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We've all heard the popular claim that the expansion of $\pi$ contains every natural number; however, as we can see from numerous sources, such as Mathematics Stack Exchange and Wikipedia, it is often stated emphatically by those in the know that this is only a conjecture.

My question is:

When did this claim become popular? Why?

Thoughts:

I have no clue why. Maybe a pop-maths author misstated the fact that it's a conjecture and the misunderstanding stuck.

I know most mathematicians believe it to be true and that the conjecture has been around for decades, but the popular understanding seems to be that it is proven.

Context:

I'm a Mathematics PhD student. I want to understand the origin of this common notion because, due to its traction, some folk I know have a hard time believing me when I say it's only a conjecture.

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    $\begingroup$ Brief comments while passing through: Maybe people don't know the difference between the property under consideration and the property of being irrational (school level math is enough to prove a number is irrational if and only if the number has an eventually periodic decimal expansion) or the property of being a transcendental number. By the way, the property under consideration is not known for any algebraic irrational number, such as $\sqrt 2$ and $\sqrt {35} - 2\sqrt{7}$ and $\sqrt[3]{13} + \sqrt 2 - \frac{7}{15}.$ (continued) $\endgroup$ Jun 7, 2023 at 10:09
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    $\begingroup$ Then there's the fact that almost all real numbers (both in the sense of Lebesgue measure and in the sense of Baire category) have the property under consideration. Possibly the "populist origin" is due to Carl Sagan's 1985 SF novel Contact. See this MSE answer for more about all these issues, in particular the distinction between the property under consideration and the property of being a normal number (these two properties are often conflated for some reason). $\endgroup$ Jun 7, 2023 at 10:09
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    $\begingroup$ Imo it's a mistake to generalize this misconception beyond mathematicians since the general public is innumerate. $\endgroup$
    – DJohnson
    Jun 7, 2023 at 11:12
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    $\begingroup$ That's a fair point, @DJohnson. $\endgroup$
    – Shaun
    Jun 7, 2023 at 11:17
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    $\begingroup$ "We've all heard the popular claim" that digits of π are random as well. Can the "general public" tell whether that's a theorem or a conjecture, or what it means exactly? Does it need someone to misstate a theorem to ignore its technical conditions? Gödel's theorem comes to mind. Maybe "general public" also heard that all combinations occur in random sequences. It sounds right, it's a theorem even (as far as the public remembers). A syllogism forms. $\endgroup$
    – Conifold
    Jun 7, 2023 at 13:10

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People without mathematical training have many mistaken beliefs. (Students studying higher math develop mistaken beliefs too, but on a higher level: see here).

It is natural to think every non-periodic pattern of digits contains all possible finite strings of digits. (Forget that it is trivial to give a counterexample when you know enough math.) This is a plausible reason people think the digits of pi contain all finite patterns, and it is not realistic to think this common misunderstanding can be traced back to a specific source.

On the other hand, the quasi-popular confusion (or even awareness) about $1+2+3+\cdots = -1/12$ in recent years has a definite cause: the ridiculous Numberphile video on that result. It is the only awful Numberphile video, in my opinion.

The math stackexchange page here will give you other settings where a false result has large first counterexamples, which might help folks you know better appreciate that a finite amount of numerical data can’t prove a result about an infinite set (like all finite strings of digits).

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