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Everything posted by wtf

  1. The log spiral has many interesting properties. For example, "the distances between the turnings of a logarithmic spiral increase in geometric progression." Perhaps this relates to something you're interested in. https://en.wikipedia.org/wiki/Logarithmic_spiral Can you draw a sketch of how 5, 17, and 85 relate to the log spiral? I think this might make your idea more clear.
  2. Why is that? Isn't there a right triangle with sides [math]5, 17[/math], [math]\sqrt{5^2 + 17^2}[/math] ? Aren't there lots of other triangles with those two sides? The angle between those two sides can be absolutely anything strictly between 0 and 180 degrees. Can you try to explain what constraints you are trying to put on the angle, and what they are based on? Remember, coordinate systems are not inherent in geometry. The geometry comes first, and then you impose a coordinate system. The coordinate system can never change the underlying geometry, it can only make the equatio
  3. I would still like to see a simple, complete worked example with lengths 2 and 3. Put the 3-side along the positive x-axis with one end at the origin; and show me exactly how you place the 2-side such that the angle is somehow uniquely determined by ... something.
  4. When they hacked the east coast pipeline I said nothing, because I live on the west coast. And by the time they hacked the US meat supply, there were no cows left to speak up for me! or something like that. https://www.washingtonpost.com/business/2021/06/01/jbs-cyberattack-meat-supply-chain/
  5. This statement is false on its face. That's why readers are put off and have no idea what you're talking about. Could you give a specific example? Say I have a line segment of length 2 and another of length 3, both with one end at the origin of the x-y plane. Can you show how you get the third side? What additional information is supplied? Please be specific. No need for extraneous discussions of the travails of your long-suffering teacher. Just show an example of two lengths and how you determine the third one. Also, what does pnp stand for? Give a specific, completely worked
  6. It's not just that China as a whole controls the vast majority of the mining. It's also that the top four Chinese mining pools control over 60% of the bitcoin hashrate, and have done so ever since I started tracking it several years ago. https://btc.com/stats/pool Basically, the conditions for a so-called 51% attack already exist and have always existed. These mining pools can wake up any morning they like and steal everyone's bitcoin. The only reason it hasn't happened yet is that they haven't done it. Not because they can't. ps -- Just to clarify what a 51% attack is, the idea
  7. A topological space does not necessarily have a metric. The definition of space is far more general.
  8. Yes. The energy cost is the cost of validating blocks. That never changes. Even after the limit of creating bitcoins is reached, blocks must still be validated whenever there is a transaction. A good question is, why would the miners keep mining when they can no longer create new bitcoins? According to Investopedia, they would still collect transaction fees. https://www.investopedia.com/tech/what-happens-bitcoin-after-21-million-mined/
  9. Thanks much. This is the answer to swanstont's question. If (in the extreme case, to illustrate the point) it takes no energy to verify a block, a bad actor can just verify false blocks at will and steal everyone's money.
  10. No that's not possible. All proof-of-work based cryptocurrencies rely on the enormous expenditure of computational energy to verify the transactions. If verifying blocks is cheap, the protocol doesn't work. There are some efforts to mitigate the problem, such as green crypto mining. But one argument for the eventual failure of bitcoin is that it's out of spirit with the age. Bitcoin wastes energy extravagantly; and we live in the age of conservation. Not endorsing this service, just a link I googled randomly to illustrate that people are thinking about the issue. https://www.perpetua
  11. [math](-1)(-1) - 1 = (-1)(-1) + (-1) = (-1)(-1) + (-1)(1)[/math] [math]= (-1)(-1 + 1) = (-1)(0) = 0[/math] So that [math](-1)(-1) - 1 = 0[/math] and therefore [math](-1)(-1) = 1[/math]. Then [math](-3)(-2) = (3)(-1)(2)(-1) = (3)(2)(-1)(-1) = (3)(2)(1) = 6[/math]. All this follows from the associative, distributive, and commutative properties of a ring, which are satisfied by the integers. https://en.wikipedia.org/wiki/Ring_(mathematics) http://www.maths.nuigalway.ie/MA416/section1-2.pdf
  12. I looked back at your comments in this thread and perhaps I missed your question, can you please repeat it? The only question I saw was that you asked what is the additive inverse of a + bi; and it's of course -a =bi. If there was another question I did not see it.
