Abstract: An attempt was made to analytically describe some basic
functions which are provided with the libraries of mathematical
functions of programming languages. This study will rely on Mathematica
in calculating all numerical results. This study provides global
methods to calculate some discrete functions, which were previously
calculated by using low-level language techniques to chop digits. Using
these definitions we can simulate digital signals without the need for
IF statements and conditions.
A comparison were made between the formulas provided by this study and
some functions provided by Mathematica such as FractionalPart function
that converges very slowly or does NOT converge at all to correct
results. The formulas derived herein were applied to complex numbers
and some interesting results were obtained, such as the integer part of
a complex number equals the Floor of its real part, and will be without
imaginary part, i.e. the integer part of a complex number is a real
number!
Full article with Mathematica evaluations can be found here:

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Integer part of a complex number is the floor of its real part???
Stop! Think! Does that make sense? You are apparently saying that the
integer part of 1+j0 =integer part of 1+j10000 etc. Your algorithm might
give this but if it does, then I humbly suggest that it is leading you down
the garden path and is not valid. Integer part being the floor of the
magnitude of the complex number- no problem. Integer part of each of the
real and imaginary components-again no problem.
I note that your results presented Int(x) =110 -j30 for x=110.1-j30.1 which
appears to be different from your claim. (the '100 is a mystery to me) which
implies that the integer function is taking the integer values of both the
real and imaginary components. This is contrary to your claim.
I feel that you have been carried away to some extent. For example,
f(x)=arctan(tan(x)) or f(x)=arcsin(sin(x)) are really redundant as the
solution f(x)=x is,by definition, known. Your algorithm requires x in order
to find the fractional part of x. If you already know x, then it is simple
to find the integer (or the fractional) part. However, If I give you a
value of tan(x) =1 there is no way that one can determine whether x =pi/4,
or some value of pi/4 +/- n*pi/2. That is the reason for the periodicity
observed in the result from Mathematica. tan(x) has the same magnitude for
an infinite number of x values and there is no way to determine which one of
these it is unless you already know x- in which case the problem is trivial.
In all your examples, x is known.
However, it appears that your intent is to get a general algorithm which is
faster than those now used while taking into account periodicity of the
function. That is good but it also appears that you have confined yourself
to rather useless cases. The speed problem is inherent in the artificial
cases that you have presented, not in the evaluation of the fractional part.
If your approach has some merit- then you will have to show this merit with
cases that are not quite so artificial. Also, please clarify what some of
your statements mean- using normal notation, not a Mathematica script.
I suggest that you go through all this and ask yourself "WHY?" "AM I SIMPLY
SPINNING MATHEMATICA(L) WHEELS?" , "DOES IT MAKE ENGINEERING SENSE?"

Believe it or not: I don't believe in that to. HOWEVER, the formula I
derived proves that.
Check the website again again. You are wrong. The output results for my
formulas clearly states that Int(110.1+30.1i)= 110+ 0(1E-19)i which
means that the imaginary part has vanished! What you are referring to
the output of Mathematica's function which gives the result you
mentioned.
You are right from the pure mathematical point of view. HOWEVER, when
you use Mathematica or Maple or any software that deals with
mathematics, the inverse functions are returned only with principal
values. They are not returned with all solutions, as in inverse there
is no one-to-one solution.
Believe it or not. When the matter involves derivatives of the
FractionalPart function in mathematica, this imposes a lot of
difficulties on the software. The main reason that the Mathematica (and
other mathematical software) uses chopping algorithms to find
FractionalPart. Truncation errors accumulate so dangerously such that
it even deceives the comparison algorithms and gives it "an impression"
that the result is correct, and convergence occured which is NOT true!
I proved that in the paper:
I have calculated the 2nd dreivative of the fractional part of the
function: y=FP(x^5) at x=12.2..
The result using Mathematica's kernel and Mathematica's FractionalPart
function was: 33582.83 WHILE the correct result is 36316.96. And guess
what? The input was given correct to 400 places of decimals, but the
output WASN'T correct even to one significant digit. FURTHERMORE:
Mathematica took about 70 seconds to evaluate this result, WITHOUT
uttering a word that the algorithm FAILED to converge. One would
believe that the result given by Mathematica is correct!
I returned the experiment using my Fr(x) function, and the result:
36316.96 was correct to 400 places of decimals within 0.11 seconds,
that means 600 times faster than Mathematica FractionalPart function!
This experiment should give you the impression that there is a serious
result that stands behind 600 times improvement in Mathematica's
performance, not to mention the precise results as compared to modest
results before.
The web page contains a link to the Mathematica notebook that generated
exactly the same webpage. You can use it and evaluate results on your
machine.
Mohamed Al-Dabbagh

