge.

Calculus - is a very large of maths.

Here is a playlist of videos introducing calculus.

The playlist contains a large number of videos, and the derivation of Euler's Identity as described in the videos above only emphasised derivatives so here is a link to some videos on derivatives

Trigonometry

The videos above used sine and cosine extensively, as well as pi. click on this linkfor a playlist on trigonometry which explains these functions.

e the exponential

The base of the Euler's Identity is e. In the last video Sal said that it comes from compound interest. Here are two videos that explain compound interest and how e emerges from it. Video one , Video two .

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