User talk:T.Stokke

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\infty - \infty = 0 * \infty = \frac{0}{1} * \frac{1}{0} = \frac{0}{0} = \Phi

\infty - \infty = 0 * \infty = \frac{0}{\infty^{-1}} = \frac{0}{0} = \Phi

\infty - \infty = \frac{1}{0} - \frac{1}{0} = \frac{1-1}{0} = \frac{0}{0} = \Phi

If a function of x when x is nearing y limits up to either the positive or negative infinity, you accept (in transreal arithmetics) that the function of y actually becomes infinity. eg;

\lim_{x \to y} f(x) = \pm \infty

f(y) = \pm \infty

Because of this we obtain that

e^{-\infty} = 0

From this we can assume the other exponentials of ex

e^{\infty} = e^{0 - (-\infty)} = \frac{e^{0}}{e^{-\infty}} = \frac{1}{0} = \infty

e^{\Phi} = e^{\infty - \infty} = e^{\infty} * e^{-\infty} = \infty * 0 = \frac{0}{\infty^{-1}} = \frac{0}{0} = \Phi

We assume all the axioms of real arithmetics are consistent, and we get in transreal arithmetics;

e^{\infty} = \infty , e^{-\infty} = 0 , e^{\Phi} = \Phi , e^0 = 1 , e^1 = e

This allows exponentiation of x^y : x \ge 0 for any transreal x, this is because as for the moment it does'nt exsist transcomplex numbers.

Since we adopt the form xy = ey * ln(x) and you can't draw a real result for negative x.

Showing that:

e^{\infty} = \infty , e^{-\infty} = 0 , e^{\Phi} = \Phi , e^0 = 1 , e^1 = e

, we also show that:

\ln(\infty) = \infty , \ln(0) = -\infty , \ln(\Phi) = \Phi , \ln(1) = 0 , \ln(e) = 1

One of the surprises as Dr. Anderson called it, was

1^{\pm \infty} = 1^{\Phi} = \Phi

1^{\infty} = e^{\infty * \ln(1)} = e^{\frac{1}{0} * \frac{0}{1}} = e^{\frac{0}{0}} = \Phi

This may be hard for someone to understand, but its simple really.

I belive that any intermediate form leads up to nullity \frac{0}{0}, and it easy to show that 1^\infty is intermediate.

If you use the common x finding formula for;

ax = b

You quickly understand that

x = loga(b)

And if you set

1x = 2,1x = 3,1x = 4

you will find that

x = \infty

therefore by common definition in standard analysis it is an intermediate form.

\sin(\infty) = \cos(\infty) = \Phi

\ln(-\infty) = \pi i + \infty

xy = ey * ln(x)

-\infty^{\infty} = e^{\infty * \ln(-\infty)} = e^{\infty * (\pi i + \infty)} = e^{\infty i + \infty} = e^{\infty i} * e^{\infty} = (i \sin(\infty) + \cos(\infty)) * \infty = \infty \Phi i + \infty \Phi = \Phi

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