Is 1 = 0.999... ? Really?

MagsJ · December 27, 2019, 12:00am

It can, but the mathematical use of a recurring number, in industry, would lose its utility at a certain point.

Magnus_Anderson · December 27, 2019, 12:03am

I made a mistake above. 0.999~ is represented differently in base-16 systems. Nonetheless, I am sure the point remains.

MagsJ · December 27, 2019, 12:05am

…and the point is?

Magnus_Anderson · December 27, 2019, 12:34am

The point is that there are numbers larger than 0.999~ but smaller than 1. For example, 0.FFF~ (in hexadecimal system) is larger than 0.999~ but smaller than 1. In base-20 system, we have 0.JJJ~ which is larger than both hexadecimal 0.FFF~ and decimal 0.999~.

MagsJ · December 27, 2019, 12:41am

…and the solution would be?

Magnus_Anderson · December 27, 2019, 1:58am

Solution to what?

MagsJ · December 27, 2019, 2:18am

I thought you posed a dilemma of sorts?

surreptitious75 · December 27, 2019, 2:56am

This thread is only about base I0 Magnus and so talking about other bases or systems is not relevant here
Also all the relevant arguments have already been made which is why the thread stopped two years ago

MagsJ · December 27, 2019, 3:01am

Yes, but bases can be consolidated, no?

Magnus_Anderson · December 27, 2019, 7:31am

I don’t agree.

Ecmandu · December 27, 2019, 4:06pm

I don’t agree either. Other bases are fair game in mathematical debates. (Btw… Magnus is right)

obsrvr524 · December 27, 2019, 8:15pm

When I was reading through this, I noticed the following that seems to bring it all to a salient conclusion.

It makes sense that “0.999…” is just an expression signifying that some ratio cant be expressed by a fixed number of digits (unlike 1). It isn’t actually a number. And neither are all of those other expressions that end with “…”. And apparently that is why the Wikipedia proofs are misleading.

To me, that seems like a “game over”.

Ecmandu · December 27, 2019, 10:05pm

Well, James was very firm that infinitesimals were useful, and usefully different than the “convergence”…

Hence, his InfA thing

obsrvr524 · December 27, 2019, 10:33pm

This seems to explain why he would -

"[url=http://www.ilovephilosophy.com/viewtopic.php?f=1&t=185999&p=2492512:

James S Saint » Wed Sep 03, 2014 8:58 am[/url]"]

Calculus
Ancient
The ancient period introduced some of the ideas that led to integral calculus, but does not seem to have developed these ideas in a rigorous and systematic way. Calculations of volume and area, one goal of integral calculus, can be found in the Egyptian Moscow papyrus (c. 1820 BC), but the formulas are simple instructions, with no indication as to method, and some of them lack major components.[2] From the age of Greek mathematics, Eudoxus (c. 408−355 BC) used the method of exhaustion, which foreshadows the concept of the limit, to calculate areas and volumes, while Archimedes (c. 287−212 BC) developed this idea further, inventing heuristics which resemble the methods of integral calculus.[3] The method of exhaustion was later reinvented in China by Liu Hui in the 3rd century AD in order to find the area of a circle.[4] In the 5th century AD, Zu Chongzhi established a method that would later be called Cavalieri’s principle to find the volume of a sphere.[5]

Medieval
Alexander the Great’s invasion of northern India brought Greek trigonometry, using the chord, to India where the sine, cosine, and tangent were conceived. Indian mathematicians gave a semi-rigorous method of differentiation of some trigonometric functions. In the Middle East, Alhazen derived a formula for the sum of fourth powers. He used the results to carry out what would now be called an integration, where the formulas for the sums of integral squares and fourth powers allowed him to calculate the volume of a paraboloid.[6] In the 14th century, Indian mathematician Madhava of Sangamagrama and the Kerala school of astronomy and mathematics stated components of calculus such as the Taylor series and infinite series approximations.[7] However, they were not able to “combine many differing ideas under the two unifying themes of the derivative and the integral, show the connection between the two, and turn calculus into the great problem-solving tool we have today”.[6]

