# Math Miscellany

## M-Heart-8

Source: I first saw the M-Heart-8 puzzle in print, perhaps in the 1980s
or 1990s, but I don't remember where. It was popularized in *The Simpsons* on
TV (1998, episode title "Lisa the Simpson") and in *The Oxford
Murders* book (2003) and movie (2008). If you know a source earlier
than 1998, please let me know.

## 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.

## 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:

i^{i} = e^{-π/2} = 0.207879576...

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

i^{i} = e^{(-π/2 + 2πn)}

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

i^{i} = e^{3π/2}= 111.317778...

Source: Euler's identity is famous. I first saw i^{i} in *Mathematics:
The New Golden Age* by Keith Devlin.