Math Miscellany

Interesting Digit Patterns

You may have seen this:

¹/₈₁ = 0.012345679012345679...

But you may not have seen this:

¹/₂₄₃ = 0.004115226337448559...

How does the pattern continue? What is special about the number 243? (Hint: find its factors.) What causes the pattern? Are there analogous numbers in other bases?
Source: I found this in Surely, You're Joking Mr. Feynmann by Richard Feynmann.

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Imaginary Powers

The concept of imaginary powers is very strange, even if you are comfortable with imaginary numbers (ex., √-1), negative powers (ex., x^-1 = 1/x), and fractional powers (ex., x^1/2 = √x).
You may have seen this famous, beautiful, and strange equation that relates the most important transcendental numbers, π (3.14159...) and e (2.71828...), with i = √-1, the imaginary square root of -1:

e^iπ = -1

This is a special case (with x = π) of Euler's formula:

e^ix = cos(x) + i sin(x)

Here's another not-as-famous strange equation involving an imaginary power:

iⁱ = e^-π/2 = 0.207879576...

This is a special case (with n = 0) of this formula:

iⁱ = e^{(-π/2 + 2πn)}

According to this formula, iⁱ has an infinite number of values! For example, another value (with n = 1) is:

iⁱ = e^3π/2= 111.317778...

Source: Euler's identity is famous. I first saw iⁱ in Mathematics: The New Golden Age by Keith Devlin.