# how do I get more numbers past the decimal? - Page 3

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## Re: how do I get more numbers past the decimal?

Jake Barnes wrote:
<snip>

And if his machine had stored 50 decimal places, he would have gotten
yet different results.

It doesn't matter what language you are using - you will NEVER get exact
results for a random number in any language.  That's what random is all
about.  You will also never get exact results for some math operations,
like (1/3) * 3 will never equal 1.  It doesn't matter what language you
use - it's a fact of digital life.

What you need to do is determine how much accuracy you really need.

--
==================
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Jerry Stuckle
JDS Computer Training Corp.
jstucklex@attglobal.net
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## Re: how do I get more numbers past the decimal?

Jerry Stuckle wrote:

Course you will if you do your maths in base 3..Or your or complier
computer understands fractions and reducing equations.

## Re: how do I get more numbers past the decimal?

Understood. I guess I'll take that question to a forum focused on
math. I've no idea what level of precision is considered meaningful in
this context, though I imagine there are forums where people can offer
some insights  into that subject.

## Re: how do I get more numbers past the decimal?

Question:  is Chaos Theory the theory of roundoff errors?  In other
words, if you do the math EXACTLY (e.g. using algebra, not reducing
everything to numbers immediately, so at least the errors cannot
cascade.  Hint:  this will be time-consuming.), do all the wierd
things described still happen?

## Re: how do I get more numbers past the decimal?

gordonb.8o3cc@burditt.org (Gordon Burditt) wrote:

You can't do the maths exactly. The equations cannot be solved, which is
what would be required.

E.g., for a universe consisting of two bodies (which orbit one another,
say), you *can* solve the equations, which means you can calculate
exactly where they both will be in a billyun years time.

However, add just *one* extra body, and you cannot solve the equations,
so you can *only* calculate the answer numerically. This means that
since you cannot know *exactly* where the three bodies are to start
with, you are inherently unable to calculate where they will be far in
the future.

For the Solar System, it's possible to calculate that a billyun years
into the future and show that the planets' orbits are broadly speaking
stable. But enough error creeps in so that you don't know where on the
orbit they are.

This is why space vehicles need mid-course corrections and why you can't
say with any certainty whether asteroid xyz123 will hit the Earth in 30
years time.

--
Tim

"That excessive bail ought not to be required, nor excessive fines imposed,
nor cruel and unusual punishments inflicted"  --  Bill of Rights 1689

## Re: how do I get more numbers past the decimal?

Tim Streater wrote:

That's one class of systems, its not the only class that falls under the
new fangled term 'chaotic'

There probably is a correct term for that set: deterministic, computable
but not by simple maths. I.e. only calculable by 'brute force' methods.

## Re: how do I get more numbers past the decimal?

wrote:

Good lord, what is your definition of the phrase "new fangled"? Chaos
theory begins to take shape with Lorenz's work in the early  1960s,
but it finally acquired a name in 1975 when James Yorke published his
paper "Period 3 implies chaos". (His paper showed that any nonlinear
equation that developed a period of 3 would also, given certain
parameters, produce chaotic results with no period at all.) And I'm
reading James Gleick's book, which was on the New York Times best-
seller list for a long time in the 80s/90s, and which was published in
1987. Not what I would refer to as "new fangled".

## Re: how do I get more numbers past the decimal?

Jake Barnes wrote:

Well old fangled is pythagoras,
Normal is Newtonian calculus ;-)

The reason I say new fangled is because its not actually new, and never
was. Equations with extreme sensitivity were known for many years, as
was the three body problem.  Then this idiot stumbles on it and thinks
he has discovered something new..

Same with the 'Gaia theory' or whatever its called.

The only 'new' bit was that computers made exploring these equations
with numerical analysis possible.

And along came Mandelbrot...

## Re: how do I get more numbers past the decimal?

Tim Streater wrote:

Actually, that's not why spacecraft need mid-course corrections.  Short
term (within a few thousand years) calculations are very accurate.
Spacecraft need corrections because angles and propulsion, while good,
are not perfect.  And even a very minor deviation from the desired
course or a minor difference in calculated vs. actual propulsion effects

Not to mention other, unknown variables such as effects of the solar
wind over long distances.

--
==================
Remove the "x" from my email address
Jerry Stuckle
JDS Computer Training Corp.
jstucklex@attglobal.net
==================

## Re: how do I get more numbers past the decimal?

Yes, that's a better exposition than mine. Related is the fact that we
don't know, and can't measure with sufficient accuracy, the initial
conditions.

--
Tim

"That excessive bail ought not to be required, nor excessive fines imposed,
nor cruel and unusual punishments inflicted"  --  Bill of Rights 1689

## Re: how do I get more numbers past the decimal?

Gordon Burditt wrote:

No.

Yes.

Chaos theory is about the incredibly complex solutions possible from
very simple equations.

