Rational or Not? This Basic Math Question Took Decades To Answer. (quantamagazine.org) 40
Three mathematicians have developed a breakthrough method for proving whether numbers can be written as fractions, solving a problem that has puzzled researchers for decades. Frank Calegari, Vesselin Dimitrov and Yunqing Tang proved the irrationality of an infinite collection of numbers related to the Riemann zeta function, building on Roger Apery's landmark 1978 proof about a single such number.
The new approach, which relies on 19th-century mathematical techniques, has already helped settle a 50-year-old conjecture about modular forms and could lead to more advances in number theory.
The new approach, which relies on 19th-century mathematical techniques, has already helped settle a 50-year-old conjecture about modular forms and could lead to more advances in number theory.
whether numbers can be written as fractions, (Score:1)
Some numbers can be written as fractions, but its going to take forever to do it.
Re:whether numbers can be written as fractions, (Score:4, Insightful)
Some numbers can be written as fractions, but its going to take forever to do it.
Well then they can't be written as fractions.
Re: whether numbers can be written as fractions, (Score:1)
They can, by G-d.
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God created the natural numbers; all else is the work of man. - Leopold Kronecker
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Some numbers can be written as fractions, but its going to take forever to do it.
I’m sorry, but that’s an irrational fear.
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not to fear, we have irrational numbers and if that set won't do, we can always fall back on the imaginary ones
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not to fear, we have irrational numbers and if that set won't do, we can always fall back on the imaginary ones
Cool, I’ve been doing imaginary trust falls my whole life.
This is one of those articles... (Score:2)
... when a lot of people (including myself) realise that their IQ just isn't up to the job. Wish I could understand maths at this level but I hit my mental limit at differentiation. Anything beyond that - forget it.
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not sure this is about IQ, just lacking knowledge on the subject matter
This. It's hard to do math if your IQ is low. But just because you can't do math doesn't mean your IQ is low.
And let's not get hung up about IQ. Remember an IQ test really just measures how good you are at doing IQ tests.
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I do understand what the article is about, and have read books about Riemann, Euler, and the Zeta function, but this article is like gee-whiz pop-science reporting in the sense that you're not really going to learn something of deep substance.
This is from the same magazine that published "Why Computer Scientists Consult Oracles" which puffed up a pretty boring, basic concept by giving it a lofty, sentimental, reverent treatment.
Here's another gem from another article: "Rithya Kunnawalkam Elayavalli’s
Re: This is one of those articles... (Score:3)
maybe you are irrational, so your IQ cannot be expressed as a rational number.
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Has not much to do with IQ.
It is either a gift, or more likely:
education at the right time in your life.
Or lets say: it is a very very special math related IQ.
Most people good at such things have a mind that can construct 3D or even 4D or 5D images of a math problem, and simply see the solution.
Irrational (Score:2)
Odd that the summary doesn't anywhere use the word "irrational," but I guess they assume the readers don't know much about math.
...readers who don't know what the word "irrational" means in mathematics will have zero interest in this, though.
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The word "irrationality" is in TFS and TFA. Does that count? [*]
[* see what I did there?]
Dumbed-Down Cutting-Edge Mathematics Quarterly (Score:2, Insightful)
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I see you also like to live on the edge. Watch this.. (looks around for math police)
e/0
Yeah, that's right, I went there.
Re:Dumbed-Down Cutting-Edge Mathematics Quarterly (Score:5, Informative)
A rational number is a number that can be expressed as a fraction involving two whole numbers. They just didn't want to get that deep into nitpicky detail.
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pi = 3 + 1/10 + 4/100 + ...
Is there some point where adding two fractions together makes an irrational number, or where the above sequence would include an irrational number?
Re:Dumbed-Down Cutting-Edge Mathematics Quarterly (Score:4, Informative)
pi = 3 + 1/10 + 4/100 + ...
Is there some point where adding two fractions together makes an irrational number, or where the above sequence would include an irrational number?
No. Fractions are rational numbers. You add two rational numbers together, and you get another rational number.
Irrational numbers cannot be expressed as fractions of integers. They must be expressed by other means, such as a sequence of rational numbers that converge to the irrational number.
To put it another way, every partial sum in the series you described above is rational, but it converges to an irrational number, namely pi.
