The Popular Misconception Of One Divided By Three

Divide one by three using a calculator, and it shows 0.333, with the number three repeating on and on.  However, 0.333 multiplied by three equals 0.999, and even the same calculator should get this much right.  This has compelled some to think that logic is not inherently true because contradictory things can be true at once, such as one being equivalent to 0.999.  They might also conclude that one divided by three must really be 0.333... (the ellipses here signify an infinitely repeating decimal).  There are other errors someone could believe about this issue.  One divided by three does not equal 2.777 or 5, for instance.  It is just that these or other examples are not the predominantly accepted delusion on the matter.  The truth is both extremely basic and extremely important.  This specific matter is not especially philosophically central on its own, but the truth of logic is supremely foundational, and since nothing can actually contradict logic and still be true, as I have repeatedly fixated on, the popular misconception is pivotal.

Because calculators reach 3.333 upon dividing one by three and arrive at 9.999 upon multiplying 9.999 by three, some people actually think that one and 9.999 are the same, or that math can deviate from the laws of logic.  Whatever the reason, there would be a logical reason why logic is false, so this is an intrinsic impossibility.  And 0.333 is a third of 0.999, not of 1.  Many people might assume that if a calculator presents an answer, it must be correct.  This is just a less overtly recognized form of an appeal to authority.  In truth, the only authority that has intrinsic correctness is logical axioms (even other necessary truths ultimately stem from them), with all other things being authoritative only to the extent that they concur with pure logical necessity or possibility.  A calculator is just a machine that might misrepresent the abstract truths of mathematics, which are themselves grounded strictly in logic.  With dividing one by three, these machines simply display the numeric symbols for a loose approximation that cannot possibly be correct.

It turns out that anyone who believes the somewhat popular idea that one divided by three amounts to a number that does not equal one when multiplied by three is wrong because the idea must be false.  Also, it could not have been any other way.  One divided into three could not possibly be 0.333, and so on, because the latter number added to itself three times (the same outcome as the number multiplied by three) logically has to be 9.999, with the decimal nine repeating infinitely.  Erroneous calculator solutions cannot falsify reason itself, and accurate calculator solutions are not a valid epistemological basis for believing anything—except that the calculator has arrived at a given number!  As an aside, logic is not primarily about numbers, though it necessitates certain truths about numbers.

No matter what, relying on calculators for anything more than the pragmatic purpose of facilitating the likes of academic work or professional calculations with a seemingly high probability of veracity is irrational.  Unless you mentally go through every single step in the calculation while looking to objective logic and making no assumptions, you would never know if a calculator (or another person, an AI, etc.) is right in any instance.  Reason alone can reveal this because reason alone grounds necessary truths, including those about numbers.  Not even memorizing an equation or the output with a given set of starting numbers means someone knows the truth of the matter.  One has to directly, actively grasp the logical necessity of why a specific number results.

Though the subject is so simple, the idiotic controversy over the real nature of one divided by three pertains to the very heart of all truth because if the popular misconception is true, then logical axioms would be false.  But logic cannot be false, because then there would be a logical reason why this is the case, which requires that reason is true either way.  One thing that follows from another cannot be anything but true, for instance, or else it would follow from the nature of reality that one thing which follows necessarily from another is untrue; or, it would follow from the nature of reality that nothing follows from anything—while these are two truths, they are both rooted in the same singular axiom that one thing which follows from another must be true because anything contrary still requires this to be so, and this is the case entirely independent of more specific examples of the axiom applying to something besides itself.  And if something contradicts any such self-necessary, self-evident truth, of which there are very few, it is automatically impossible.  One and 0.999, with the nine repeating, are not the same number, and, inevitably, 0.333 multiplied by three does not become one.