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#37
Sep2311, 10:48 PM

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#38
Sep2311, 10:53 PM

P: 113




#39
Sep2311, 10:57 PM

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#40
Sep2311, 11:03 PM

P: 113

does q_i mean "q sub i" if so then, if I have shown that f(q_i)=q_i1.. and since the function is continuous for numbers..then a sequence of rational numbers exists to form an irrational number as the number of terms in the sequence approach infinity?
edit: and this would show that..as i>>infinity...f(q_i)=q_i1...which means the formula is true for the irrational numbers 


#41
Sep2311, 11:10 PM

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#42
Sep2311, 11:17 PM

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edit: is it because an irrational number is a sum of decimals? so like pi=3+.1+.04+.001.. 


#43
Sep2311, 11:25 PM

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#44
Sep2311, 11:27 PM

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#45
Sep2411, 12:28 AM

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And now for extra credit
What happens if you remove the assumption that f(1) = 0? 


#46
Sep2411, 06:28 PM

P: 113

So now can I use induction to prove that the formula is true for all real?



#47
Sep2411, 06:29 PM

P: 113

haha.. does it become an infinite number of functions?...like since a linear function is f(x+y)=f(x)+f(y)...then would removing the assumption give me all linear functions +1? 


#48
Sep2411, 08:38 PM

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#49
Sep2411, 09:17 PM

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I show that f(nx)=nf(x)+(n1), where n is an integer. Then, setting x=1 and using the initial condition that f(x)=0, I show that f(n)=n1. Then, setting x=m/n I show that f(m)=nf(m/n)+n1..since f(m)=m1, this becomes m1=nf(m/n)+n1.. so f(m/n)=m/n1 (m and n are integers). Since the function is continuous, it exits for all points and every irrational number has a sequence of rational numbers that approach it and so f(q_i)=q_i1, where q_i is the sequence of rational numbers approaching an irrational number. Since this formula can be derived for all real numbers i can use induction to show that f((n+1)x)=(n+1)f(x)+n equals the original formula 


#50
Sep2411, 09:22 PM

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#51
Sep2411, 09:32 PM

P: 113

hmm.. Ok I know that since its continuous a point exists everywhere which guarantees the existence of there being a rational number right?..I guess what Im trying to say after that is that since an irrational number is a sequence of rational numbers..then I can take 1 minus those rational numbers. Like pi is (3, .1, .04, .001..) = pi so I can do (31, .11, .41, .0011...) = pi1 by using the formula right?



#52
Sep2411, 09:34 PM

P: 113

maybe if i add like... f(3)+f(.1)+f(.04)+f(.005)....=pi1?



#53
Sep2411, 09:41 PM

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