Monday, February 27, 2017

Difficulty in accepting that 0.999... = 1

There are many common psychological phenomena that I find interesting, from a semi-academic perspective. One of them is the sheer difficulty that some people seem to have in accepting the notion that 0.9 recurring (ie. the decimal representation of a number with "0." followed by an infinite string of 9's) is exactly equal to 1. Both decimal expansions represent the exact same number.

Some people will go to incredible lengths to try to argue why they are not equal, up to trying to invent new mathematics, and new kinds of numbers, just to forcefully try to make them unequal. (With very little success, of course, because invariably these are people with very little experience and understanding of mathematics, especially math related to the set of real numbers, and their decimal representations.)

There are a few quite simple counter-arguments to their assertion that the two decimal representations are not the same number. For example, if 0.9... is not equal to 1, then by definition there have to exist numbers between them, ie. numbers that are larger than 0.9... and smaller than 1. Ask for an example of such a number.

Another equivalent counter-argument is: If 0.9... is not equal to 1, then what is their difference (ie. 1-0.9...)? If the difference is 0, then they are equal by definition. If it's not 0, then what is it?

One common misunderstanding these people have is that they don't really grasp what it means for the decimal representation to have an infinite amount of digits '9' in it. They will often only be thinking of it in terms of a finite amount of digits, and will argue that no matter how many digits you add to it, while it approaches 1, it never reaches that value. (I have seen some people in online conversations write miles and miles of text about this, always assuming a finite amount of 9's.) However, the number 0.9 recurring is not a function; it's not a limit statement; it's not an expression that says "0. followed by an arbitrarily large, but finite, amount of 9's". It's literally infinitely many 9's.

When pressed on enough, sometimes they will come up with the argument that the difference between the two numbers is, essentially, "the smallest real number that's larger than 0". Of course such a number doesn't exist. This is actually very trivial to prove mathematically, and one of the classical examples given in high school and university courses about proof by contradiction. (Incidentally, "the smallest rational number larger than 0" likewise doesn't exist, for the exact same reason.)

One attempt some try, after seeing some YouTube video after the subject, is to appeal to so called "infinitesimals". This is an expansion of the set of real numbers. In other words, they want to expand the very set of reals in order to make 0.9... not equal to 1. However, even this doesn't work, because even in this new set 0.9... is still equal to 1. Infinitesimals, even if we accepted them, do not change this.

And, of course, when everything else fails, but they still can't accept the fact, some will start attacking the very notation of "0.9...", claiming that you can't represent an "infinite amount of 9's". They attack the very notion of infinite decimal expansions, and how they "don't really exist". In other words, that the number "0.9..." doesn't exist, and that you can't write infinite decimal expansions.

Of course no mathematician has any problem with the notion of infinite decimal expansions. As an example, Cantor's diagonal argument relies on infinite decimal (or other base) representations. Most rational numbers, and all irrational numbers, have an infinite decimal expansion, and this is absolutely non-controversial in mathematics, and it causes absolutely no problems anywhere. (And, because of the above, in fact all integer numbers also have an alternative decimal representation: one that has an infinite amount of 9's as decimal places.)

But these people, sometimes very seriously, want to go against pretty much the entirety of the mathematical and scientific community, and rewrite mathematics, just to make 0.9... not equal to 1. No other reason, really. It's actually incredible, when you think about it.

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