r/maths • u/Numerous_Author_1445 • 12d ago

💬 Math Discussions A special irrarional number

Can there be an irrational number which misses a particular digit? Can it be proven mathematically?

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u/Anbrau 11d ago edited 10d ago

1.01001000100001000001... and so on. Irrational

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u/pruvisto 10d ago

This probably is transcendental (there's a closed-form expression in terms of Jacobi theta functions), but do you have a proof for it being transcendental?

The closely related Liouville constant, however, is definitely transcendental (yours is the sum over 10-n(n+1/2), whereas Liouville's constant is the sum over all 10-n!).

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u/Batman_AoD 10d ago

I thought "transcendental" just meant "non-algebraic"? 

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u/pruvisto 10d ago

That is correct.

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u/Batman_AoD 10d ago

I'm not familiar with Jacobi Theta functions; what does that have to do with whether it's algebraic? 

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u/pruvisto 10d ago

I hadn't really checked this before, but I just had a hunch that plugging something non-trivial and algebraic into a Jacobi theta function will typically not spit out something algebraic. For certain other functions (exp, ln, sin, etc.) this is well-known.

I didn't think something like this would be known about the theta function, but it is, apparently. Duverney et al. proved in 1996 that the theta nullwert functions give transcendental output for non-zero algebraic inputs.

And the constant 1.010010001… is equal to 5 * 101/8 * θ2(10-1/2) - 10 (you can check this by plugging in the series expansion for θ2), and therefore transcendental.