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Math Quirks:...Anybody (else) experienced strange Date coincidences ?

**SiliconWizard**:

--- Quote from: helius on July 14, 2023, 07:10:36 pm ---The number theoreticians just speak of uniform distributions of digits (not "randomness") because it has nothing to do with random probabilities. Every digit of pi is calculable from simple formulas: it has zero entropy.

--- End quote ---

Yes, that's an interesting question actually.

Yes, all decimals of pi are calculable iteratively. So in that regard, they are not "random".

Does the sequence of decimals of pi provide a reasonable pseudo-random generator though?

More generally speaking, if we admit that everything in the universe is a succession of causes and consequences, then "randomness" is just a fiction anyway.

But then comes the trickery of quantum physics and you don't really understand what a cause is anymore. Or randomness. Or just about anything, actually.

Small digression maybe.

Fun exercise: can you find irrational numbers that do not have a uniform distribution of their decimals?

**helius**:

--- Quote from: SiliconWizard on July 14, 2023, 09:11:55 pm ---Fun exercise: can you find irrational numbers that do not have a uniform distribution of their decimals?

--- End quote ---

We could construct a number that didn't use all 10 digits in its expansion.

For example, along the lines of Champerowne's constant, suppose we have a decimal sequence

1.101001000100001000001...

It's easy to see that this is irrational. Whether such numbers arise in other contexts, I do not know.

**Nominal Animal**:

Interestingly, binary (and quaternary, octal, hexadecimal, etc.) digits of Pi can be calculated directly using Bailey–Borwein–Plouffe (discovered in 1995) and Bellard's (discovered in 1997) formulae.

**helius**:

--- Quote from: Nominal Animal on July 15, 2023, 03:33:32 am ---Interestingly, binary (and quaternary, octal, hexadecimal, etc.) digits of Pi can be calculated directly

--- End quote ---

It's always interesting when there is a "closed form" for a function that is usually defined iteratively or recursively.

The Fibonacci function is a favorite of mine, which is always taught in programming textbooks, but never as a closed form.

Yes, it is possible.

**Nominal Animal**:

Yup:

$$F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}, \quad \text{where} \quad \varphi = \frac{1 + \sqrt{5}}{2} \quad \text{and} \quad \psi = \frac{1 - \sqrt{5}}{2}$$

and \$\varphi \approx 1.61803\$ is the golden ratio, and \$\psi = 1 - \varphi = \frac{-1}{\varphi} \approx -0.61803\$ is its conjugate. Funky!

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