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But surely we can do a proof by induction that at each step of the way the sequence pi = 3 + 1/10 + 4/100 + ... is a rational number, and the next step will be rational too. And mathematical induction is supposed to prove from finite n all the way to infinity, so it must be rational to an arbitrarily large/infinite number of digits? I don't understand where the magic is supposed to happen -- if we have to stop at some point then there would be a "biggest rational number" which is nonsense, if we don't stop
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I understand your dilemma. It's one that every math major faces and resolves at some point, myself included.
A proof by induction would only show that every partial sum is rational. And they are. But that doesn't mean the series converges to a rational number. It doesn't.
I think the fact that a sequence of rational numbers can converge to an irrational number is what's hanging you up. The sequence doesn't "suddenly" become irrational at some point. The sequence is always rational, but it gets closer and clos
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What I'm hung up on is, if this sequence is arbitrarily close to pi, what's the difference? Can you give me a non-zero number epsilon such that (pi - [3 + 1/10 + 4/100 + ...]) >= epsilon ?
Actually, that would be a very interesting number, it should be irrational but its sequence would converge to zero.
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What I'm hung up on is, if this sequence is arbitrarily close to pi, what's the difference? Can you give me a non-zero number epsilon such that (pi - [3 + 1/10 + 4/100 + ...]) >= epsilon ?
The short answer is no. Your implied infinite sum of terms gets arbitrarily close to pi as each term is included in the sum. There is no epsilon you can choose that can't be defeated by just taking enough terms in the sum. That was the point I was making in my previous post.
Mathematicians usually dispose of the pedantry of saying the series approaches pi to arbitrary closeness, and just say the infinite series equals pi. You can't claim it is un-equal to pi by some epsilon, because, as I said above, it's ea
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Right, so the sequence is always rational, there's no point at which it becomes irrational, and it equals pi (or as close to equals as you like).
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Right, so the sequence is always rational, there's no point at which it becomes irrational, and it equals pi (or as close to equals as you like).
As I said, the sequence of partial sums of the infinite series gets arbitrarily close to pi, but mathematicians just say the infinite series equals pi. Maybe think of it like this: arbitrarily close means there's no way to get in between pi and the limit-value of the infinite series. But that essentially means they're equal.
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But why distinguish between rational and irrational, if the difference is zero?
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How about this: you cannot pick a rational number and an irrational number that have a difference of zero. That would mean it is the same number, and it is both rational and irrational. There is a definite distinction between rational and irrational numbers. One can be written as a fraction, and the other cannot.
But a sequence of rational numbers can approach an irrational number with arbitrary closeness. In fact, we can consider this sequence of such numbers to represent the irrational number that is the
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Right, it's just it seems to me the difference really is zero. Eg let's define R = [the rational number that minimizes R - pi], and epsilon = R - pi. It seems to me that epsilon ought to be a non-zero irrational number, but also must be arbitrarily close to zero. Also I have the vague feeling that this definition somehow cheats the rationals, but no one can point to a place to stop despite the rationals are countable.
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It sounds like you think there is a rational number that is "right next to" pi. There is no such number. You can show me a number you think is "right next to" pi and immediately I can show you a closer one. And then if you just say "well I choose that one then" I can just do the same thing, and choose a number that is closer still. So you can't give me a rational number that is "right next to" pi.
All you can do is provide a sequence of rational numbers that get progressively closer to pi, but never equal it
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Fixing the last line:
the rationals are densely ordered. That's just a fancy way of saying that for any two distinct rational numbers x and y with x < y, you can find another rational number z that is between them: x < z < y.
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But there is a number that is arbitrarily close to pi, and that number is pi. As far as I know, pi is the only number that is arbitrarily close to pi. And since I can always find a rational number arbitrarily close to pi, I can't see why that number would be different to pi.
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pi = 3 + 1/10 + 4/100 + ...
Is there some point where adding two fractions together makes an irrational number, or where the above sequence would include an irrational number?
No. But the sum of an infinite number of rational numbers can converge to an irrational one, e.g. https://en.wikipedia.org/wiki/... [wikipedia.org]
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No irrational number can be expressed as a finite series of sums of rational numbers. The formula you gave for pi is in an infinite series.
Legendary (Score:5, Funny)
With little explanation, Apéry presented equation after equation, some involving impossible operations like dividing by zero. When asked where his formulas came from, he claimed, “They grow in my garden.”
This is a legendary response if I've ever heard one.
Just mention irrational in the title. (Score:3)
Also: i/1, there it's a fraction.
Main result under discussion (Score:2)
(this space left intentionally blank) (Score:2)
Nerds.