Modern
“The calculus was the first achievement of modern mathematics and it is difficult to overestimate its importance. I think it defines more unequivocally than anything else the inception of modern mathematics, and the system of mathematical analysis, which is its logical development, still constitutes the greatest technical advance in exact thinking.” —John von Neumann[8]

In Europe, the foundational work was a treatise due to Bonaventura Cavalieri, who argued that volumes and areas should be computed as the sums of the volumes and areas of infinitesimally thin cross-sections. The ideas were similar to Archimedes’ in The Method, but this treatise was lost until the early part of the twentieth century. Cavalieri’s work was not well respected since his methods could lead to erroneous results, and the infinitesimal quantities he introduced were disreputable at first.

The formal study of calculus brought together Cavalieri’s infinitesimals with the calculus of finite differences developed in Europe at around the same time. Pierre de Fermat, claiming that he borrowed from Diophantus, introduced the concept of adequality, which represented equality up to an infinitesimal error term.[9] The combination was achieved by John Wallis, Isaac Barrow, and James Gregory, the latter two proving the second fundamental theorem of calculus around 1670.

Newton didn’t publish work on it until 1687, 17 years after Wallis, Barrow, and Gregory had done the conceptual theorems in modern language.

Sometimes you just have to ignore religious fanatics, especially the kind who claim to not be religious yet can’t explain why they know what they think they “know”.

I’m no maths genius but isn’t calculus just the sum of the infinitesimals? Where would science be without calculus?

Ecmandu · December 27, 2019, 10:40pm

Obsrvr,

I’ve been in many threads with James …

So I’m certainly putting my own words in his mouth:

James would say that both infinitesimals and approximations are useful in their own right and for different reasons.

MagsJ · December 27, 2019, 10:47pm

Calculus is awesome… it spans multi-disciplines… where would industry be without it.

Magnus_Anderson · December 28, 2019, 1:44am

I am not sure why you think it’s not a number i.e. a symbol representing some quantity. It appears to me that it clearly is. It has many properties that numbers have e.g. it’s greater than some numbers and less than others.

What it isn’t is a finite quantity, that’s for sure, and that’s why it can’t be 1.

Magnus_Anderson · December 28, 2019, 1:45am

Not really. I just expanded upon Gloominary’s post.

obsrvr524 · December 28, 2019, 3:16am

If it isn’t a “finite quantity” I imagine that it isn’t a number.

Silhouette · December 28, 2019, 3:55am

The whole question boils down to a confusion between the qualitative and the quantitative.

“1” is clearly a precise quantity, but as soon as you profess 0.(9) you add in the quality of “endlessness” to describe the repetition of the quantity of “9” for each decreasing power of 10 (or whatever base you’re using).

0.(9) is an attempt to restate the quantity “1” in a way that involves endlessness. As is 0.(3) to restate 1/3 when one divides 1 by 3. It’s an admission that one cannot denote 1/3 etc. entirely quantitatively without the use of the quality of endlessness. Multiplying 1/3 again by 3 is obviously 1 (3/3), yet multiplying 0.(3) by 3 is not so obviously 1 (0.(9)) precisely because of the injection of the qualitative into the otherwise entirely quantitative.

Subtracting 0.(9) from 9.(9) to get the exact quantity of 9 requires the same confusion.
As soon as you allow the notion of the qualitative into the quantitative you invite possibilities such as ε as an epsilon number and so on.

This is the same kind of mistake that every extended or “new” number set allows - much to the advancement of mathematics and other utilities… but not truths. Experientialism highlights the distinction.
So we see how useful it is to make particular types of mistakes that are not true, but are useful: such as the notion that 1 =/= 0.(9)
Is it really? No.
But that’s the wrong question.
The more useful question is whether any new knowledge can be gleaned from the possibility that 1 =/= 0.( 9)