‘Proof that .999… is NOT equal to 1’ – by Kenneth Ahistrom is Mathematically in Error.
(i.e. It’s comically wrong)
In 2018 a software developer by the name Kenneth Ahistrom wrote a blog entry on Medium that still unfortunately makes its way into the never ending online “debate” of if .999… = 1. Why this blog is still up is beyond me, as over 5 years (since the time of this blog entry), you would think this person would have read the numerous comments which correctly explain why he was completely and utterly wrong. But alas it is still on Medium, and is still being used as a source for those who are not very mathematically literate to objectively incorrectly argue that .999… < 1. This is my response blog that people can use to “counter” the bizarrely wrong arguments by Kenneth, should his blog come up in the course of a discussion.
Kenneth states that “It seems that .99999… really does = 1”, but that is immediately followed by:
“That is, at least, until one applies true analytical rigor to the proofs and finds that they are all fallacious”. (Spoiler: Ironically Kenneth uses absolutely no actual rigor of any kind, more over any “true analytical rigor”). He also explicitly states that Polymathmatic, Wikipedia, PurpleMath, and Khan Academy are “all wrong”. Kenneth also recognizes he is an “amateur”, has an “argumentative streak”, and that he has a burden of proof to show that mathematical experts in their field are wrong on the matter “as it should be”.
Kenneth’ first claim is that “Everyone agrees that 1/3 = .33333…”, but that is just an “approximation”. This is quite a specious argument, as it appeals to the common understanding of what it means to approximate. We often for ease of calculations use approximations, based upon the need for precision. We can approximate pi for most general calculations as 3.14159, or 3.1415927 for more precise engineering usages, or use even more decimal places such as “3.141592653589793238462643383279502884197” for calculations for estimating the measurement of the circumference of the observable universe to within one hydrogen atom. All of these are “approximations” of pi since pi has an infinite number of decimal places, we can in theory be as precise as we want, but we can never write out the entire decimal expansion of pi. So we use a shorthand to represent pi…which is the symbol of “π”, but even the symbol of “pi” is just a shorthand notation that represents the entire infinite decimal expansion of pi.
Equivalently, “.333…” is a shorthand symbol that represents the infinite decimal expansion of ⅓. We can not write out an infinite string of numbers, but we can write something that represents that infinite string. So what is being approximated here? .333 would be an approximation of ⅓. .33333 would be an approximation of ⅓. .333333333333333333333333333333333333 would be an approximation of ⅓. But .333… is not approximating anything. “.333…” is symbolically representing the same exact value as ⅓, just expressed in a different form of notation called an “infinite decimal expansion.”
Kenneth then proceeds to ask the reader to “Try to convert .33333… into ⅓”. Ok, let’s convert .33333… into ⅓” using the same exact method that Euler used to prove that 9.999… = 10 in “Elements of Algebra” (circa 1795)
Let x = 0.333…
Multiply both sides by 10:
10x = 3.333…
Subtract x from the LHS and what it equals (by initial assumption) .333… to the RHS:
10x – x = 3.333… – .333…
Simply:
9x = 3
Divide each side by 9:
x= 3/9
Simply:
x= 1/3
QED
But wait! Kenneth seems to take issue with this method and says “In order for two numbers to be exactly equal to each other, you must be able to convert its visual representation both ways”. What rules is this? What does this even mean? He then reiterates that “.33333… is just a ‘lazy’ approximation of 1/3. The two numbers are not actually equal.”. So what would be an exact way to represent ⅓ to Kenneth? He initially promised us “true analytical rigor”, but doesn’t ever show us his non-lazy way to expressly represent ⅓ in infinite decimal expansion.
But wait even more! Kenneth has foreseen this proof and calls it “especially insidious”. Euler, one of the, if not the greatest mathematician who ever lived being, charged with being “insidious”? Kenneth claims it is “a simple case of algebraic trickery”, and attempts to show this supposedly mathematical sophistry by arguing:
10x = 9.999…0
10x – x = 9.999…0 – .999…
9x = 8.999…
His conclusion happens to be correct since 8.999… = 9 and thus x=1, but his reasoning is flawed. In what decimal place does “0” exist here after an infinite decimal expansion? It is like saying you take an infinite trip and after that you can rest. In .999… there is simply no point in which a decimal places fails to be a “9”. Every decimal place is a “9” as there is an infinite number of 9’s. There simply can not be a “0” after a string of infinite numbers. This is what Kenneth believes is constitutive of “true analytical rigor”?
Kenneth then tries to argue that .999… is not 1 by geometric progression. He actually gets it right that 9/10 + 9/100 + 9/1000 + 9/10000 is a geometric progression, but fails to note that geometric progressions have a common ratio. A geometric progression can be expressed by the formula:
S=a/(1-r) where (S) is the Sum of the geometric ratio, (a) is the initial term, and (r) is the common ratio.
Given:
(S) = .999… (sum of the infinite geometric progression)
(a) = 9/10 (initial term)
(r) = 1/10 (common ratio)
999… = 9/10 /(1- (1/10)) = 1
QED
But wait again! Kenneth seems to think that because .999… never “exceeds” .999… that somehow means it can not be 1 because of “infinitesimals”? The problem with this is that the set of real numbers (ℝ) does not contain any infinitesimals other than 0 due to the Archimedean property. There simply is no real number between .999… and 1. It is like asking what real number exists between ½ and .5? Or what number is the highest number next to 1, when no such number exists as ℝ is dense and for any two given unequal real numbers you can find an infinite set of numbers between them. In other words, I can get arbitrarily closer to 1 all I want numerically, but I can always find a real number that is even closer to 1. So how do we eventually reach one? We reach one by taking what is called an “infinite summation” for a series (Σ .9 + .09 + .009 + .0009 +…) or a limit of a sequence (.9, .99, .999, .9999, +…).
Formulated:
lim n=1 to ∞ .999…9 = 1 where “.999…9” n times.
However I think this is easier to understand by Zeno’s paradox by using an infinite summation. An arrow in flight to a target at some point in time travels ½ the distance to the target. Then at another point in time it travels ½ from that distance to the target. Then ½ again, and again, and again giving us the summation of ½ + ¼+ ⅛ + ¹⁄₁₆ +…
We know that the arrow eventually strikes the target and we can express that mathematically by using geometric progression of a=½ (initial term) and r=½ (common ratio):
S = (½)/(1-(½)) = Σ ½ + ¼+ ⅛ + ¹⁄₁₆ +… = 1
But wait again once more! Kenneth now attempts to invoke the “hyperreals” (ℝ*) to try to make his argument. The hyperreals are an extension of the real numbers and therefore has all the same rules as ℝ and .999… = 1 must be true in ℝ* eo ipso. This fact that all statements for ℝ applies to ℝ* is referred to as the “transfer principle”. Moreover, invoking a ℝ* takes us out of real analysis and into non-standard analysis which wouldn’t therefore change anything on how .999… = 1 in real analysis.
Kenneth then erroneously concludes that “.99999… was never exactly equal to 1”. As they say Garbage In. Garbage Out. Kenneth starts with a profound misunderstanding of a multitude of mathematical misconceptions, which leads him to objectively wrong conclusions. He advises the reader to “Question everything and everyone, even the experts. If something feels wrong and it’s <sic> ‘proofs’ seem insufficient, do more research”. To which I advise Kenneth to take his own advice and seriously do some more research, a lot more research…and perhaps retract his blog from the internet so save anyone else from having to refute it ever again.
