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given that pi is transcendental
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given that pi is transcendental

and e is transcendental

can we rigorously prove that their sum and product are transcendental? better yet, can we prove that they HAVE to be transcendental if pi and e are?
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>>7789443
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>>7789443
maybe you could use Euler's formula in some way to disprove it?
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This is still an unsolved problem. However it IS known that π + eπ, πeπ and eπ√n are all transcendental.
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>>7789545
Should be π + e^π, πe^π and e^(π√n)
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>>7789529
i had a go at that but got nowhere
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Can it be proven that the set of transcendental numbers is infinite? If pi and e can be represented by an infinite series, are there transcendental numbers which cannot? Are there numbers which cannot be represented by any process, finite or infinite, or is some form of process representation necessary for a number to exist?
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>>7789545
>eπ√n are all transcendental.
>>7789572
QED
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>>7789572

Yes, the set of transcendental numbers is infinite. In fact, there are more transcendental numbers than there are algebraic numbers (non transcendental).
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so does n(pi) have a finite number of decimals?
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>>7789597
You have to multiply all of pi's decimals by n so I don't think so.
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>>7789599
then all multiples of n(pi) are transcendental

I assume its the same with n(e)

thus e(pi) is also
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Given: a and b are transcendental.

Prove: at least one of a+b, ab is transcendental

Suppose not. Then, a+b and ab are both algebraic. But then (x+a)(x+b)=x^2+(a+b)x+ab, an obvious contradiction.

So pi+e or epi are transcendental.

Obviously the transcendental numbers are infinite. e^a is transcendental for all numbers a that are algebraic so theres an infinite set of transcendental numbers right there.
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>>7789654
are they of a higher cardinal than the reals?
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>>7789687
Of course no, but transcendental numbers are in bijection with them
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>>7789443
>can we prove that they HAVE to be transcendental if pi and e are?
No. Multiplying any two transcendental numbers does not guarantee a transcendental result.

Proof:
[math]\pi \cdot \frac{1}{\pi}[/math]
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>>7789710
what if you make an axiom that excludes their inverse?
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>>7789603
What is n? Integer? Real? What if n=1/pi?
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>>7789654
Why is this an obvious contradiction?
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>>7789654
I don't get it. If a+b and ab are not transcendental, where is the contradiction?
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>>7789712
Why would you do that? 1/pi is just as transcendental as pi, you might as well call it by another letter. Then, you'll have two transcendental numbers which when multiplied give a number which isn't. That's the question isn't it? Proving that their products are/have to be transcendental?

Also, you couldn't prove the sums have to be transcendental. For instance, if you gave me the number pi, I could create a number where every digit, when added to pi, sums to 9. Or 8, or 7 or whatever. I could construct a number so that the addition gives an infinitely repeating, but not transcendental number. You could argue that it would take an infinite amount of steps to construct such a number, but would it also take an infinite amount of calculations to add two transcendental numbers also?

So to answer both questions, no and no.
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>>7789712
Excluding inverses, or any non-transcendental divided by the transcendental in question, is equivalent to assuming what you are trying to prove.
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>>7789772
Exactly. Creating arbitrary axioms for special cases that happen to disprove a theory doesn't prove the theory.
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>>7789727
I >>7789599 was assuming n was an integer. In retrospect I should have mentioned that.
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>>7789572
Every number can be represented as an infinite sum - just take its decimal digits and sum up the appropriate powers of 10.
There are numbers, though, where given an abstract definition, we cannot calculate the actual digits of the number, or any kind of representation.
For example, Chaitin's constant is the probability that a randomly chosen program halts. Because there is no algorithm that can decide if a program halts, there is no algorithm for the digits of this number.
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>>7790223
>Chaitin's constant is the probability that a randomly chosen program halts.
*.*
I love this thread.
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>>7789572
>Are there numbers which cannot be represented by any process, finite or infinite, or is some form of process representation necessary for a number to exist?
yes.

the number of possible processes is only countably infinite, while there are uncountably many transcendental numbers (proven by cantor)
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>>7789443
[math]e^{\pi} - \pi = 20[/math]
This is a well known fact and modern processors still do not get this result right.
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>>7790223
>Chaitin's constant

nice
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>>7790364
Lol, did you get that from xkcd?
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>>7789735
Because then -a and -b (so therefore a and b) are roots of a polynomial, which says they are algebraic.

>>7789687
They are IN the reals. So no. However, the algebraic numbers are countable. That makes the transcendentals uncountable. So they actually have the same cardinality as the reals.
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>>7790364
19.9991 != 20
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>>7789772
>>7789827
oh so when the axiom of choice does it it's hailed as the foundation of set theory, but when i do it i get yelled at for ignoring inconsistencies. totally not contradictory.
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>>7790866
Yes, because saying you've proven something for every case except where it has been proven wrong means you didn't prove the original theorem.

It's like me saying "All numbers are integers if we ignore all the cases with decimals in them."
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>>7789747
a and b are transcendental
a+b and ab are algebraic
but the sum of x^2 + (a+b)x + ab must be transcendental because of (x+a)(x+b)
but a sum of algebraic numbers is algebraic
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>>7790364
There's a better one that works out to like 0.999979... or something like that but it's a really simple equation. Supposedly, it was an inside joke at Intel. They would tell the new guy, "hey run these tests on the CPU" and he would poop his pants thinking that Intel's floating-point division logic was broken.
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