investigation of binary numbers

If I could have just one more book gentlemen, it would be something that describes....
a three tiered architecture or way of looking at a math system such that the math simulation is in layer 2, operators and operator are in layer 1, and meta-operators would be in Layer 0?

Mentor
 Quote by HermyTheCrab Mark44, I'll have to read Thomas for technical writing? and another for engineering calculus? Is that right? Thanks. Any particular favorite? I prefer the writers be "long winded".
No, Thomas writes calculus books.

Mentor
 Quote by HermyTheCrab If I could have just one more book gentlemen, it would be something that describes.... a three tiered architecture or way of looking at a math system such that the math simulation is in layer 2, operators and operator are in layer 1, and meta-operators would be in Layer 0?
I'm not aware of any such book.
 Roger that Sir!!! Thomas, Calculus. HUAH! Thanks to the forums input and pointers i'm feeling like the next year or so has a light at the end. I'll google for technical writing as well as on the subjects ordered for study. Train, learn, repeat. (I need a short cut. hint hint) The more you train the less you bleed. And this is a tough arena. Huah! I feel like a kid in a candy/hardware store. Where the big boys have all the killer tools and the tools are free... you just have to know what to ask for... DOH! identifies a serious trainng deficiency on my end. Thanks Again. This is GREAT.

 Quote by mfb I do not see anything new, "outside the box" or whatever here. In a similar way, you can consider 0.10012 = 1/2 + 0/4 + 0/8 + 1/16 = 9/16 where the subscript 2 denotes a binary number and the other parts are decimal.
 Quote by Mark44 What mfb showed has nothing to do with the harmonic series. It is how floating point numbers that aren't integers can be represented. The representation mfb showed is exactly the same as a decimal fraction, except that the numbers to the right of the "binary point" (not decimal point) are coefficients of (negative) powers of 2. In a decimal fraction, the digits are coefficients of negative powers of 10. An example: 3/8 = .37510 = 3 x 10-1 + 7 x 10-2 + 5 x 10-3 3/8 = .0112 = 0 x 2-1 + 1 x 2-2 + 1 x 2-3
 Quote by Mark44 The series that mfb and I wrote are power series, which generally look like this: $$\sum_{n = 0}^{\infty} a_nx^n~=~a_0 + a_1x + a_2x^2 + ... + a_nx^n + ...$$ The examples that mfb and I wrote were finite series, where the base was 1/2 (for binary fractions) or 1/10 (for decimal fractions). The harmonic series, which is one of many kinds of series, looks like this: $$\sum_{n = 1}^{\infty} \frac{1}{n} = 1 + 1/2 + 1/3 + ... + 1/n + ...$$
Mark44. I need to generalize "decimal point" and "binary point" to seperate the context of point and base. Is there a way to extend the point concept in generic symbolic form (using mfb notation and zula's extension) such that the "point" and base indicate a very specific subspace defined by the properties of "location","digit width", "base", and "value" in the form:

0.10012.9910.etc3.etc2

Sir, what symbolic definition would be acceptable to you that I can use as a base proposition in an argument? or reference?

Mentor
 Quote by HermyTheCrab Mark44. I need to generalize "decimal point" and "binary point" to seperate the context of point and base. Is there a way to extend the point concept in generic symbolic form (using mfb notation and zula's extension) such that the "point" and base indicate a very specific subspace defined by the properties of "location","digit width", "base", and "value" in the form: 0.10012.9910.etc3.etc2
I don't understand why you would need to do this. The purpose of the "point" (decimal, binary, whatever) is to separate the integer part of a number from the fractional part. For a given base, all of the digits to the left of the point are coefficients of nonnegative powers of the base. All of the digits to the right of the point are coefficients of negative powers of the base.

Some examples:

152.37510 means 1 X 102 + 5 X 101 + 2 X 100 + 3 X 10-1 + 7 X 10-2 + 5 X 10-3

The terms in blue, above, represent the integer part, or 152. The terms in red, above, represent the fractional part.

