Recent content by Michael Hardy

  1. Michael Hardy

    I If the mean of x is 0, is the mean of x-squared also 0?

    The variance of a real-valued random variable is the mean of the square of something whose mean is ##0##. If the proposed proposition were true, then every variance would be ##0,## and that is nonsense. The example already pointed out, where the two possible values of ##x## are ##\pm1,## each...
  2. Michael Hardy

    I What does it mean when an integral is evaluated over a single limit?

    No, that's backwards. The harmonics would be over submultiples, i.e. divisors, of ##2\pi##, i.e. ##2\pi/2,\, 2\pi/3,\, 2\pi/4,\, 2\pi/5,\,\ldots## They have higher frequencies, hence shorter periods. Thus all of them would have ##2\pi## as a period, but not necessarily as a shortest period.
  3. Michael Hardy

    What is the Greatest Common Divisor of Two Polynomials?

    In MathJax and LaTeX, don't write \text{gcd}; instead write \gcd. Unlike \text{gcd} this will yield proper spacing in expressions like ##8\gcd(a,b)##, whereas with \text{gcd} you'll see ##8\text{gcd}(a,b)## instead, with the conspicuous lack of spacing. And the amount of space to the left and...
  4. Michael Hardy

    I How did the Greeks deal with sqrt(2)?

    You seem to be working under a strangely widespread and persistent confusion about what irrational numbers are. There is nothing in the definition of irrational numbers that says anything at all about decimals or decimal digits. DEFINITION: An irrational number is a number that is not a...
  5. Michael Hardy

    B A question about Greatest common factor (GCF) ?

    This is wrong. Euclid's very efficient algorithm for finding GCDs does not require doing anything at all with prime numbers.
  6. Michael Hardy

    B A question about Greatest common factor (GCF) ?

    You don't need prime factorizations to find GCDs. There is an efficient method not requiring any knowledge of prime numbers: Euclid's algorithm. This is the oldest algorithm still in standard use, dating back to Euclid's writings in the 3rd century BC, and it is very efficient. $$ \begin{align*}...
  7. Michael Hardy

    B A question about Greatest common factor (GCF) ?

    But you don't need to know anything about prime factorizations to find the GCD; you can use Euclid's algorithm, which is very efficient.
  8. Michael Hardy

    How Many Police Teams Are Needed for Optimal Area Division?

    $$\begin{align*} 91 & = 7 \times 13 \\ 39 & = 3\times13 \end{align*}$$ Either prime factorization, as above, or Euclid's algorithm, will reveal that ##91## and ##39## have no factor in common bigger than ##13##. So ##1.3\text{ km}\times1.3\text{ km}## squares will go ##3## times into ##3.9\text{...
  9. Michael Hardy

    I How to evaluate the Greatest Common Divisor?

    If the prime factorization is ##2\times2\times2\times2\times2\times2\times2\times2\times3\times3\times3## then you can discover in a matter of seconds via only paper and pencil or maybe in your head that that's what it is. If it's ##13679\times18269## then you probably cannot. But Euclid's...
  10. Michael Hardy

    I What does it mean when an integral is evaluated over a single limit?

    This is a far more difficult question to answer completely than it may appear. It can be answered incompletely, including the following. First suppose ##\int_0^{2\pi} |f(x)|^2\,dx < \infty.## (The absolute value sign is needed since ##f(x)## need not be a real number; it is a complex number, so...
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