I've been told authoritatively that 0.999999... equals 1. Unfortunately, I have never been that good at listening submissively. When looking at repeating nines, I see a rich universe of values approaching one. I agree that–for all practical calculations–we can treat endless decimals with nothing but nines as one, yet I am unwilling to say with absolute certainty that convergent sequences reach their point of convergence. Repeating nines stand in that mysterious transition of a converging sequence becoming one. I see within that transition itself a great mysterious universe filled with intrigue.
By repeating nines, I am referring to an infinite decimal in base ten. An infinite decimal is the summation of an infinite series of the form a1/10, a1/102, a3/103, ..., an/10n, ... where a is a number from 0 to 9. Each entry in the series is a full magnitude smaller than the previous entry. All infinite decimals converge. Theorists have speculated that infinite decimals can converge to any point on the number line. For example, the decimal expansion of pi might look something like: 3.14159265358979323846... I've been told that the full infinite expansion of the decimal actually equals pi. With great temerity, I still hold that any decmimal expansion is never exactly equal to pi. The decimal expansion is simply an aproximation of pi.
Note, that if c is equal to the first n digits of pi, then 1 - (pi - c) contains a string of (n-1) repeating nines. With each additional digit in our calculation of pi, we extend our series of nines and get closer to the real value of pi. However, a million digit decimal is still not equal to pi. If we subtract the difference of the absolute value of pi from a finite decimal representation of pi, we will get a decimal of repeating 9s. 1 - (pi - 3.14159265358979323846...) = 0.99999999999999999999... Expanding pi indefinitely still leaves us with a mysterious logical entity between pi and the decimal.
Saying that 0.99999... equals 1 is the same thing as saying that converging sequences actually reach the point of convergence. Although the quantities are essentially the same, absolute equality requires a major metaphysical leap.
One of the best places to start a discussion on repeating nines is to simply try to subtract an infinite decimal with repeating nines from 1. Try to perform the following subtraction:
The first thing I hope you notice is that there is essentially nothing between the two numbers. On the other hand, it is impossible to actually do the subtraction. We write decimals from left to right, but do subtraction from right to left.
Infinite decimals are unbound on the right. There is not a furthest right hand digit. That means we cannot do standard subtraction.
We can try doing a left to right subtraction. In the first digit to the right of the decimal we see we are trying to subtract 9 from 0. So we borrow the 1 from the left of the decimal. This gives us 10 - 9 which leaves a 1. We can borrow this new 1, and subtract the next 9 from a 10 leaving 1. We can repeat this process of borrowing ones forever. These are both infinite strings, and we are in an infinite loop. Each iteration of the infinite loop leaves us with a smaller and smaller remainder. Yet there is always a remainder. Subtracting 0.99999... from 1.0000... leaves 0.00000... plus a remainder that has been infinitely dimished.
Some might try calling the remainder an infinitesmal, but clearly there is no longer any appreciable space or value between one and repeating nines. The diminished remainder is simply the creation of our logic. I will call it a logical entity. As repeating nines and the unit whole have essentially the same value, are they the exact same thing, or does the existence of this dimensionless logical entity play a role in things?
Perhaps, I should preface the question with an even more fundamental question: Should the existence of this strange logical entity even be acknowledged? Acknowledging the existence of the question brings to light that there may be more than one possible theory for the nature of calculus. Should we not just say authoratively that repeating nines equal one reject the question?
For transfinite theorists, the question itself is troubling: Repeating nines have essentially the same value as 1. In this regard it belongs to the set of things that equal one. From outward appearance, 0.9999... belongs to the set of things less than one. Having a number that appears to belong to both the set of things less than one and belonging to the set of things equal to one is extremely troubling to transfinite theorists as it breaks down the division between less than and equal. How can something be both less than and equal? The methods used by Eudoxes demands a clear distinction between less than and equal. This group chooses to say repeating nines are 1. Repeating nines are not part of the set of things less than one.