The same number in hexadecimal (base-16) is 98.616, or
9 X 161 + 8 X 160 + 6 X 16-1

The same number is binary is 10011000.0112.

To save myself some typing I will omit the terms where the coefficient is zero, and dispense with the colors.
10011000.0112 = 1 X 27 + 1 X 24 + 1 X 23 + 1 X 2-2 + 1 X 2-3

As a sanity check, the numbers above are 128 + 16 + 8 + 1/4 + 1/8 = 152 + 3/8 = 152.375

To get back to your question, what would your example (below) mean?
 0.10012.9910.etc3.etc2
In any representation of a number, the meaning of a particular digit is tied to its position in the string of digits. A number to the left of another number is the coefficient of a higher power of the base, with nonnegative powers of the base to the left of the point, and negative powers to the right of the point.

 Quote by HermyTheCrab Sir, what symbolic definition would be acceptable to you that I can use as a base proposition in an argument? or reference?

 Quote by Mark44 I don't understand why you would need to do this. The purpose of the "point" (decimal, binary, whatever) is to separate the integer part of a number from the fractional part. For a given base, all of the digits to the left of the point are coefficients of nonnegative powers of the base. All of the digits to the right of the point are coefficients of negative powers of the base. {...}To get back to your question, what would your example (below) mean? In any representation of a number, the meaning of a particular digit is tied to its position in the string of digits. A number to the left of another number is the coefficient of a higher power of the base, with nonnegative powers of the base to the left of the point, and negative powers to the right of the point.

The purpose is to as you said "seperate" the integer part from the fraction part, i seek the same seperation of the subspaces in a matrices weights? in the polynomials coefficients? for the purpose of identification and analysis and to graph behavior. Fractions of fractions of fractions that preserve unit of measure? relation?

Mark44, I often get in the "trap" of being asked to explain myself when someone I am asking then questions and I then wind up "verbalizing" to extreme and trying to explain myself. After Dr. Tashi's statement... I learned something important. I'm an incoherant writer. As Halls of Ivy directed... I must very carefully define terms I have invented or combined from different areas of study.

When the concepts that I comprehend are in a grammar that is the intersection of the grammar sets: programming, mathematics, and physics. I have difficulty expressing myself... The wrong word is recalled. so I have taken to using your definitions before I make statements. (my apologies, i just realized i barreled through to the goal and am being rude.)

So, please be forgiving of the grammar and don't whack me for attempting expression.

Given your confirmation or support on the extension of "fixed point" to a "relative point" and the physical properties of limit encoded therein, I would have and now am noting the "Benefits to Myself":
#1) I could then argue about subspaces, space time, and subsests of subsets.
#2) I could then discuss a position formula on this axis.
#3) I could then extend the principle to multiple dimensions using the concept of interleaving.
#4) I could then "parametize" the nDimensions in the same manner as we did above with the base parameter.
#5) I could then identify very specific quanta at very specific locations and times in these matrix subspaces.
- extend operators to these relative quanta (conjunctive, disjunctive, etc).
- extend calculus to these ".
#6) I could then use geometry to aide a proof in relation to #2. maybe win some cash, go back to school forever fantansy. you know... dream.
#7) I could then use calculus and physics to illustrate the rates of change of and in these subspaces during the process of "training".
#8) I could then use Graph Theory (a tree) to graph this relation and show the geometrical relations.
#9) Use the tool so defined to illustrate the difference between counting and measuring.
#10) And finally, figure out a method for compiling a NxM matrix from the composition of these relative... "quanta?".

"Everest? because it's there" - unknown

The above really makes no sense without context. I wish I could go back in time and remove that last.

using item #2 on my wish list, a position formula on this axis, I was to extend the tool to include the mathematician as part of the equation. hence the "relative" comment about points that are relative to each other...and the mathematician/observer.

by using two of these relative points A and B, I "triangulate" the mathematicians position C on a line parallel to the axis we defined...
 "with nonnegative powers of the base to the left of the point, and negative powers to the right of the point."
Such that.... I could note when "my" velocity on this line changed in relation to that of the expression I was studying, using the multiple points to "pin" down values under observation at the same time I'm recording my own actions.

