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Euler
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Offline3ReputationRep: Follow
 1
 13102004 11:10
Last edited by LadySmythe; 20112010 at 13:32. 
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 2
 13102004 11:42
Surely (1) is not correct. I.e. the equation x^24x+4=0 only has 1 repeated root.
(2) Follows from the definition of the exponent.
a^n = a*a*a*a...*a
so a^x*a^y = a*a*a...(x times)*a*a*a...(y times) = a^(x+y) 
Euler
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 13102004 12:30
(1) We count the repeated roots as two...
for your information it is correct as its called "the fundamental theorem of algebra"
(2) your argument does not prove e^x.e^y = e^(x+y) as u dont know what e is...is it it an arbitrary number? ur argument merely shows the result but doesnt prove it using first principles... 
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 4
 13102004 12:37
Righto, I and III are equivalent then aren't they. As they are both formulations of the FTA.
i.e. if the polynomial has n roots then you can always express it as (xa1)(xa2)...(xan), and as the complex roots come in complex conjugate pairs you can multiply those together to get a quadratic with real coefficients.
But for II, I can't see your point.
e can be any number you want and e^x*e^y = e^(x+y), if you take it to be THE e then it is just as valid as if you take e to be 1. 
Euler
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 13102004 12:46
First you are certainly right that (3) is formulated from (1)....and the proof of (3) depends on the proof of (1), so if u can prove (1) thenas u said that multipying conjugate pairs of complex roots will give a quadratic with real coefficients thus it can be reduced in linear and quadratic factors with real coefficients (however u might have to prove that complex roots occur in conjugate pairs)
For (2) i mean ur argument SHOWS the desired result but cannot be taken as a valid PROOF...
surely a^n = a.a.a....a (n times)
but u cannot say that exp(x) is the same as e^x we know that it is as exp(2) does equal [exp(1)]^2 but we need a definition of exp(x) and then we can start the proof...so a hint is find how exp(x) is defined and then embark on a proof... 
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 6
 13102004 13:08
I am not sure about whether it is neccesary to prove what you are asking. We can define e^x as an infinite series, but e is still a real number and as such e^x has the same properties as any other exponential function.

Euler
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 13102004 14:52
(Original post by AntiMagicMan)
I am not sure about whether it is neccesary to prove what you are asking. We can define e^x as an infinite series, but e is still a real number and as such e^x has the same properties as any other exponential function. 
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 8
 13102004 15:23
(Original post by IntegralNeo)
infinite series is a good definition, use thsi to prove (2)
the details are gone from my brain but basically you assume f(z)=0 has no solutions and define g=1/f so g is defined and continuous since f(z) does not equal 0 for any z. using some complex analysis it is shown g is bounded
and using liouville's theorem g is constant.hence only polynomials with no roots are the constant roots.
let a be a root of f(z)=0
then f(z)=(za)h(z) for some poly h of degree <f
then apply above to h to show h is constant or has root
so if b is a root
f(z)=(za)(zb)j(z) etc 
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 9
 15102004 13:42
take the infinite series for exp(x), defined as 1+x+(x^2)/2!+(x^3)/3!...., multiply that by exp(y), collect terms, it comes out to exp(x+y)....
(meeheehee ) 
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 10
 25102004 00:59
Count me in too

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 11
 26102004 01:16
(Original post by IntegralNeo)
================================ ==========================
Problems: (green is solved, red is unsolved)
(1) Prove that every polynomial of degree n with real coefficients has n roots in complex field.
(2) Prove that exp(x).exp(y) = exp(x+y) , where exp(x) is exponential function
(3) Prove that every polynomial of degree n with real coefficients can be reduced to the product of linear and/or quadratic factors.
1st i'll get back to you.
2nd. we define exp (x) = 1 + x + x^2/2! + x^3/3! + ....
so exp (x) * exp (y) =
(1 + x + x^2/2! + x^3/3! + ....)(1 + y + y^2/2! + x^3/3! + ....) =
1 + x + x^2/2! + x^3/3! + .....
+ y + xy + yx^2/2! + yx^3/3! + ....
+ y^2/2! + xy^2/2! +x^2 y^2/2!2! + ....
+ y^3/3! + .....
etc. looking at the terms diagonally, you have:
1 + (x+y) + 1/2! (x+y)^2 + 1/3! (x+y)^3 + ...
= exp (x+y)
QED.
