Ask a Mathematician

In order to take something akin to a photograph of a planet in another solar system, you'd need to build a network of telescopes with a diameter on the order of the size of Earth's distance from the sun. And in addition to being spread apart, they'd need to be pretty huge in order to capture enough photons from the target. (though of course there could be clever ways around this) Here's a great explanation of the physics involved: http://www.askamathematician.com/2013/04/q-what-kind-of-telescope-woul…

"Ask a Physician" has something to say about this: http://www.askamathematician.com/?p=6657

>Although "Quantum Teleportation" could that theoretically be used to "teleport" a spacecraft FTL? Everything I've ever read about quantum teleportation says no [0..2]. Even if state changes are instantaneous, the transmission of information itself is still limited to the speed of light (and a ship counts as "information" in this case), and the process of quantum teleportation would require some classical communication channel to transmit state. All that notwithstanding the amount of energy you would need to "entangle" an entire ship in the first place. [0]https://www.quora.com/Is-the-any-possibility-that-Quantum-Teleportatio… [1]https://medium.com/starts-with-a-bang/ask-ethan-can-we-use-quantum-ent… [2]http://www.askamathematician.com/2013/01/q-what-is-quantum-teleportati…

Apparently man has long wanted to know about the universe, God as maker of the universe, etc. Well, now we know much more about the universe, our solar system, earth, and life on earth than just 100 years ago. Here is a little of what we know: At http://www.askamathematician.com/2017/06/q-where-is-the-middle-of-nowh… is Q: Where is the middle of nowhere? Posted on June 13, 2017 by The Physicist with at http://www.askamathematician.com/wp-content/uploads/2017/06/Nearsc.jpg a map, amazing, of our local universe, within about 500 million light years from earth. Amazingly there are large voids. Can see several voids 100+ million light years across on the map. Off the map there are voids maybe 1 billion light years across; considering that the big bang was about 13.7 billion years ago, a void 1 billion light years across is surprisingly large. Where there are stars and galaxies, there are strings, sheets, walls, etc. Weird stuff. But we know much more than in such a map. We must ask: Q 1. What the heck was the propose of creating the universe? Q 2. Then we ask, do we know enough now to start to guess a purpose? Maybe the most amazing thing we know about so far in the universe is human life. Well, even if human life now is a bit messy to serve as a guess of the purpose of the universe, the amazing parts of human life have increased a lot in just the last 100 years. So, we ask, Q 3. Give human life another short time to develop, say, another 100…

http://www.askamathematician.com/2010/12/q-what-does-00-zero-raised-to… It might be a mistake to try to argue that defining 0^0=1 is the best answer or the most consistent answer. The problem is that there are conflicts. There is no such thing as more consistent, if any rule breaks, they all break. Some math rules lead you to 0^0=0, some to 0^0=1, and some to 0^0=undefined. So math doesn't have an answer for this, and what really happens is we decide that 0^0=1 by convention. There are other examples of this happening in math, and it works reasonably well. But let's not pretend it's "right", let's accept that it's a choice. We're choosing to say that the rule x^0=1 takes precedence over the rule 0 * x = 0. Just to play devil's advocate, I like the choice of 0 for intuition consistency, not math consistency. The positive side limit is 0, so if 0^x is defined at x=0, how about we try to make the function more continuous, instead of trying to make it discontinuous? 0^0 is also saying 'start with zero, then multiply it by itself ... Wait, on second thought, don't start' There's nothing, because you don't multiply zero by anything. The best number we have to represent nothing is zero. Saying that not raising zero to a power is one feels like a random number. But, I'm wrong. ;)

If you're interested in this, you should read up on Bell's theorem. [1] [2] The explanation of it I like best is this one (I don't remember where I originally read it): It involves a two player game, the players completely separated from each other. Each is independently shown X or Y. Each player must then give either answer A and B. They win the game if: - when both are shown X, they choose different answers - otherwise, they choose the same answer They can choose a strategy to follow before, but after they're physically separated, each player's decision can only depend on what they're shown (X or Y). Given this restriction, even allowing the players to have (pseudo-)non-deterministic strategies, they'll win with a chance of 75%. However, if the players' strategy involves each having half of an entangled pair of photons, they can make measurements of its spin at one of two predetermined angles (P1X = 0, P2X = 67.5, P1Y = 45, P2Y = 22.5), based on if they see X or Y. Their answer is based on whether they get a spin-up or spin-down measurement. With this strategy, they win about 85% (cos^2(tau/16)) of the time. [1] http://en.wikipedia.org/wiki/Bell%27s_theorem [2] http://www.askamathematician.com/2010/06/q-how-it-is-that-bells-theore…

yeah this is simplified a lot. What it truly means is that some of the energy emitted at the core takes thousands of years to get "out". The energy is converted multiple times, getting released as photons, then absorbed again, etc. https://www.askamathematician.com/2013/08/q-why-does-it-take-thousands…

