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Motivated by the common unproven claim in memes that every finite sequence of digits appears in the digits of $\pi$ (see Does $\pi$ contain all possible number combinations?), I started wondering what other similar common claims are out there?

Criteria

  1. A claim about math that's "commonly" stated as true by math enthusiasts who are non-mathematicians. For instance, something that one might find posted somewhere as a meme.
  2. The claim must be either false or unproven.
Avi Steiner
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    What's "commonly" in this context? I'm not nitpicking, but it's not like the general public is interested enough in mathematics for there to be an abundance of such claims – Yuriy S Feb 03 '21 at 20:58
  • You mean like [freshman's dream](https://en.wikipedia.org/wiki/Freshman%27s_dream)? – J. W. Tanner Feb 03 '21 at 21:08
  • @YuriyS I've edited the question to clarify. – Avi Steiner Feb 03 '21 at 21:10
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    I once knew a guy who thought 1 and 6 were less likely on the roll of a die because they were extremes and not closer to the average. He was an inveterate gambler. Statistics/probabilty is probably full of this stuff. – leslie townes Feb 03 '21 at 21:12
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    Second thought: almost everything involving the Fibonacci sequence would fit in this category. – leslie townes Feb 03 '21 at 21:14
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    Probability is definitely a good example – Yuriy S Feb 03 '21 at 21:16
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    In addition to *false* and *unproven*, there's *misinterpreted*, e.g., $1+2+3+\dots=-1/12$ and $1-1+1-1+1\cdots=1/2$. – Gerry Myerson Feb 03 '21 at 22:05
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    @leslietownes: scary. He was probably confusing that with 2 and 12 genuinely being less common in the sum of 2D6 since they are extremes (which is Binomial Theorem). – smci Feb 04 '21 at 08:08

5 Answers5

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Let's assume playing with a fair coin.

Common claim (false) In a long coin-tossing game each player will be on the winning side for about half the time, and the lead will pass not infrequently from one player to the other.

Contrary to this false claim we have with probability $\frac{1}{2}$ that no equalization occurs in the second half of the game regardless of the length of the game. Furthermore, the probabilities near the end point are greatest. This is a consequence of the Arc sine law for last visits.

epi163sqrt
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Most people (probably outside mathematics) think that continuous functions are those that can be drawn without lifting the pencil from the paper.

But this is actually false. For example it is well known that the set of continuous nowhere differentiable functions is prevalent (its complement has, in some sense, of infinite-dimensional-measure equals zero), residual (its complement is of first Baire category), it is spaceable (it contains, except for the null-function, a closed infinite dimensional vector space) and many more properties that suggest to us it is a VERY BIG set.

And what have this to do with being drawable? Well, any of these functions cannot be drawn with a pencil: If you draw some (enough) small line, you do it upwards, downwards or horizontally, so it must have positive, negative or null derivative (on its whole domain, say a small interval).... BUT the functions is NOT derivable.

I used to call this, "The great lie of continuity".

Joseph Sible-Reinstate Monica
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Tito Eliatron
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    Not to mention continuous functions where you must lift the pencil because you're not allowed to draw anything at points that don't belong to the domain of definition. Like $f(x)=1/x$ for $x \neq 0$. – Hans Lundmark Feb 03 '21 at 21:29
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    By the way, you can of course draw non-differentiable functions like $f(x)=|x|$, so it's not true that anything you can draw must be differentiable. But for things like the Weierstrass function you would have to have infinitely shaky hands. And an infinite amount of ink/lead, since the curve length of the graph (over any interval, no matter how short) isn't finite! – Hans Lundmark Feb 03 '21 at 21:32
  • If you draw a little upwards line, you're drawing a function with positive derivative on the whole segment. – Tito Eliatron Feb 03 '21 at 21:57
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Many people don't believe that $$0.\overline{9}=1$$

They will argue $0.999999999...$ cannot equal $1$.

Math777
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I add this as an answer, instead of a comment, for it gains more visibility by people. So please, treat this as a comment.

https://mathoverflow.net/questions/23478/examples-of-common-false-beliefs-in-mathematics

Laplacian
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A lot of things in cryptography are like this. When people say that something like Blum Blum Shub is a provably secure pseudorandom number generator, what is really meant is that cracking the generator is provably equivalent to factoring. Since $P\neq NP$ is unproven, it's often necessary to make that kind of hardness assumptions to "prove" that something is secure. I imagine this confuses a lot of laypeople. (Of course, it's not always necessary. One time pads actually are provably secure, in a completely $P=NP$-proof way.)

Other examples:

"There is no such thing as a formula for primes."

"$|\mathbb{R}|=\aleph_1$"

"$P(X\text{ and } Y) = P(X)P(Y)$" (You need independence for this to be true. Easy to pick up the error by watching someone do this, and not realizing they were using independence.)

"If the random variables $X, Y, Z$ are pairwise independent, then $X, Y, Z$ are independent of each other."

"The Godel sentence for Peano Arithmetic is necessarily true." (There exist very strange models of PA where it's false.)

Ricky Tensor
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