3. I'll type up the proof tomorrow... it's about 1am atm! 
ConfidentGenius
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 25112004 11:12
what is this all about?
i want to join... 
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 27112004 23:40
(3) Prove/Disprove that square of an integer a is either 0 or 1 mod 4
Quadratic residues?
a mod 4  a^2 mod 4
0  0
1  1
2  4 = 0 mod 4
3  9 = 1 mod 4 
Hags
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Offline0ReputationRep: Follow
 14
 28112004 01:32
(3) Prove/Disprove that square of an integer a is either 0 or 1 mod 4
It seems joe1212 has beaten me to it and I don't even know what quadratic residues are. I don't know much about modular arithmetic so I hope I'm not missing something but here's an attempt:
suppose a² = 2mod4
a² = 4b + 2 (where b is an integer)
a² is even => a is even => a² is divisible by 4
(a²  2)/4 = b
since a² is divisible by 4 (a²2) is not => b is not an integer, which is a contradiction, so a² is never 2mod4
similarly, suppose a² = 3mod4
a² = 4c + 3 (where c is an integer)
a² is odd, so a is odd. Let a = (E+1) where E is even
(E+1)² = 4c + 3
E² + 2E + 1 = 4c +3
(E² + 2E  2)/4 = c
since E² and 2E are both divisible by 4, (E² + 2E  2) is not => c is not an integer => contradiction => a² is never 3mod4.
a² is either 0 or 1 mod4.
Also, please could I join? 
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 15
 28112004 02:17
Proving that a^2 is 1mod4 or 0mod4 is pretty simple.
Any integer a can be generated by one of these equations:
a=4k
a=4k+1
a=4k+2
a=4k+3
This means that we can write a modular congruence as,
a=0mod4,
a=1mod4,
a=2mod4,
a=3mod4
Squaring each of the above equations, we get (respectively):
a^2=0mod4
a^2=1mod4
a^2=4mod4=0mod4
a^2=9mod4=1mod4
So all a^2 is either 0mod4 or 1mod4.
And please add me to the list 
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 16
 18122004 13:29
Tour of perfect squares starting from 4 involves adding the next odd number, so add 5 to get the next perfect square, then 7 to get the next, then 9, and so on.
Since we are interested in proving that the next perfect square is NEVER 3q1, convert all of the above into an expression relating the number to multiples of 3, and then discard the multiples of 3. 4 becomes +1 (multiple of 3 +1), 5 becomes –1, 7 becomes + 1, 9 becomes 0, 11 becomes –1, and so on.
The sequence of odd numbers will then become (starting with 5) –1, +1, 0, 1, +1, 0, 1, +1, 0, and so on. Provable if req.
This sequence is added, in turn, to the starting perfect square 4 (which is converted into +1) You can also start with 1(which is also +1)
Adding each converted odd number in turn gives 0, +1, +1, 0, +1, +1, 0, +1, +1….i.e. always 3q or 3q+1. Interesting.
Please add my name to your list.
Aitch 
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 17
 18122004 20:45
(Original post by Aitch)
The sequence of odd numbers will then become (starting with 5) –1, +1, 0, 1, +1, 0, 1, +1, 0, and so on. Provable if req. 
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 18
 18122004 22:39
I think it looks OK 
Dividing odd numbers from 5+ by 3 leaves remainders of
1, +1, 0, 1, +1, 0... etc.
Does this need a formal proof? I suppose it might...
Aitch 
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 19
 19122004 05:49
(Original post by Aitch)
I think it looks OK 
Dividing odd numbers from 5+ by 3 leaves remainders of
1, +1, 0, 1, +1, 0... etc.
Does this need a formal proof? I suppose it might...
Aitch
So this is exactly what you have to prove. 
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 20
 19122004 13:09
Sequence 5,7,9,11,13,15,17,...
If we divide the list into blocks of 3 terms, then since for any term T(n) we know that T(n) = T (n3) + 6, we can reduce the list to 3 general terms:
5+6b
7+6b
9+6b
where b is some + integer.
So, since adding 6b to any of these does not change the relationship of the expression to multiples of 3 (as 6b is a multiple of 3), it should be sufficient to prove that
5=(multiple of 3)1
7=(multiple of 3)+1
9=(multiple of 3)+0
and since all following blocks of 3 terms are of the form
5+6b
7+6b
9+6b
this sequence will be repeated ad infinitum.
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