Take two real numbers, for example 1 and 2. There are an infinite number of real numbers between 1 and 2. So, you can always increment 1.5 to get a new number that is also between 1 and 2. e.g 1.5 + 0.01 = 1.51. Take (1.999 …) what real number can you add to (1.9999 …) to get a number that is also between 1 and 2? There is no number. Therefore (1.9999 …) is either less than 1 or greater than 2. We know that both of those cases are absurd, so it must equal 2. Right?
I am not a particularly math-centered person and I thoroughly enjoyed this article. It is a great leg up over that stumbling block that lays before Calculus, where people must come to terms with math not being an infinitely precise, binary thing.
My biggest issue with saying 0.9… doesn’t equal one is the base thing (how would this apply to binary 0.1… and hexadecimal 0.F…). My second-biggest issue is that any “new” arithmetic dealing with 0.9… would have to allow for numbers of the form 0.5…2, etc. (That is, five-tenths plus five-hundredths plus five-thousandths, going on forever, somehow followed by a two. And it’s somehow a “bigger” number than the same thing followed by a one instead of a two.)
If we allow for this, then all sorts of patterns in decimal arithmetic break. For example, one can normally derive one-twelfth by multiplying the decimal forms of one-third and one-fourth, and this in turn may be done through extrapolation. 0.3*.25=0.075, 0.33*.25=0.0825, 0.333*.25=0.08325, 0.3333*0.5=.083325, and so on, adding a “3” to the middle of the answer. By extrapolation, we find that one-twelfth should equal 0.83333…25. Under normal arithmetic we quite correctly ignore the “25” part (it could be replaced with any other finite sequence and remain equally meaningless), so one-twelfth simply equals 0.83333…
But in a hypothetical new arithmetic, we would be wrong in ignoring that “suffix”! Which would in turn mean that there is an actual difference in the “value” of one-twelfth in decimal, depending on how we calculate it! For example, ordinary long division applied to 1/12 never involves any sort of “25 at the end”. Yet if 0.999… is less than, not equal to, 1, than 0.83333… would be less than, not equal to, 0.83333…25, right? Madness! 
This is analogous to saying 1/2+1/4+1/8… ≠ 1 because you’ll never reach one.
Well, most 0.9… “deniers” would consistently deny that equality as well, for exactly that reason. Above, I posited the question (in different terms) of whether these two sums are unequal to one another, in addition to being unequal to one? I think a consistent denial would involve those being three separate numbers, instead of the same number expressed three ways.
Another problem is saying you’ll “never” reach one implies a process linked to time, or to some other variable. If 0.9… will “never” reach one, can we say say far it is as of noon today? Has it reached ten digits, or 10^10^10 digits, or more than that even? Obviously, that’s silly. 0.9… is “all the way there”, just like 7/14 is “all the way” to 1/2.
This IS really interesting, let’s see if I can add something even more mind breaking:
1=0.9…+h
so 1/3= 0.3…+1/3h using an arbitrary close precision, then
also 1.9…=1+0.9…+h and is different from
2*0.9…=1.9…8 where the difference is exactly h to get 1.9… and 2h to exact 2.
Seems to work:)
Even when using infinite numbers, look:
h is an infinitesimal to 0.9… and multiplying them by infinite gives an infinite number and h(infinitesimal)*infinite=a number, the practical difference is still infinite among them.
There’s now the problem that 1/3h or any other of is fractions is smaller than h, a supposedly infinitely small distance…
@15 0.12341234… is not irrational. It is equal to 1234/9999, and .989898… is 98/99, or 9898/9999, etc.
Likewise, any repeating decimal is a rational number which can be expressed as a fraction, with the numerator being the repeating digits, and the denominator being the same number of nines, or to put it a bit more mathematically:
0.abc…nabc…n… = abc…n/10ⁿ-1
where n is the number of recurring digits, or the length of the numerator in the fraction.
Similarly, 0.999… is equal to 9/9 or 99/99 or 999/999, etc., all of which = 1
Also, the problem with the graph with an infinitesimal difference between the .999… graph and the 1 graph, is that you didn’t go on to infinity. You stopped at a finite value. This is analogous to saying 1/2+1/4+1/8… ≠ 1 because you’ll never reach one. Clearly this is false.
Infinity is undefined and any process of calculation involving infinite concept is likely undefined.
Infinite floating numbers can only be approximated with the smallest tolerance possible but it cannot be exactly represented as a whole number or as a fractional form of number. Even when dealing with numbers in different bases some numbers that are exactly represented in one base system would turn out to be not exact when converted in another base system representation. That problem is also true when dealing with different number forms. The fraction 1/3 (base 10) is exactly 1/3 in fractional form but it is only approximated as .3333… in decimal form. Same problems occur with other fractions that result in repeating decimals when we attempt to convert them.
