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Gloominary wrote:However, if your hand is any closer to the table than 1 cm away from it, even just the teeniest, tiniest bit, you will touch both the 1 cm domino, and the 0.99> cm domino, as you wave your hand over the table, knocking them both over, because 0.999> is as close to being 1 as you possibly can come without being 1, so anything closer than 1 cm away, even if it's only 0.0000000001 cm closer less is going to be within its range to interact with it.
Magnus Anderson wrote:I made a mistake above. 0.999~ is represented differently in base-16 systems. Nonetheless, I am sure the point remains.
surreptitious75 wrote: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
surreptitious75 wrote:This thread is only about base I0 Magnus and so talking about other bases or systems is not relevant here
Magnus Anderson wrote:surreptitious75 wrote:This thread is only about base I0 Magnus and so talking about other bases or systems is not relevant here
I don't agree.
James S Saint » Mon Jun 05, 2017 3:30 pm wrote:0.999... is not a quantity or fixed value. It is an endless series of diminishing decimal values. The ellipsis "..." means "infinitely" - "never ending". The decimals begin but never end - open ended. The fraction is never satisfied by the series.
1 is an obvious quantity and fixed value with a beginning and an obvious end.
James S Saint » Mon Jun 05, 2017 4:32 pm wrote:the only alteration required is the acknowledgement that infinitely diminishing decimal series are technically not numbers or quantities. Such can be merely stated in a foot note. Nothing very serious changes anywhere.
Ecmandu wrote:Well, James was very firm that infinitesimals were useful, and usefully different than the “convergence”...
Hence, his InfA thing
James S Saint » Wed Sep 03, 2014 8:58 am wrote: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".
obsrvr524 wrote:It isn't actually a number. And neither are all of those other expressions that end with "...".
MagsJ wrote:I thought you posed a dilemma of sorts?
Magnus Anderson wrote:obsrvr524 wrote:It isn't actually a number. And neither are all of those other expressions that end with "...".
I am not sure why you think it's not a number i.e. a symbol representing some quantity. .
'
'
What it isn't is a finite quantity, that's for sure, and that's why it can be 1.
Silhouette wrote:0.(9) is an attempt to restate the quantity "1" in a way that involves endlessness.
obsrvr524 wrote:Silhouette wrote:0.(9) is an attempt to restate the quantity "1" in a way that involves endlessness.
That would be "1.000...", not "0.999..."
Magnus Anderson wrote:Not really. I just expanded upon Gloominary's post.MagsJ wrote:I thought you posed a dilemma of sorts?
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