The Three points A,B,C form a plane or "frame" of temporal reference. When the velocity of point C changes to a positive or negative value.. from zero or back to zero.... I've got interesting observations to share. When the distance between A and B change (without and corresponding change in B to C).... interesting observations again.

The result was such that it identified my choice of base in the polynomial had influenced the outcome of the results in a similar manner to my choices in rounding or threshold (floor, 1/2, ceiling) affecting the outcome or results. You know, how varying the pattern of my rounding can obfuscate an answer such that... another mathematician will not get the same answer unless he applies the exact same pattern of rules. Varying the sizes and types of these spaces between points I define added the same ability to confuse when considering a process off alternating the dimesionality, value, and "type" of the spaces between the points by another random factor. The converse.... extracting a series of rounding patterns is possible...

A quick experiment in varying the "type" of coefficient in a matrix (integer, whole, and real, binary, and bipolar) showed additional subspaces I could leverage. It followed that if one point caused a division of subspace, that it should be possible to extend that to n points.

The extension of fixed points with the varying of base and dimension just give me additional partitioning ability with "fine tuning"? I'm at the limit of my grammar.
"Is there not an implicit "point" between EACH and every digit?" holy SMOKES!!! I'm almost there... I ALMOST get it. If you could just push this over the finish line for me...ARGH!
 This would almost be funny, if the image of you as lion, and me as a rabbit didn't scare the heck out of me. I've been shot at, held up, and almost blown up... and you scare me more. I'm so sensitive to the .... implications, that I cannot write coherently. A real landmine has more appeal to myself at this very second. I can clear that... but you? I'd rather face the taliban without the grunts protecting me, with one round in my magazine, than do this again. You sir, are intimidating me to the point I gibber. Caught between the briar patch and briar wolf with nothing left but... exhaustion. My stomach hurts... "sick call!!!!"

Mentor
 Quote by HermyTheCrab The above really makes no sense without context. I wish I could go back in time and remove that last. using item #2 on my wish list, a position formula on this axis, I was to extend the tool to include the mathematician as part of the equation. hence the "relative" comment about points that are relative to each other...and the mathematician/observer. by using two of these relative points A and B, I "triangulate" the mathematicians position C on a line parallel to the axis we defined... Such that.... I could note when "my" velocity on this line changed in relation to that of the expression I was studying, using the multiple points to "pin" down values under observation at the same time I'm recording my own actions.
Even with this additional explanation, this doesn't make much sense to me. You have two points A and B on an axis. What are the values being observed? Why is your velocity changing (which implies that you are accelerating or decelerating).

Points on a line are static. We can have particles that move along a straight line, a curve in the plane or in space. There's a topic in calculus, parametric equations, in which this is studied.
 Quote by HermyTheCrab The Three points A,B,C form a plane or "frame" of temporal reference. When the velocity of point C changes to a positive or negative value.. from zero or back to zero.... I've got interesting observations to share. When the distance between A and B change (without and corresponding change in B to C).... interesting observations again. The result was such that it identified my choice of base in the polynomial had influenced the outcome of the results in a similar manner to my choices in rounding or threshold (floor, 1/2, ceiling) affecting the outcome or results. You know, how varying the pattern of my rounding can obfuscate an answer such that... another mathematician will not get the same answer unless he applies the exact same pattern of rules. Varying the sizes and types of these spaces between points I define added the same ability to confuse when considering a process off alternating the dimesionality, value, and "type" of the spaces between the points by another random factor. The converse.... extracting a series of rounding patterns is possible...
I don't understand at all what you are saying here.
 Quote by HermyTheCrab A quick experiment in varying the "type" of coefficient in a matrix (integer, whole, and real, binary, and bipolar) showed additional subspaces I could leverage.
Bipolar? Your other categories are not mutually exclusive. A whole number is an integer, but not necessarily the other way around. A number (real or integer) can have any number of representations, including binary.

