# A Curious π Pattern

A curious pattern appears when calculating the value of π to the power of a negative even number:

Reminder: The above formula, π^-n , is calculated as 1/π^n.

When n is zero, the value of π^-n is 1.

As expected, the value gets smaller as the exponent n increases.

As n is increased by 2, each value becomes one order of magnitude smaller. That is, for the following powers, the first significant digit of the calculated value is a 1, followed by a 0:

Of course the above values are truncated and not precise.

For example, for n = 2, the value is more precisely 0.101321…

As n increases from 2 to 14, the third significant digit of the calculated value increases from 1 to 2, then 4, 5, 6, 8, and 9 (3 and 7 are skipped). Thus, when n is greater than 14, the pattern stops appearing.

The next switches in order of magnitude will happen approximately when the exponent n is around 16.008, 18.1, 20.108, 22.12, etc.

This curious pattern happens because the square of π is close to 10, thus the value of its inverse is close to 0.10. The value of every second integer exponent of π becomes about one magnitude bigger (e.g., π^4 is about 97), and their corresponding inverse values become one magnitude smaller.

The square root of 10 is 3.16…, and with this number the calculated values follow precisely the above pattern for all even values of n. That is:

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