2^1/2 is exactly 2^1/2 but when converted to decimal form it can only be approximated to be 1.41421…
It is therefore ridiculous to say that .999… is exactly equal to 1. The correct statement should be .999… is APPROXIMATELY equal to one and we should just be fine with that in our real world calculations where numbers are applied.
@Madison: Great question. I think this gets to the heart about what a number is. Traditionally, numbers were a count of something (fingers, rocks, sheep, etc.) so everything was straightforward, or in chunks. Over time, numbers evolved to be fractions, decimals, and real numbers – becoming “continuous” in a way. The problem with continuity is that you can keep subdividing it (seemingly forever), and things like 1/3 = .33333… means that “well, we can keep subdividing our smallest counting unit (tenths, hundredths, thousandths) into 3”. Rounding is an easy way to “deal” with this (“just call it .33 and ignore the rest”) but it brings up issues with what we mean by precision, etc. It’s a really interesting issue and something I want to explore more! Appreciate the comment.
Why is it that we were raised to just simply round the number?
When doing a mathmatical euqation that .00001 that you take out can make all the difference. If decimals and negative numbers are really not numbers at all, why must we deal with them?
Rational or irrational, INFINITELY repeating numbers are just approximations of the fractional number form it’s trying to represent.
.999… has no exact fractional representation in our base ten numbering system and it is NOt equal to 1
Repeating numbers should only be treated as the final approximated answer value in any calculation. Using it to process an algebraic equation and assuming( falsely) that it could be an equivalent representation to some other number in order to simplify or cancel out terms would result in an incorrect and imprecise proof.
And your statement, “not all decimal forms can be exactly represented in fractional form” is absolutely ludicrous.
“2^1/2 is exactly 2^1/2 but when converted to decimal form it can only be approximated to be 1.41421…
It is therefore ridiculous to say that .999… is exactly equal to 1.”
2^1/2 is not rational and cannot be represented by a fraction. .999… is a repeating decimal and therefore IS rational.
1.00…1 doesn’t mean anything. Increase it by 0.00…9, and it’s 1.00…1 again. So
1.00…1 + 0.00…9 = 1.00…1? That makes no sense.
@Anonymous: Sort of. Do we have infinite precision in anything, and if you do, do you have infinite computer storage space to store that infinitely-precise number, like 1.20000000… (exactly)? If we can get 2 numbers beyond our knowingly-finite error tolerance, very, very close (too close for us to measure) = there.
The fact that we say it’s “infinitely” repeating implies that it would never end and it would never reach finality. Coming up with a final result is just an imaginary concept of infinity.
0.9999… is just an approximation of 1 in pure terms and it is no different from saying 1.000…1 is an approximation of 1.
This problem arise because in every number system not all fractional forms can be exactly represented in decimal form just as not all decimal forms can be exactly represented in fractional form.
Just because we can round off a number to its closest value when we calculate real world problems does not mean the rounded number is exactly equivalent to its closest value form.
This discussion is really more of a discussion on how much tolerance in number
value is good enough for a calculation process.
Lastly, is it even possible to come up with an exact value when a number that is dependent on the concept of infinity is multiplied with by number?
After all, any nonzero number multiplied by infinity would still be infinity.
Lenoxus, you’re partially right.
The “alternative system” talked about here is called ‘trans-finite arithmetic’. It’s about as crazy as Cantor’s cardinal calculus. Anything relating to infinitesimals is rather crazy. But it’s terribly useful. The normal rules of arithmetic aren’t completely broken in this language, but they take more care.
And it is very useful when doing calculus and functional analysis. Suppose you have a function y(x). If you think of dy and dx as an ‘infinitesimal’, rather than as a limiting process, you suddenly allow yourself to do thing like 1/(dy/dx) = dx/dy and other tricks which physicists have been doing for more than a century, with great ease. True, it needs care. But it can be made rigorous, and it’s as useful as it is entertaining.
“Very, very, very,… close”, is not equal to “there”.
0.999… = 9/10 + 9/100 + 9/1000… = 1.
A simple proof to disprove that 1 is NOT equal to .99999…
If you assume that .9999… = 9/10, and 1 = .9999…
It must also follow that 9/10 = 1
However, that is not true because claiming that 9/10 = 1 would lead us to believe that 9 =10 . That is of course FALSE and mathematically unsound, therefore 1 is NOT equal to 0.999…
The weakness in all the proofs presented so far to “prove” that 1 =.999… lie in the FALSE and erroneous assumption that fractions can always be exactly and perfectly represented in decimal form without degrading their true values.
Should have read: " A simple proof to DISPROVE the claim that 1 is equal to .99999…