Even if they are exact solutions.

The bullet that misses by a hairsbreadth and doesn't kill you, and the
one that gets a slight gust and does..

Its also about the mathematics of edges, because edges are binary things
in an analogue world.

If you like the simplest example of a chaotic system is balancing a
pencil on its point, and working out which way it will fall.

Never mind the maths, you know it depends on the draught that sets it in
motion, or the precision with which you can balance it.

If a set of equations has a widely varying solution depending on small
variations in its terms, you know its a very unstable description.
Plenty of life features things like that. Just not for long..:-)

## Re: how do I get more numbers past the decimal?

On Aug 1, 7:12=A0pm, gordonb.v6...@burditt.org (Gordon Burditt) wrote:

Interesting stuff. So for the multi precision arithmetic, programmers
switch to the 64 bit or 80 bit floating point formats?

## Re: how do I get more numbers past the decimal?

Jake Barnes wrote:

For very high precision, programmers use binary coded decimal (BCD).
It's slower - but can be as accurate as you want.  That's what they use
to calculate pi to multi-billion decimal places, for instance (of
course, they don't use a PC for it! :-) ).

--
==================
Remove the "x" from my email address
Jerry Stuckle
JDS Computer Training Corp.
jstucklex@attglobal.net
==================

## Re: how do I get more numbers past the decimal?

This page makes it seem like there are several variations on BCD:

http://en.wikipedia.org/wiki/Binary-coded_decimal

I'm curious which of these variations is considered high precision?
I'm confused by this bit, which sounds like it lacks precision:

"BCD is very common in electronic systems where a numeric value is to
be displayed, especially in systems consisting solely of digital
logic, and not containing a microprocessor. By utilizing BCD, the
manipulation of numerical data for display can be greatly simplified
by treating each digit as a separate single sub-circuit. This matches
much more closely the physical reality of display hardware=97a designer
might choose to use a series of separate identical 7-segment displays
to build a metering circuit, for example. If the numeric quantity were
stored and manipulated as pure binary, interfacing to such a display
would require complex circuitry. Therefore, in cases where the
calculations are relatively simple working throughout with BCD can
lead to a simpler overall system than converting to binary.

The same argument applies when hardware of this type uses an embedded
microcontroller or other small processor. Often, smaller code results
when representing numbers internally in BCD format, since a conversion
from or to binary representation can be expensive on such limited
processors. For these applications, some small processors feature BCD
arithmetic modes, which assist when writing routines that manipulate
BCD quantities."

## Re: how do I get more numbers past the decimal?

Jake Barnes wrote:

You could say that - some hardware will support bcd natively, some
won't.  For instance, IBM mainframes have special operations just for
BCD arithmetic.  It supports up to 255 digits.  Intel chips also support
bcd natively, but only up to 7 or 8 digits.

However, there are also software implementations which effectively have
no length limit (other than available storage); digits are typically
stored in a string and operations performed on the string.  As you can
imagine, this will be much slower - but accuracy can be very high.

Yes, you'll need a math group.  What precision you will want will depend
a lot on the algorithms you're using.

--
==================
Remove the "x" from my email address
Jerry Stuckle
JDS Computer Training Corp.
jstucklex@attglobal.net
==================

## Re: how do I get more numbers past the decimal?

Jake Barnes wrote:

Dont worry. Jerry was talking out of his arse. BCD is no more and no=20
less accurate on generic numbers than anything else., its virtue is its=20
exact reproduction of decimal numbers like 12.3456.

Fair comment really.
With BCD you generally use one byte per significant digit, so its as=20
precise on the mantissa as the number of bytes allocated.

## Re: how do I get more numbers past the decimal?

The Natural Philosopher wrote:

Which once again shows you're stupidity.  ROFLMAO!

Processors use one nibble per digit (two digits per byte).   But if you
really were an engineer like you claim, you'd know that.  Or, if you
ever coded in assembler, you'd know that.

The idiot has been caught (again)!

--
==================
Remove the "x" from my email address
Jerry Stuckle
JDS Computer Training Corp.
jstucklex@attglobal.net
==================

## Re: how do I get more numbers past the decimal?

Jerry Stuckle wrote:

Like you claiming that only BCD was *completely* precise?

Pots and kettles.

Of course in this case you are correct. Its a nibble.,  However at my=20
age and not having dealt with BCD since about 1986, memory is a little=20
rusty.

## Re: how do I get more numbers past the decimal?

The Natural Philosopher wrote:

Can't read, either, can you?  I never complained it was *completely*
precise.  I claimed it was *more* precise.

And it has nothing to do with your memory.  It has everything to do with
you being caught in another lie.

--
==================
Remove the "x" from my email address
Jerry Stuckle
JDS Computer Training Corp.
jstucklex@attglobal.net
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s/'re/r/