Thursday, March 16, 2017

Bad Mathematics on Π Day

Every March 14 you can expect a flood of stories about the famous number π = 3.14159... . And you can also expect that most of them are bad, written by people who know very little mathematics and don't bother to get their facts checked by a mathematician.

Here's an example: the writer, Tia Ghose, says that π is "an infinitely long, nonrepeating number". Later she says "Because pi is an infinite number...".

This is not correct. More than that, it illustrates a very, very common misunderstanding: confusing a real number with its representation in some base.

Real numbers do not have "lengths" and are not "infinitely long". π is not an "infinite number". (All real numbers are finite.) Instead, one could speak about the base-10 representation of a number (or, more generally, about the base-k representation of the number for some integer k ≥ 2). This representation is "infinitely long", but so is the representation of every real number! For example the base-10 representation of 1/3 is 0.3333.... and the base-10 representation of 1/2 is 0.5000.... . The thing that makes π different from these examples is not that π's representation is "infinitely long", but rather that it is not ultimately periodic. "Ultimate periodicity" means that, after some point, the representation consists of a single block of digits that repeats forever, so the representation looks like x.uvvvvvv.... for some blocks x, u, v.

That's what the author apparently means by "nonrepeating", but "nonrepeating" is such a vague term that it should be avoided. Some kinds of repetitions are inevitable in the base-10 representation of any number. For example, in the base-10 representation of π we quickly find the repetitions "33" and "88" and "99". The first group of three consecutive identical digits to appear is 111; the first group of four consecutive identical digits to appear is 9999. At the same position we even get 999999 ! Now, not every real number will have a base-10 representation with blocks of consecutive identical digits, but there are beautiful theorems, like Dejean's theorem whose proof was recently completed, that say specifically what kinds of repetitions cannot be avoided.

More bad math follows in the same article. The writer claims that "normal numbers" are those "numbers that have the same frequency of all the digits". This is not correct. A normal number to base 10 satisfies a much stronger property: namely, that every block of digits occurs with the expected frequency: if the block has length t then it must occur with limiting frequency 10-t. An example of a number that "has the same frequency of all the digits", but is not normal, is .01234567890123456789... = 13717421/1111111111. No rational number can be normal, because the number of distinct blocks of length t that appear in a rational number is eventually constant.

What happens when bad mathematics journalism like this gets reposted by the World's Worst Journalist™, Denyse O'Leary? Why, she just quotes it verbatim with no understanding. Despite the fact that nobody knows for sure whether π is normal, and despite the fact that nearly all mathematicians suspect it probably is normal, she insists that it is not! Even when commenters try to set her straight, she still doesn't seem to understand that "normal" is a term of art in mathematics, and does not correspond to the ordinary English meaning.

Bad mathematics all around.

Sunday, March 12, 2017

Evangelical Persecution Complex

This interesting article in the Atlantic, about how evangelical Christians perceive that they are the victims of persecution, is very revealing.

And so is a comment chain on Facebook about it by a guy whose name I will omit. Bracketed comments are inserted by me for clarity. It's like a greatest hits of creationist foolishness.

"They are [indeed discriminated against] and it mostly comes from the left. Which seems to be embracing islam."

"OK if my son who is also a Christian desides he would like to pray or read his bible on his free periods during the day he will be stopped and told that he can't do that in public schools. The same school will turn a blind eye to a Muslim. The poor kid has to listen to and be tested on the theory of evolution which is just a theory because there is no more proof of it than there is creationism. That's pretty big to a Christian. Another thing is Christians are refused certain venues and of course the reasons for that aren't stated because those venues could be sued. If you are a Christian and you want to be a journalist you will have a tough time getting a job at mainstream media outlets other than fox. The list goes on and on. There is a war against the sis white male and that include Christians. Don't act like you don't see it."

"There is not ample proof of evolution. At least macro evolution. There is and will never be proof of one species evolving to another. No such missing link has been found. And yes the sic white male is under attack and will eventually be downtrodden if leftist have anything to do with the laws"

"Whether it is legal or not it happened. I am not mixed up in the slightest bit. You are."

"There is no concrete evidence of it. When scientist discover anything that contradicts the theory of evolution they are laughed at and defunded. They have been standing on that falsehood for years and they refuse to let any evidence prove them wrong. The fact is they have never produced a missing link, they try to prove the earth is millions of years old with radiocarbon dating and calibrate that to striations and when asked how they know how old the striations are they refer back to radiocarbon dating. Circular logic. Most scientist worth their salt will tell you the if the planet was even a million years old there wouldn't be any carbon 14 left."

"Oh no Tim you just showed me a picture of fossils. Sorry that does not prove macro evolution. I do not reject science, in fact I have an engineering degree. Science is very important. The lack of proof of macro evolution is why I have a problem with it. And science has never been wrong about anything, right? I suggest that you look into it farther before you jump on the bandwagon. Here's a question that you won't read in natgeo because they can't answer it without having to throw evolution out the window. There are tons of fossils that are supposed to be millions of years old that still have soft tissue attached to it. By their own guidelines that tissue is not supposed to exist unless of course the animal was alive less than 1000 years ago. So to me I feel sorry for you for being suckered into believing lies. That's fine that you believe them but they don't show proof of any animal in transition from one to another. Fact"

"None of them [the examples given by another commenter] display macro evolution."

"So when they found a t-rex fossil with soft tissue on it. They can't explain why though."

"I did not make that up. Look into it."

"You are a fool. It is useless to argue with someone that has already convinced himself of the lie of evolution."

"Just for the record you have produced no facts of macro evolution."

"That's funny. I don't see an scientific understanding coming from any of you trolls. Your lack of evidence is solid proof of your ignorance"

"I haven't seen any substantial evidence of how old the earth is or real evidence of macro evolution. You guys on here are just trolling. Why is it so important to you that my son learns some thing that is still and will always be labeled as a theory, not scientific fact? Is it because you hate God for some reason?"

"I have studied it and I am not in the least bit humiliated."

"No Ann. I was indoctrinated with the theory of evolution from as long as I can remember. When I was saved I realized that to be a believer I couldn't just pick and choose what parts of the bible to believe. This was a big hurtle I had to get over so I started researching the so called evidence and I have found enough hole in it. I would like for my son to learn about all theories but I don't want him to be miss led into believing the theories are actual facts. They aren't."

"John you are a God less idiot"

"If you all are so smart than why did you start the name calling?"

"Tim, it's hard to find evidence that simply don't exist!"

"John, I wonder if you will say that when you are face to face with Jesus? All knees shall bow."

"I have wasted enough of my time arguing with Godless Sodomites for the night. I will pray for you all."

It's like they exist in a different universe. No fact can penetrate a brain this deluded.