Quote from Becky Rose :

Technically the difference is 0.0(0)1.

Conveniently, that's the exact 'mass' I have decided to award all numbers in Becky-math.

^ Yay. Someone who actually gets the point I am trying to make

This is the last I will say on this.. I can almost hear the relief from the crowd. I hope it actually highlights the point I am trying to make because I do actually understand math.

Right... here we go.

1) The world is NOT in base 10
2) Humans currently have decided on base 10 to describe the world.

It is analogue vs digital if you will, as such 1/n where n is base -1 will ALWAYS presents a "problem".

Maths is just a tool we use to describe the around us. It is NOT perfect but full of compromises that are used to improve the usefulness of the said tool.

Let me give some examples.

Infitesimal numbers might as well be 0. Cool, yet pi has infitesimal precision and the more digits employed in the calculation the "more correct" your result. pi is not 3.14 but for many calculations that is all that is required.

It is all about the precision required for the task at hand, no more.

We accept the 0.0(0)1 from Becky maths has 0 "significance" because it allows our base 10 number system to have a “consistency” which allows it to be a better "tool" for our needs. Does not mean it is right, just a compromise.

There is actually NO 1, 2, 3 etc. They are all just markers on a line that represent a "mass" for want of a better word (thanks Becky) that we have attached a value to. Trouble is that line does not actually have markers, it is analogue. We see fit to divide it into factors of 10 between markers we have called 1,2,3... because that allows us to have a frame of reference.

When we changes from imperial to metric we also changed all the sizes of nuts and bolts etc. did 3/8" cease to be a valid size of bolt because it did not fix perfect into mm? Did 1/1000" cease to be a valid tolerance? No it is just those markers were in the "wrong" location for our new system so we discarded them.

"John, pass me that 9.525mm socket will you". Do the math

Accept, embrace and acknowledge the compromises we employ in the name of base 10 maths and it becomes a more powerful tool! Try and hide behind 0.9~ is 1 because 0.0(0)1 is so small we will call it 0 and sooner or later it will bite

If you try and travel millions of light years that infitesimal might become the difference between landing or smashing into a sun. It is all just about the precision required for the task at hand, no more.


As means of a sidetrack..... There is a Savant that quoted pi to 80000dp calculated in realtime. There were teams of people checking off his numbers as he spouted them out at machine gun speed stopping only for drinks etc. He only stopped at the 80000th dp as the point was proved, that took long enough and their books of pi didnt go much further anyway.


Luckily the Savant is HSA so is able to communicate in semi mormal terms unlike many Savants. He “sees” numbers as shapes and patterns, nothing else. I remember him saying 9 appears as a big shape. To test this they presented him with simple maths where the size of each number was opposite to how his brain worked and it caused a “number” blindness that made it very difficult for him to “see” the solutions.


Base 10 numbers are not “real”, real numbers they are a continuous stream of values in infitesimal steps that DO not fit base 10. It is imperial vs metric, analogue vs digital!


Thats me done. Feel free for the “last word” and trash what I have said

Quote from Woz :

It is analogue vs digital if you will, as such 1/n where n is base -1 will ALWAYS presents a "problem".

base... 10... 10-1=9... 1/9 ... 1/9 is a problem? i think it's a fraction. it's a rational number. hardly what i would call a "problem".

Quote from Woz :

Maths is just a tool we use to describe the around us. It is NOT perfect but full of compromises that are used to improve the usefulness of the said tool.

and have you studied... how much, exactly? ODEs? PDEs? vector analysis? multivariate? tensors? have you actually studied enough to know where math is lacking? considering your previous thoughts on the matter of simple arithmetic, you lack even the most basic knowledge on this subject. sorry, you are not one who will be taken seriously when you talk about the shortcomings of math. Even if you have studied almost anything that can be studied, you will gain the knowledge that math is built-in in this world when you study the advances made in quantum mechanics and particle physics the last 100 years.

Quote from Woz :

Infitesimal numbers might as well be 0. Cool, yet pi has infitesimal precision and the more digits employed in the calculation the "more correct" your result. pi is not 3.14 but for many calculations that is all that is required.

infinitesimal numbers are small, but they are not equal to 0. Pi is accurate. it is the ratio of the circumference of a circle (ANY circle) to its diameter. If you start writing out 3.14159... then it is an approximation. but 'Pi' is accurate. much like when you say '1'. it's absolutely accurate.

Quote from Woz :

It is all about the precision required for the task at hand, no more.

Quote from Woz :

We accept the 0.0(0)1 from Becky maths has 0 "significance" because it allows our base 10 number system to have a “consistency” which allows it to be a better "tool" for our needs. Does not mean it is right, just a compromise.

becky-math might have compromises, but 0.333(...) is not a compromise. it is an other way to write 1/3. much like 2/6, -1/-3, etc.