It’s nothing to do with old light. It’s to do with the relativity of simultaneity. Have a look at the Wikipedia article I linked to. Or read these posts on Ask a Mathematician / Ask a Physicist: - this one explains the basis of the phenomenon: https://www.askamathematician.com/2010/12/q-according-to-relativity-tw… - this one explains the implications for FTL travel (and why it can result in time-travel): https://www.askamathematician.com/2011/09/q-hyperspace-warp-drives-and…

This actually isn't known to be true afaict: https://www.askamathematician.com/2009/11/since-pi-is-infinite-can-i-d…

FWIW, the hidden-variable theory, that the universe is deterministic and we just lack all information to predict it, has been considered and rejected by physicists. https://www.askamathematician.com/2009/12/q-do-physicists-really-belie… https://en.wikipedia.org/wiki/Hidden-variable_theory

A few explanations I've collected: https://www.askamathematician.com/2012/07/q-how-does-instantaneous-com… http://www.physicsmatt.com/blog/2016/8/25/why-ftl-implies-time-travel http://www.theculture.org/rich/sharpblue/archives/000089.html The important bit is that it doesn't matter HOW information is transmitted FTL – it can be via wormholes or magic. For observers in another reference frame it will still result in causality violations.

GR and QM are incompatible. https://www.askamathematician.com/2009/12/q-howwhy-are-quantum-mechani…

This is appropriate: https://www.askamathematician.com/2012/04/q-can-you-poke-something-tha…

https://www.askamathematician.com/2013/01/qhow-do-lenses-that-concentr… https://scienceblogs.com/builtonfacts/2010/05/27/solar-heat-thermodyna…

> As pi is infinite, it also stores any imaginable information. Whether that is true is related to the normalness [1] of the number pi, but this property has not been proven. See also: [2]. [1] https://en.wikipedia.org/wiki/Normal_number [2] http://www.askamathematician.com/2009/11/since-pi-is-infinite-can-i-dr…

It could also mention that fire can be considered plasma[1]. 1. http://www.askamathematician.com/2013/05/q-is-fire-a-plasma-what-is-pl…

A side benefit of the movie was that it prompted me to read Ted Chiang's story again. The first time, I had skipped past the bit about Variational Calculus. Now, I'm intrigued by Fermat's Calculus of Variations. http://www.askamathematician.com/2011/08/q-why-does-light-choose-the-p… I wonder: given that a Photon travels at the speed of light, then everything happens instantaneously, from its point of view. So, is choosing the path that takes the shortest time like a standing wave?

It's Oberth effect, basically? :). http://www.askamathematician.com/2013/01/q-how-does-the-oberth-effect-… By the way, I only know about it because I played KSP a lot ;).

Another concern with CERN's activities: http://www.askamathematician.com/2012/04/q-will-cern-awaken-the-elder-…

Is this actually proven? Pi is irrational, but is it proven to be random (or normal)? http://www.askamathematician.com/2009/11/since-pi-is-infinite-can-i-dr… Also, assuming that it is, if 'start as position X and read Y bits from pi' produced an illegal image (top secret document, abuse images, etc), what would be the legality of trading such information?

The standard answer is that the magnetic field protects us from part of the radiation that comes from the space. I tried to find a good source with more information. I'm still not sure, perhaps it's only a myth. This page says that if there were no magnetic field, the atmosphere would absorb the radiation anyway. (I vote for this, but I'd like to see more info.) http://www.sciencebits.com/NotReallyAProtection These two pages say that the magnetic field provides protection: http://www.windows2universe.org/glossary/plasmaspheric_gain.html http://www.askamathematician.com/2014/07/q-how-does-earths-magnetic-fi…

I posted an article about 0^0 to HN sometime back: http://www.askamathematician.com/2010/12/q-what-does-00-zero-raised-to… HN discussion: https://news.ycombinator.com/item?id=7519827

The Ask A Mathematician page linked is quite a bit better. http://www.askamathematician.com/2012/03/q-is-the-quantum-zeno-effect-…

Has anyone tried the experiment with something obviously larger and not quantum-scale, such as table tennis balls? (Search finds lots of pages using table tennis balls as a help for visualising atoms or electrons or molecules.) Here's a page (with many puns) about what you'd need to try the experiment with cats. (http://www.askamathematician.com/2010/12/q-can-you-do-the-double-slit-…

Light has no mass so it does not experience time. Everything happens instantly for it. While it takes us 10 years to wait for light to travel 10 light-years, the light takes no time at all. This explains it perfectly: http://www.askamathematician.com/2011/07/q-does-light-experience-time/

https://www.askamathematician.com/2018/03/q-with-entangled-particles-c… > Tricks that use entanglement to instantly send information generally involve collapsing the state of the distant particle (by measuring your own), and assuming that the person in charge of that particle will somehow notice. Unfortunately (and this is the answer), there’s no “new-particle-smell” for particles that are still in superpositions and there’s no big flash when that superposition collapses. If you measure an electron’s spin the result is simply “spin up” or “spin down”. That’s all you get. But that doesn’t tell you what the state was before, or if the original state was a superposition of several states, or even if the particle was entangled with something else. > If you only have access to one particle, then the two situations, 1) the other particle is inaccessible or 2) it has been quietly measured, are indistinguishable in every physical sense. Regardless of whatever else is involved, you can never notice the weirdness of entangled pairs until you directly compare them with each other.