  13. ps -- I don't think my previous explanations were very good. I found a much better page explaining this matter. If you google "why can't you distinguish i from -i" you get hundreds of totally irrelevant hits no matter how you alter or rephrase the question. Took me a while to find this. https://math.stackexchange.com/questions/177594/how-to-tell-i-from-i/177601#177601 The right answer is that there's an automorphism of [math]C[/math] that takes [math]i[/math] to [math]-i[/math]; namely, complex conjugation. In other words the difference between the two amounts to a relabeling with no
  14. -a -ib. Why are you asking such an elementary question whose answer you perfectly well know? If z is a complex number, -z is its reflection through the origin. Perhaps you'll find this helpful. There are no positive or negative complex numbers because it's not possible to put a total order on the complex numbers that is compatible with their addition and multiplication. https://math.stackexchange.com/questions/788164/positive-and-negative-complex-numbers Or https://en.wikipedia.org/wiki/Complex_number#Ordering
  15. I'm afraid I didn't see where you said what it is. There are no positive or negative numbers in the complex numbers, that's why you can't unambiguously use the sqrt sign convention that works in the real numbers.
  16. OP never came back? After crossposting this to two different discussion forums?
  17. Yes. The notation [math]\sqrt{-1}[/math] is often used casually, but it's imprecise in the branch of science, profession, or art of mathematics.
  18. Me being the picky type, let me point out that [math]\sqrt{-1}[/math] is not good notation and is technically not correct. In the case of a nonnegative real number [math]x[/math], we can define [math]\sqrt x[/math] as the positive of the two values whose square is [math]x[/math]. However in the complex numbers there is no concept of positive or negative. That is, we can't algebraically distinguish between [math]i[/math] and [math]-i[/math]. So we define [math]i[/math] as a complex number such that [math]i^2 = -1[/math]. We pick one of the two possible values and call it [math]i[/mat
  19. pps -- I looked at your handle and found this. You most definitely do know how to do math markup. Can you please do us a favor and mark up this linear algebra post?
  20. ps -- I follow everything up to this line. If you can please put in proper parens and show exactly how you got this I'd find it very helpful. Also please note that the [math]y_i[/math]'s are presumably taken to be all distinct from each other, else you can't be sure they contain a linearly independent subset. And also note that when ask us to consider the equation you need to specify that at least one of the [math]c_i[/math]'s is not zero. There's no reason they couldn't all be. When you say, "We cannot have all c_i=0 [individually]in an exclusive manner since that would ma
  21. I saw your similar post on another site where it didn't get any traction. May I suggest a couple of minor notational changes that will improve clarity? Since [math]e \in V \setminus W[/math], I'd call it [math]v[/math]. Likewise I'd call the [math]y_i[/math]'s [math]w_i[/math]. These minor changes would decrease the cognitive burden on the reader; and (if you don't mind my saying) your exposition here and on the other site are already a little convoluted and the reader can use all the help they can get. This notation's hard to figure out. You did convince me on the other site that [mat
  22. I don't believe this is true. sqrt(1) = 1 by definition, assuming by Root(1) you mean [math]\sqrt{1}[/math]. The square root of a positive real number is the positive of the two numbers whose square is the number. So if someone asks, find [math]x[/math] such that [math]x^2 = 1[/math], the answer is {1, -1}. But if someone asks what is [math]\sqrt{1}[/math], the answer is 1. There is no solution to the question in the title. What is true is that [math]- \sqrt{1} = -1[/math].
  23. I happened to run across this terrific talk called The Secret Life of Quarks, well worth watching. https://www.youtube.com/watch?v=H_PmmMkGyx0 Like others, I'd always heard there are 3 quarks inside the proton. Turns out it's not really true. 3 is the number of quarks minus the number of antiquarks. But it's not a matter of counting and subtracting. Rather, you integrate something called the quark density function, and when you do, you get the answer of 3. The actual number of quarks and antiquarks depends on the scale at which you look. So there could be millions, zillions, whatever
  24. Thanks for the info. As a suburbanite I'm just getting over the shock of learning that cheeseburgers are made of chopped up dead cows. I had no idea.
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