Quote from Woz :

There is actually NO 1, 2, 3 etc. They are all just markers on a line that represent a "mass" for want of a better word (thanks Becky) that we have attached a value to. Trouble is that line does not actually have markers, it is analogue. We see fit to divide it into factors of 10 between markers we have called 1,2,3... because that allows us to have a frame of reference.

'1' is analog? well, '1' has a certain significance, unrelated to any numbering system. actually, representation systems are unrelated to what you can do with numbers. in number theory anyway, '1' symbolizes a number with certain properties. etc.

Quote from Woz :

Try and hide behind 0.9~ is 1 because 0.0(0)1 is so small we will call it 0 and sooner or later it will bite

0.999(...) does not end with a '9'. it doesn't 'end'. there is no ...0001 in the 'end'

Quote from Woz :

If you try and travel millions of light years that infitesimal might become the difference between landing or smashing into a sun. It is all just about the precision required for the task at hand, no more.

someone take the calculator from this guy's hands. we are not talking about engineering here. we are talking about math. in math there is no 'just about the precision'.

Quote from Woz :

Base 10 numbers are not “real”, real numbers they are a continuous stream of values in infitesimal steps that DO not fit base 10. It is imperial vs metric, analogue vs digital!

this is marginaly pseudoscientific. there is no "real" in math. we define "real numbers" to be a set with very strict definition. we don't make up words or characteristics.

talking about the "Reality" of a number is as serious as is talking about the "politeness" of them. 8 is more polite than 4 because it has more curves, it is not angled and angry like 4.

Quote from Woz :

Thats me done. Feel free for the “last word” and trash what I have said

yeah. math loves you too. but you make me sad that instead of learning new things or correcting your mistakes, you spout unscientific nonsense and ignore whatever knowledge comes your way, saying "that's me". there is a word, in greek, for what you do. Εθελοτυφλείς. "being willingly blind"

But at the level of 0.999...=1 we're not really considering base 10 or the actual number system itself, but the whole concept of mathematics itself. Regardless of what base you use, or even if you use bases, the concept itself holds true.

in base 3, 1/3(10) is written as 0.1(3)

heh heh heh

( 1/3(10) = 0.1(3) )

much like 1/2(10) is written as 0.1(2)

at least that's what i think at the moment

which means that

0.333(3)(10) = 0.1(3)

an infinite number of digits in the decimal system ... needs only one digit in an other system.

Quote from george_tsiros :

infinitesimal numbers are small, but they are not equal to 0.

Quote from tristancliffe :

Because lets say you get to a billion, trillion, dodechedronillion 9s. The difference is now tiny. No divide that tiny tiny difference by a billion, trillion, dodechedronillion. The difference is even tinier. Divide the new difference by a billion, trillion, dodechedronillion. And keep divided the new difference by a billion, trillion, dodechedronillion an infinite number of times.
You then end up (if ending up were possible when using infinities) with NO DIFFERENCE WHATSOEVER

Come on then which is right lol

And dont say it depends on context because that is what I have said already.

I got in from a night on the pop about 10 mins ago, and i've just finished reading this thread from start to finish.
I'm now going to pop 800mg of Ibuprofen and go to bed, where i will have a nightmare about maths, in which i'm sure i will be chased by giant calculators or something...
:nut:

Sorry for O/T

Quote from Woz :

Come on then which is right lol

And dont say it depends on context because that is what I have said already.

strictly speaking, tristan is wrong. he is describing 0.999(9) as a sequence. see point #3 .

Agreed. I was trying to get the point across to you. I suppose technically it's not zero, but it's not as big as a number (an infitesimally small number I mean)

Quote :

We accept the 0.0(0)1 from Becky maths has 0 "significance" because it allows our base 10 number system to have a “consistency” which allows it to be a better "tool" for our needs.

Quote :

becky-math might have compromises

It's not a compromise dammit, it's the future of maths, it's ... it's ... TO INFINITY (of zeros) AND BEYOND (a one) !

Binary is superior.

Quote from herki :

Binary is superior.

Not with floating points it isnt.

I did read up once on a way to store floating points in binary, i've forgotten now how it works, something to do with either square roots or power of. Anyone know?

I bet 0.(9) doesn't work too well in binary

Quote from tristancliffe :

Agreed. I was trying to get the point across to you. I suppose technically it's not zero, but it's not as big as a number (an infitesimally small number I mean)

And the point I was trying to get across is that 0.9~ must equal 1 BUT only because of the round trip of the render in base 10

To truly represent 1/3 in base 10 we really need to add 1/3 of 0.0(0)1 to 0.3~ but we cant and so 0.3~ is a good enough render that is accepted to mean the same. As an infitesimal value is a value (although we need to conceed it is not) to solve the render issue.

