Ask a Mathematician

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Re: Ask a Mathematician

Postby Windy » Wed Apr 19, 2017 3:14 am

If I have integers x and x-1, are these two integers always coprime?
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Re: Ask a Mathematician

Postby jbobsully11 » Wed Apr 19, 2017 3:43 am

Windy wrote:If I have integers x and x-1, are these two integers always coprime?

Yes (the end of Daniel Goldman's answer has a succinct proof).
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Re: Ask a Mathematician

Postby Windy » Wed Apr 19, 2017 3:58 am

What's the easiest way to find a coprime of a number that has 100 digits and isn't ridiculously large?

Also, what are the chances that for any number, it's coprime with at least one of the first five Fermat numbers?
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Re: Ask a Mathematician

Postby aviel » Wed Apr 19, 2017 5:19 am

Windy wrote:What's the easiest way to find a coprime of a number that has 100 digits and isn't ridiculously large?

You know the factors of that number. Just grab a few random small primes that aren't factors of that number and multiply them together.
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Re: Ask a Mathematician

Postby Windy » Wed Apr 19, 2017 7:05 am

What's the easiest way to find a in this equation:

a = x⁻¹ mod y
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Re: Ask a Mathematician

Postby AboveGL » Wed Apr 19, 2017 12:54 pm

Not quite related to maths problems and apologies in advance if you've answered this in another thread, but what's your favourite mathematical visual? I mean, like the Mandlebrot set or bifurcation diagram of the Logistic Map (I find the latter especially fascinating and there's a pretty awesome animation showing how it's one dimension of the Mandlebrot set), or any animated visual of maths functions like the circle GIF you posted. That kind of thing.
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Re: Ask a Mathematician

Postby Matthew Notch » Wed Apr 19, 2017 6:55 pm

Not to step on Kim's question but I had another question that I had to make sure hadn't already been posted:

I saw in a comment section for a Numberphile video an assertion that higher maths were more or less unnecessary at some point, since as theories coalesce and sublimate, they will eventually be too complex for a human brain to calculate in a timely fashion (i.e. before somebody dies). So, the assertion continued, at some point any math beyond the basic stuff for everyday life--like adding and subtracting and basic fractions--would be handled entirely by AI, and mathematicians would cease to be useful. They even made a joke that maybe we'd see a lot more mathematician buskers, which seems ridiculous until you consider that the only mathematician I'm familiar with is such because he has a YouTube series. So what do you think? I mean I expect some level of bias in the answer, but at a distance how useful are mathematicians going to be to the world at large in just a few years?
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Re: Ask a Mathematician

Postby Windy » Fri Apr 28, 2017 7:22 am

Accurate depiction of math exams

Image
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Re: Ask a Mathematician

Postby thedogknows » Sat Oct 07, 2017 4:43 pm

Matthew Notch wrote:So what do you think? I mean I expect some level of bias in the answer, but at a distance how useful are mathematicians going to be to the world at large in just a few years?


Now, I'll start this off with the disclaimer that even though I love math, I'm not a mathematician, so if a real mathematician has a view about this, listen to them instead. With that said no one else has answered this yet, so my view is that it won't be for quite some time that AI is able to totally usurp the roles of mathematicians. I think that the comment you read, and most of the public in general, seems to think about math as simply performing calculations with numbers. While it's true that computers are widely employed to perform calculations that no human could feasibly do by hand already, the vast majority of mathematical research today simply uses those calculations to supplement their research and verify results. What this comes down to is that math has a much larger creative element than what most people learned in school, and making a creative program is loads more difficult than making one that crunches numbers.

In other words, I don't think computers will be able to take over humans in mathematics until we manage to make programs that are essentially sentient. I'm not sure how far off that will be from today, but my hunch is it won't be for a while, since most contemporary AI's rely on there being external sources of related information, rather than coming up with new information by scratch. Additionally, although the boundaries of mathematical knowledge have gotten more complicated throughout the ages, there have also been lots of paradigm shifts and new mathematical techniques that crop up and simplify math. There's a reason modern high school students regularly learn math that would have blown the minds of the ancient Greeks, and it's not because Brad from calculus class is smarter than Archimedes. Rather, there's tons of stuff that we take for granted, from algebraic symbols, to decimal notation, to the point-set definition of a topology, that greatly simplify math and allow us to learn more math with less effort. I see no reason that more of these simplifying concepts won't be discovered in the future.

Windy wrote:What's the easiest way to find a in this equation:

a = x⁻¹ mod y


Basically, run Euclid's algorithm and then go through it in reverse. Note that this method only works when x and y are coprime, and indeed you can prove that an integer has a multiplicative inverse mod y if and only if it is coprime with y.
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Re: Ask a Mathematician

Postby DamianaRaven » Sun Oct 08, 2017 12:28 am

Is this post mathematically correct? I never was good with percentages.
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Re: Ask a Mathematician

Postby thedogknows » Sun Oct 08, 2017 1:28 am

It is correct, assuming your measure of tragedy is directly proportional to the number of lives lost.
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Re: Ask a Mathematician

Postby DamianaRaven » Sun Oct 08, 2017 1:41 am

Thank you, yes... and for clarifying that it's only correct if equal value is assigned to every life lost... which society is usually not really wont to do. It's kind of understandable - a pediatric cardiologist objectively provides more benefit to society than a homeless drug addict but in situations of mass tragedy, there really isn't time to calculate the actual "net value" of each person's life. Plus, these people were killed seemingly at random, so EVERY life lost is equally horrifying to me!
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Re: Ask a Mathematician

Postby jbobsully11 » Sun Apr 15, 2018 4:03 pm

In the context of why mathematicians seem to want to prove simple things in a complicated way, this post lists a bunch of counterintuitive results. One of them is that if you flip a coin three times and get heads three times in a row, there isn’t an equal probability of getting heads or tails the next time. Why would that be?
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Re: Ask a Mathematician

Postby A Combustible Lemon » Sun Apr 15, 2018 6:13 pm

There is an equal probability next time. The lower number is just the probability of getting four heads in a row. They're not the same thing. It's just a difference of taking the probability from the beginning, which is 1 case in 16, or the probability for the single test, which is 1 case in 2.
Think about it this way:
HHHT and HHHH are equally likely because the coin is fairly distributed, but HHHH is only 1/16th of the possible results in flipping four coins in a row.
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Re: Ask a Mathematician

Postby thedogknows » Mon Apr 30, 2018 1:55 am

I'd also like to point out that the list of things the person wrote for that question are "obvious" intuitive arguments about math that are false. So the point of the answer was that to many people, it seems as though the 21st coin flip after 20 heads is more likely to come up tails, but in reality it's still a 50-50 chance-- hence the need for complicated proofs about "obvious" ideas. Just wanted to clear that up, since I could see it being confusing to read if you aren't already pretty well versed in the ideas he's talking about.
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