Those who studied logic in the tradition of Aristotle tend to say that the repeating nines approach the unit whole but never equal it. Such a view permits us to jump from repeating nines to one. Yet this wishy-washy-ness is a disconcerting. A third view sees repeating nines like the barber of Seville. The Barber of Seville is the only barber in town. He states that everyone in town either gets his hair cut in his shop, or they cut their own hair. The barber's clear distinction falls apart when asked which group he belongs to. He belongs to both groups.
There are different views as to how we should handle this strange logical entity dividing repeating nines and the unit whole. Saying that repeating nines equals the unit whole leads to a nice internally consistent model. Because mathematicians like this model they reject the question. I believe that there are many different legitimate models that we can use to describe space and time. Following John Cougar Mellencamp's example, I fight authority, knowing authority always win.
My thoughts on this subject begin at an interesting point: I've been told authoratively that every number on the the number line can be expressed by a distinct infinite decimal. This leads to a problem: Let's say that q is an irrational number. As a member of the reals it can be expressed as a distinct infinite decimal. If I understand the term "distinct" to mean that q differs from all other infinite decimals at a finite digit from the decimal point (I will call this digit n), then I can construct a real number that exists between q and all real numbers less than q. To do so, I simply truncate q at a digit somewhere to the right of digit n.
For my concept of infinite decimal to be complete, I have to modify my understanding of distinct decimal. That is I have to accept that there might be two distinct decimals a and b, where a and b do not begin to differ until after an infinite number of digits from the decimal point.
Accepting such an idea means that I cannot establish absolute equality simply by starting at the decimal point and comparing digit by digit. In other words, I might have two decimals with repeating nines, but the decimals themselves are different.
For example (1 - 1/2n) where n approaches infinity is equal to 0.99999. Also (1 - 1/3n) = 0.99999... Comparing digit by digit, they appear the same, but they still might be distinct numbers.
To an extent these converging sequences are distinct. Each operation converges at a different rate. Converging sequences may have the same ultimate value, but still having some subtle distinctions. We could compare the results digit by digit and find no difference and even treat them as practically equal. Yet there still might be a difference at the absolute level.
Repeating nines seem to offer the same problem as multiplying and dividing by zero 0 * 5 = 0 * 4. Dividing both sides of the equation by 0 we get 5 = 4. There really is not a moral precept against dividing by zero. It is just that information gets lost when we perform division by zero...causing problems in our reasoning process. Converging sequences lets us get around some of the problems of dividing by zero. Is this in part because the sequences preserve information otherwise lost?
Fractions behave in an interesting way when converted to infinite decimals. The infinite decimal represenation of fractions all end in repeating digits. For example 1/3 = 0.3333... The three will repeat indefinitely. Sometimes, several digits will precede the repeating digits. For example 1/6 = 0.1666... The 6 is the repeating digit.
In base ten, When the denominator of the fraction has no other factors than 2 or 5, the decimal representation will end with repeating zeros. For example 1/2 = 0.5000..., 1/5 = 0.2000..., 1/20 = 0.050... All other fractions end in repeating digits.
In the last section of this essay, I suggested that two numbers might have the same decimal representation for an infinite number of digits and still not be the same number. I have wondered at times if perhaps the infinite decimal 0.333... and 1/3 are actually the same thing.
As I do the long division that produces 0.3333..., I notice that each iteration of the long division leaves me with a remainder. Certainly, each step in the expansion of the decimal gets me closer to the true value of 1/3, but there is still a remainder. Expanding the decimal 1000 digits gives me a number extremely close to a third, but there is still a remainder of 1/(31000).
Here is the really strange thing: Clearly, three thirds equals a whole. 1/3 + 1/3 + 1/3 = 1. If I add up the infinite decimals, I get repeating nines:
0.333333... 0.333333... 0.333333... ============ 0.999999...