Quote from tristancliffe :

But at the level of 0.999...=1 we're not really considering base 10 or the actual number system itself, but the whole concept of mathematics itself. Regardless of what base you use, or even if you use bases, the concept itself holds true.

0.9~ has EVERYTHING to do with 1/n where n = base - 1. There can be NO disputing that. It has nothing to do with mathematics itself, just the base chosen to render a given value in which causes a "mismatch" in what can be rendered well in that base. It is the compromises that is required to make maths work.

1/3 in base 3 is 0.1 which is a perfect render (George, I pointed this out very early on btw). It becomes a problem rendering in base 10 for the simple fact 3 into 10 causes a recursion which means you can NEVER resolve the carried 1 so you end up with a problem if 0.0(0)1 as it were. Does not mean that 1/3 is not a valid number it is just being rendered in the best way given the limitations.

Analogue vs digital! The render of LFS by an NVidia card vs ATI card if you will. Both a view on the LFS world but both slightly different yet no less correct.

1 / 3 * 3 = 1, it can never be anything else and as all we can render 1/3 in base 10 as is 0.3~ then it stands to reason that 0.9~ = 1 as 0.3~ * 3 must be 1.

It IS a render issue of our chosen base. I think that is as concise and I can ever get

Quote from Woz :

I hope it actually highlights the point I am trying to make because I do actually understand math.

That's a bit arrogant of you. There are five pages of posts in this thread, some by people who are well-learned in maths.

[EDIT]

I didn't have time to post a more extensive reply this morning, but now I do.

So...

Quote from Woz :

When we changes from imperial to metric we also changed all the sizes of nuts and bolts etc. did 3/8" cease to be a valid size of bolt because it did not fix perfect into mm? Did 1/1000" cease to be a valid tolerance? No it is just those markers were in the "wrong" location for our new system so we discarded them.

That has nothing to do with the problem at hand.

3/8 = 0.375
1/1000 = 0.001

No problems.

And more relevantly for us:

1/3 = 0.333...
1/3 * 3 = 0.333... * 3
1/3 * 3 = 1
0.333... * 3 = 0.999...
therefore 0.999... = 1

Quote from Woz :

If you try and travel millions of light years that infitesimal might become the difference between landing or smashing into a sun. It is all just about the precision required for the task at hand, no more.

For 0.999... = 1 it isn't about the "precision required". The end result between a calculation that uses 0.999... and 1 will be identical. There is no difference.

Quote from Woz :

Base 10 numbers are not “real”, real numbers they are a continuous stream of values in infitesimal steps that DO not fit base 10. It is imperial vs metric, analogue vs digital!

0.999... = 1 has absolutely nothing to do with bases. You could be using binary, and the result will be the same.

In binary: 0.111... = 1.
In octcal: 0.777... = 1.
In hexadecimal: 0.FFF... = 1.

Quote from Woz :

To truly represent 1/3 in base 10 we really need to add 1/3 of 0.0(0)1 to 0.3~ but we cant and so 0.3~ is a good enough render that is accepted to mean the same. As an infitesimal value is a value (although we need to conceed it is not) to solve the render issue.

Again, you try to place a finite limit to a value of infinite precision. 0.333... is not merely "good enough". 0.333... is 1/3. There is no difference between 0.333... and 1/3.

The difference between 0.999... and 1 is 0.000... . And 0.000... is 0. Therefore they are the same value: 0.999... = 1.

Something that's infinitely small is nothing. That is a concept not tied to any kind of numerical base representation. So your argument that 0.999... = 1 is some sort of compromise or a make-do is invalid.

Quote from samjh :

Tristan is right.

no, he is isn't and i gave an explanation which tristan accepted

Quote from samjh :

0.333... is 1/3.

i've said the same thing. he can't get it. seems it is beyond his levels of comprehension. that, or he just won't admit he was in error. those reluctant to admit their ignorance can't learn math.

Quote from george_tsiros :

no, he is isn't and i gave an explanation which tristan accepted.

My bad. That will teach me to read quoted posts properly before replying. It was written in haste.

That portion of my reply has been removed.

Quote from george_tsiros :

i've said the same thing. he can't get it. seems it is beyond his levels of comprehension. that, or he just won't admit he was in error. those reluctant to admit their ignorance can't learn math.

I have to admit I struggled with the concept of limits and infinity when I had to apply them in later years of education. In high school it wasn't a problem, I excelled in maths and physics. At university though... it was a new level.

bump, for those who feel like reading a little bit and thinking a little more