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A formula can feel like a locked door: e to the i pi, plus

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one, equals zero. Five symbols, one line, and somehow growth knows about circles.

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Start with ordinary exponential growth. Put a point at one,

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then let time increase. The point does not add distance;

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it multiplies, stretching away from the origin smoothly.

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The number e is the natural growth rate: the

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one where the function's height and its rate of

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change keep matching each other at every instant.

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Now make one strange change. Point that growth

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direction sideways, into the imaginary axis. Instead of

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running outward, the motion begins to rotate.

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The rotating point is e to the i t. Its

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horizontal shadow is cosine, and its vertical shadow is sine,

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synchronized as the point sweeps around the unit circle.

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So Euler's formula is not decoration. It says this

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exponential is secretly a coordinate pair: cosine t on

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the real axis, plus i times sine t upward.

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Lift time into a third dimension. The same rotating motion becomes a

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helix, a spiral staircase whose shadow is still the unit circle.

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From the side, time rises. From above, the height disappears. The shadow

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left on the complex plane is exactly e to the i t.

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Now replay only the shadow, starting at one and sweeping

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through the upper half of the circle. By the time

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t reaches pi, the arrow lands at negative one.

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At that moment, the sine part has vanished and

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the cosine part is negative one. The whole rotating

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exponential has become a plain real number.

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So e to the i pi equals negative one. Add one

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to both sides, and the two opposite points cancel into zero.

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That is Euler's identity: not a coincidence, but exponential

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growth with its direction turned sideways into rotation.