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I saw that- the result you get is meaningless. That is why I said "THINK"
Mathematica gives the integral part of both the real and imaginary values-
each of which, taken alone, is a real number. The only other meaningful
interpretation is to determine the integral part of the magnitude of the
complex number. It is your formula that is giving unrealistic values in this
case. It may be in the way that mathematica handles the tan and arctan of a
complex number (and your formula is based on "real" arithmetic ).
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Agreed but there is actually no way to differentiate between tan(pi/4) and
tan(pi(1/4 +n)) For all n, the value is 1. How then can the inverse
function determine the difference between 1 and 1? Note that in your
evaluation of INT(x), you have, in addition to the trig functions, used the
value of x. It seems that old fashioned conversion from floating point to
integer arithmetic which is at the machine language level or hard wired, is
a lot simpler than the approach that you have, with rather more complex
computation.
Secondly, consider how a trig function is evaluated. An interpolation table
or more generally a series approximation is used. There are limits to a
table (likely only to the 0 to pi/2 range) and a series approximation loses
accuracy rapidly for large x. Hence, buried in the software- at an
elementary level (often machine language level) any evaluation of a trig
function is reduced to the range 0 to pi/2 with a sign change in order to
get a fast but accurate answer. This requires several "if" decisions and a
mod function based on pi.
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Possibly there is a problem in "what is the fractional part?" as I see it
and as you see it.
Is it the fractional part of y ? evidently not
Is it the change in y due to the fractional part of x? Apparently so.
This is not clear in your paper.
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I calculated the second derivative using an obsolete version of APL*PLUS.
This agrees with your result but the time taken, including screen output was
less than 0.06 seconds for 100 evaluations including loop time -on an
interpreter. Mind you I did not carry the solution beyond 64 bit FP accuracy
and there is no practical reason to do so as 400 decimal places is
meaningless. I also calculated the fractional part of a set of numbers using
your formula and it was correct except for integer x (approximately 60ms for
1000 numbers). The reason is, of course, roundoff and other numerical
errors. Have you tried your formula in Mathematica for integer x? Have you
compared the speed and accuracy for a set of real floating point numbers,
with the Mathematica INT function?
What you have shown is that the Mathematica trig functions are very well
optimised while the fractional part function used to calculate a derivative,
has a heavy overhead. This seems to be a problem with mathematica, not a
general advantage of your routine. Mathematica can give you, quickly, the
analytic form of the second derivative of x^5 and should be able to solve
this case quickly for any x. I am surprised because of the inherent
overhead of trig functions. Certainly less than the 70 seconds quoted.
Possibly Mathematica should review their fractional part routine.-this may
well be due to the need to handle analytic solutions as well as purely
numerical solutions- but you will have to show many more examples to
convince them of the errors of their ways.
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The link doesn't lead to the notebook. It does lead to fortunecity

First of all thank you Don for your review.
I think my formula is working well as the floor of a complex number
should logically has a derivative of zero. Complex number means that
you are considering components of sines and cosines the matter that
necessitates the existance of infinite number of derivatives. Also, you
have to remember that there is no specific agreement on the meaning of
Floor(a + b i). What makes me feel comfortable about my results is that
it is all done using trigonometric quantities,and complex numbers are
well coherent with the use of trigonometry.
Yes... I know that very well.. However, what has drawn my attention are
the dreivative of my formulas, the results were completely correct
within the continuity of these functions. This has lead me to
experiment things numerically. My choice was Mathematica which is the
best in numerical evaluation using arbitrary precision.
Mathematica handbook defines fractional part being the absolute value
of the degits to the right of the decimal point. And this is how
mathematica calculate it. What encourages me is that I used the SAME
Mathematica to evaluate my results.
Mathematica is even faster than you have. If you have Mathematica you
will see how swiftly it goes in these things.The purpose of 400 places
of decimals I used was to measure the speed and accuracy.
I have tried with many other examples. Some of which taken 30 minutes
to evaluate ERRATICALLY! But taken less than a second to evaluate
CORRECTLY using my formulas.
You are right, there is a problem with the server.However, you may give
your email I will send it over. No problem! Use my email in the
website. Thank you for your patience.
Mohamed Al-Dabbagh