 Quote by HermyTheCrab It followed that if one point caused a division of subspace, that it should be possible to extend that to n points.
The term "subspace" is precisely defined in mathematics, and is studied in Linear Algebra. You are using this term in what seems to be a completely different way.

Extend what?
 Quote by HermyTheCrab The extension of fixed points with the varying of base and dimension just give me additional partitioning ability with "fine tuning"? I'm at the limit of my grammar. "Is there not an implicit "point" between EACH and every digit?"
No. There is an explicit point between the integer part and the fractional part (if any).
 Quote by HermyTheCrab holy SMOKES!!! I'm almost there... I ALMOST get it. If you could just push this over the finish line for me...ARGH!

Hermy <= drinks his first glass of shut the hell up... not his last.

NOT a Lion, WORSE.... you are a drill seargent.
I'm going to like you instead of fear you one day. train... bleed...

You have forced me to identify one of the concepts I was badly describing... to do the research to name the concept I was looking for. It is "radix point"... YES!!! and you made me look for it... It was worth it. So MUCH information.
A decimal point or binary point IS A radix point IS A point.

Point B was THE "radix point" in my poorly worded example.
Point A was the relative point a digit width up or down this axis I want you to define for me. (man up Hermy!, quit whining you baby.) In the space I have no name for.
Point C is the relative point locating the observer on a plane perpendicular to the darn “unnamed” axis in the dang “unnamed” space.

That was fun, challenging...ah! doh!
 clearly define
i heard and failed to understand.

Another mistake I made was in confusing the difference between the symbolic representation of a number and the number. Beginners mistake... Thanks for the pointer. I didn't realize there was a difference.

SIR!!! I'm on it and will find the next definitions and terms required to define the point one digit (base) to the left of the radix point as -A, the point one digit (base) to the right of the radix point as +A. I will find the term to descibe point B and C as relative to each other on the "unamed axis" that point A and -A define with B at the origin. I will try and define the plane perpendicular to B on this axis.

Maybe figure out how to describe to you what it means when B and C are not coplanar and not at a right angle to the darn "unnamed axis"
Please give me just a little more time before you drop the next bomb. I have to find the right terms... clearly define!!! AH! references, etc.

Do you have hint on where to start on the correct term for the relation between the radix point and the observer? I know, go look it up, right?

Mark44.
After last night, I've come to the conclusion that you are by far the most patient man I have ever met. If I had blabbed like this to a drill seargent...I'd be a dead man.

Instead, I feel like the biggest ... joke ... on the planet.
I can't believe I missed or forgot the differece between the digits and the numbers.
Sigh.
 Quote by Mark44 The series that mfb and I wrote are power series, which generally look like this: $$\sum_{n = 0}^{\infty} a_nx^n~=~a_0 + a_1x + a_2x^2 + ... + a_nx^n + ...$$ The examples that mfb and I wrote were finite series, where the base was 1/2 (for binary fractions) or 1/10 (for decimal fractions). The harmonic series, which is one of many kinds of series, looks like this: $$\sum_{n = 1}^{\infty} \frac{1}{n} = 1 + 1/2 + 1/3 + ... + 1/n + ...$$
NOW, I understand what you were saying.
I was ... confused and dazed.
Right there in front of me, i bet you got a good laugh. The digit is the space and the value is the number. I was simply watching a ratio change. I was chasing this "thing" and when I bit it, it hurt.... my own damn tail. I thought is was a STEAK!
You gave me the symbolic representations of the two functions... i'll find the plane where the first function is the x, and the second is the y for the other idea. I'm so sorry I took so long to "get" what you "said".

And... the pointer to engineering calculus...<drool> the p series... OMFG!!! the geometric series... ZOMG!! <drool> You were DEAD on the money. Thomas is the MAN!... I've quite a bit of studying to do now... THANKS