You are likely to come across math texts claiming that: if three thirds equals a whole, and repeating threes add up to repeating nines; Then repeating nines equals a whole. The flaw in this proof is with the stipulation that repeating threes equal a third. The same mysterious logical entity that stands between between repeating nines and a whol stands between repeating threes and a third. After then nth iteration of the expanding a third into decimal form, I still have a remainder 1/(3*10(n+1)). A third equals repeating threes plus the logical entity. Adding three thirds equals repeating nines plus the strange logical entity.
The demonstatration that three repeating threes add up to a repeating nine adds to my belief that repeating threes might be different from the absolute value a third.
There are some strange things going on with fractions. Let's say I had a fraction of the form a/b where b is has factors other than 2 or 5. a is a number between 1 and b. Such a fraction would be between 0 and 1.
The digits in the decimal expansion of a/b and (b-a)/b will add up to repeating nines. Here are a few samples:
1/7 = 0.142857... (142857 repeats) 6/7 = 0.857142... 2/7 = 0.285714... 5/7 = 0.714285... 3/7 = 0.428571... 4/7 = 0.571428... 1/6 = 0.166666... (6 repeats) 5/6 = 0.833333... (3 repeats) 5/11 = 0.454545... (45 repeats) 6/11 = 0.545454... (54 repeats) 1/11 = 0.0909090... (90 repeats) 10/11= 0.9090909... (09 repeats)
I find it interesting that the infinite decimals of a/b and (b-a)/b will always add up to repeating nines.
Earlier in this brain fart, I mentioned that the (1 - 1/2n) converges at a different rate than (1 - 1/3n). Altough both numbers produce an endless string of repeating nines, I am willing to accept that they still are different numbers. The fact that the infinite decimal expansions of a/b + (b-a)/b always creates an string of repeatings nines in one step makes me wonder if the repeating nines produced by adding 0.333... and 0.666... is absolutely equal to the repeating nines produced by adding 0.090909... and 0.909090...
In Everything and More, David Foster Wallace presents an interesting trick to convince the world that repeating nines "equal" the unit whole. He starts with x = 0.9999... He then subtracts x from 10*x as follows:
9.99999... 0.99999... =========== 9.00000...
10 - 1 = 9. Hence 9x = 9.000... implying that x=1.0. We start with x = 0.999... and conclude that x equals 1.0. In the essay above we noted that a digit to digit comparison from left to right does not prove that numbers are actually the same thing. This leaves the possibility that (10 * 0.9999... - 9) is not actually equal to 0.9999... despite the fact that we can match the digits from left to right.
I do not like limiting myself to base ten. All bases behave similar to base ten. Working in base six, I would find that fractions of the form a/b, where b has factors other than 2 or 3, then a/b + (b-a)/b will add up to repeating fives.
Base two is perhaps the most interesting base. All fractions would be represented as infinite decimals with the digits 0 and 1. For example 1/3 = 0.010101... In base two, repeating 1s play the same role as repeating 9s in base ten. Does 0.11111... = 1? In base two, the question of repeating nines is restated as: Is a zero followed by repeating 1s the same as a 1 followed by repeating zeros?
Base two is interesting in that you can do all of your mathematics with bit functions. For example, when b is not a power of 2, a/b turns out to be the bit not of (b-a)/b.
Repeating nines and the unit whole are essentially the same quantity. However, I have never been able to say with absolute certainty that they are the same thing. As we are unable to complete any infinite task, we do not know for certain that converging sequences ever actually equal the point of convergence. Certainly, we can make mathematical models that assume repeating nines equal one. Yet other mathematical models are equally valid.
When considering the density of the set of all infinite decimals, I am left with a strange puzzle. Let's say a is a distinct infinite decimal. When I think of a as a member of the set of infinite decimals, I cannot find a finite point where it becomes distinct. That means that I could have two distinct infinite decimals, but not be able to prove they are distinct by starting at the decimal point and comparing digit by digit.
The only conclusion I can derive from such thought experiments is that we do not know anything conclusively about infinite numbers.
A Tale of Two Paradoxes ~ ~ Salt Lake City Music