The significance of knowing some function to have a zero derivative in
all of its domain means that it was constant before derivation! Now can
we actually consider a complex number to be a constant? IMHO, I cannot
consider complex number to be a FULLY CONSTANT. The reason is obvious,
complex number has an infinite set of equivalents in terms of phase
angle. You can only differentiate between different complex numbers if
and only if you write them in polar form.
I don't really know if I am right in considering Mathematica one of the
best algebraic applications in the market. However, I am 100% sure that
they want to excel in using the most reliable definitions. Mathematica
maintains a website that considered one of the most complete
encyclopedias online. There is something said there about our
discussion:

formatting link

You will see that.... yes! many people interested in these weird
functions.
I didn't say second derivative of x^5 !! Maybe you misunderstood: I
said the FP''(x^5) that means the second derivative OF the fractional
part OF x^5....!!! The study has proved that the second derivative of
fractional part of x^5 is equal to the derivative of x^5 (ie
FP''(x^5)=f''(x^5). This proof is also included in my study using the
definition of fractional part formula (8). The result of this study was
used then in the numerical example.
You are right concerning the method by which the Sgn is calculated.
Just search for the sign bit and voila. But if you read further in the
study, you will see that Sgn function analytic form was a great aid in
deriving unit step function, from which one is capable of representing
ANALYTIC FORM of piecewise functions! Further one is also capable of
finding the derivatives of them or even integrate them (of course we
should notice the discontinuities).
What you forget here is that the experiment was done to 400 significant
digits. What is sought is an output that is at least 390 digits
correct. The output precision cannot exceed input precision. When you
tell arbitrary precision software to do a certain math you need to
specify how much input digits are available. There is a working
precision and precision goal. In my experiment my formulas lead to an
answer correct to 400 figures in practically NO time, while Mathematica
internal formulas lead to a figure that is ERRATIC in 70 seconds
without even complaining of divergence.!!
Mohamed Al-Dabbagh

I feel compelled to jump in, but unfortunately lack appropriate tact..
The above is totally false. Stupid even..
False again... Makes no sense at all... Where are you getting
this silliness??
adieu

Let me rephrase: can you remind me of one constant in mathematics or
physics which is a complex number (with BOTH real part and imaginary
part DO NOT equal zero)? Can you for example add a complex number as a
constant to an indefinite integral?
If you write r(cos theta + sin theta i) then if theta is 0 it may also
be 2Pi and 4Pi , etc. and they will all indicate the same number if you
write it in the form (a+bi).
Mohamed Al-Dabbagh

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The floor of a number is a single number. It's domain spans a single
integer. Rather uninteresting. If you want to represent something by a
series of steps, then you have a discontinuous function. derivative =0
except at the step where it becomes infinite. The only significance is that
if you are looking for the derivative of a function, then f' (xo +x) =f' (x)
where xo is the floor - if the region is limited to xo>Mathematica apparently
Thanks. On the basis of the definitions as used (either one) for the Int and
fractional parts (which are what i always knew) there is a great gap in your
evaluation of the second derivative- much is left out. Eq 17 hints at part
of it.
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The report has not proven what you claim. You have since clarified part of
this but the fractional part of x^5 is by definition
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And we couldn't do this before without calling on transcendential functions?
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Possibly Mathematica's routine needs work or there is a difference in the
way that the problem was presented. I said this before. However, my point
is still valid. In real life, the luxury of 100 to 400 decimal points is not
available and is meaningless.
If you want, you can continue with this as personal mail.

A mathematical constant: 1+1i
Why is it a constant? Because it never changes!!!
Sure, why not: 1+1i + INTEGRAL(xdx)
This is silly. A constant is still a constant even if if you compute
it using trig functions.
e.g. Is ONE a constant? By your logic its not since
1 = sin(pi/2 + 2pi*n)
What's you point to all this nuttiness?

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