> There is absolute angular speed. Not really. If you have two bodies in hydrostatic equilibrium arranged so that you can extend the rotational axis of one such that it overlaps completely with the rotational axis of the other, how do you come up with "absolute rotation"? You could consider the case where one is exactly spherical and the other is highly oblate. While you're free to choose a system of coordinates which keeps the highly oblate one non-rotating with respect to the coordinates, you have to appeal to fictitious forces to explain the oblateness of the coordinate-stationary body and the exact spherical symmetry of the coordinate-rotating body. Physics typically takes a simpler form if you decide the spherical body should be non-rotating against a set of coordinates that covers both bodies. (On the other hand, if these are large bodies -- planets, for example -- and you can put a lab on the surface of each, you get the Special Relativistic form of physics for practically all possible experiments wholly contained within each lab). However, a more typical case is that neither body is exactly spherically symmetrical, in which case you might want to appeal to the view of the motions of deep sky objects observable on each body. One might be ultra-Machian and say that "absolute rotation is determined by the movement of fixed stars", but really you probably should be more interested about whether you need to resort to pseudotensors in your write-down of…

If I understand right, this starts to touch on Bell's Theorem. The description as is means that the situation is identical if there's "spooky action at a distance" or if there's none and you just don't know what's in each box until you open one (and therefore can guarantee what's in the other). There are predictions from this however that when tested just don't work. There is something else going on. I would go further but I'm sure my explanations would contain huge inaccuracies that probably would cause more problems with understanding what's happening so will link you to a couple of things: https://en.wikipedia.org/wiki/Bell%27s_theorem http://www.askamathematician.com/2010/06/q-how-it-is-that-bells-theore… I'm sure there are others here who are well versed in this who can chip in.

I don't know about (1), but (2) could be achieved by scanning through the first n digits and building a lookup tree. If we restrict ourselves to Pi then (3) could be achieved by either just keeping all the digits in memory or using the Bailey–Borwein–Plouffe formula [1], which allows the nth digit to be determined without needing to calculate the preceeding digits. Your're essentially describing a compression mechanism. As you might expect, this isn't a new idea [2] and it generally fails because (as you said) the index quickly gets large as the size of the block of data increases - basically destroying any advantage for non-trivial block sizes. There is some discussion of the space trade-offs in [3], and I wouldn't be surprised if the speed of this approach is poor when compared to conventional techniques. Another problem is that it is still not known whether number like Pi actually contains every possible finite sequence of digits [3], so not all input blocks can be compressed. In practice, this shouldn't be a problem for small blocks though. [1] http://en.wikipedia.org/wiki/Bailey-Borwein-Plouffe_formula [2] https://www.google.co.uk/search?q=pi+compression [3] http://penduin.blogspot.co.uk/2006/10/pi-compression.html [4] http://www.askamathematician.com/2009/11/since-pi-is-infinite-can-i-dr…

Cache: http://webcache.googleusercontent.com/search?q=cache%3Ahttp%3A%2F%2Fww… Unfortunately the author seems uninformed. He claims that 0^0 = 1, not indeterminate which is what Wolfram Alpha claims (and as anyone knows, if you use indeterminates you can prove any number equals any other number, which the author does in uncached screenshots). The problem though is that 0^0 changes its meaning depending on the context. If you have an function f(x,y) = x^y, it's not continuous at (0,0) because the limit of x -> 0 is 0 and the limit y -> 0 is 1, making the value undefined! (Which is still different from 1, 0, or indeterminate.) I used to know of a better blog post on the issue, but here's http://mathforum.org/dr.math/faq/faq.0.to.0.power.html Oh wait, I found it: http://www.askamathematician.com/?p=4524 "Zero raised to the zero power is one. Why? Because mathematicians said so." Sort of in the same character of dispute is the fact that 0 and 1 aren't probabilities: http://lesswrong.com/lw/mp/0_and_1_are_not_probabilities/ I'm also unsure what the author is entering since the screenshot isn't cached. If I try http://www.wolframalpha.com/input/?i=summation+x^n%2Fn!+from+n%3D0+to+… wolfram gives the correct e^0 = 1. Edit: Ah, he's using http://www.wolframalpha.com/input/?i=summation+0^n%2Fn!+from+n%3D0+to+… (substituting 0 in explicitly instead of instructing Wolfram to substitute on x) which is quite…