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.
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., x1/2 = √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:
eiπ = -1
This is a special case (with x = π) of Euler's formula:
eix = 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 = e3π/2= 111.317778...
Source: Euler's identity is famous. I first saw ii in Mathematics: The New Golden Age by Keith Devlin.