12 Jun 2010 22:00
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Web sites about mathematics should help people understand and appreciate mathematics, not confuse the crap out of them with misinformation. Unfortunately, Wolfram Mathworld does the latter.
Example 1. MathWorld explains here that “The numbers of palindromic numbers less than a given number are illustrated in the plot [below].”
So the left plot tells us that there are about 100 palindromes less than or equal to 20. But there are only 21 nonnegative integers less than or equal to 20, so there can’t be 100 palindromes among them. In fact, there are 11 palindromes less than or equal to 20: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 11. My guess is that the left plot illustrates the n-th palindromic number as a function of n. In any case, it’s not what MathWorld describes.
MathWorld begins its list of the “first few palindromic numbers” with 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (these 10 numbers are palindromes and are all less than 10), but in the next paragraph, MathWorld states that the number of palindromic numbers less than 10 is 9. There are 9 if you don’t count zero for some strange reason, but if you don’t intend to, give a definition that excludes it (MathWorld’s definition is less than clear), and then don’t list it.
Still confused? Read the Wikipedia article.
Example 2. Pascal’s Triangle shouldn’t be hard to screw up, right? Wrong. Here’s MathWorld’s Pascal’s Triangle:
This triangle needs to go to the shop for an alignment. The numbers are neither lined up in columns nor staggered (the latter being the usual presentation). What are the numbers in the column containing the rightmost 4? What numbers are along the diagonal through the top? (1, 1, 1, 1, 1, 5, 6?) As shown, MathWorld’s anyway-ill-worded “each subsequent row is obtained by adding the two entries diagonally above” is meaningless.
Example 3. In its article on Mersenne numbers (numbers that are one less than a power of two), MathWorld attempts to explain why “[i]n order for the Mersenne number [2n-1] to be prime, n must be prime.” MathWorld’s justification: “This is true since for composite n with factors r and s, n = rs. Therefore, 2n-1 can be written as 2rs-1, which is a binomial number and can be factored.” That’s sloppy to say the least. First, if a composite number n has factors r and s, it’s not necessarily the case that n = rs. Furthermore, the fact that a number can be factored doesn’t prove it’s composite. Every Mersenne number 2n-1 can be factored. It’s just that when n is composite, there’s definitely a factorization into positive integers neither of which equals 1. Explaining it isn’t hard: In order for 2n-1 to be prime, n must be prime. For if not, n = rs where r and s are integers greater than 1 and less than n; then 2n-1 = 2rs-1 has a factor between 1 and 2n-1, namely 2r-1.
Example 4. MathWorld describes prime numbers as “numbers that cannot be factored.” Prime numbers, like all integers, however, can be factored, and elsewhere, MathWorld gives the factorization of several prime numbers, such as 7: 7 = 7×1.
Example 5. Any of MathWorld’s articles on statistics.
In the article on the Central Limit Theorem, what is lowercase n? What is f? The “limiting cumulative distribution function” of Xnorm is limiting in the sense of what approaching what? (It’s not clear to me that MathWorld’s statement of the theorem is even correct, but it’s clearly unclear.)
The article “explaining” the p-value has perhaps the worst definition of p-value I’ve ever seen when not grading exams. MathWorld says it’s “[t]he probability that a variate would assume a value greater than or equal to the observed value strictly by chance: P(z > zobserved)” (wrong). Wikipedia says “In statistical hypothesis testing, the p-value is the probability of obtaining a test statistic at least as extreme as the one that was actually observed, assuming that the null hypothesis is true” (right).