"The identity 0.999... = 1 is a convention" (Timothy Gowers).
"0.999... does not represent a number, but a process" (The Straight Dope).
"The argument that finally convinced me was this: they must be the same [1 and 0.999...] because there is no gap to fit any other decimal between them" (Steven Strogatz)
The Paradox
It is admitted in mathematics that 1 = 0.999.... (in short, 0,9). The paradox consists of: How is it possible that the same number has two different expressions in the same numbering system (base 10)? How is it possible that a finite number (1) is equal to a number expressed with infinite decimal places (0.999...)?
There are many arguments or demonstrations in favor of this equality:
Since 1/3 = 0.333..., multiplying by 3 both sides of the equality, we have: 1 = 0.999....
Analogously, since 1/9 = 0.111..., multiplying by 9 both sides of the equality, we have: 1 = 0.999...
If x = 0.999... and 1 were different, there would be an intermediate number between them, which seems not to be the case, since (1 + x)/2 = 1.999.../2 = 0.999.... = x
If x = 0.999..., then multiplying by 10 both terms, we have:
10x = 9.999... (the decimal point is shifted one position to the right)
But 9.999... = 9 + 0.999... = 9 + x
Therefore, 10x = 9+x and x=1.
A pure periodic decimal number is nothing but a disguised form of an infinite arithmetic series. If we have the arithmetic progression a, ar, ar2, ar3,.... with r<1, their sum is x = a+ar+ar2+ar3+... Multiplying both sides by r, we have:
xr = ar + ar2 + ar3 +... = x−a x=a/(1−r)
The number 0.999... is the sum of the elements of the infinite geometric progression 0.9, 0.09, 0.009, ..., whose ratio is 0.1 and whose sum is 0.9/(1 − 0.1) = 1.
There is also a geometric proof due to Benjamin G. Klein and Irl C. Bivens [1988]:
In the figure, triangle ABC is similar to triangle ADE. Therefore, x/1 = 1/(1 − r), being.
x = 1 + r + r2 + r3 +....
This is easily generalizable to
x = a + ar + ar2 + ar3 +....
The expression 0.999... = 1 is generalizable. Every finite decimal number has a twin with infinite nines. Examples:
This paradox has been presented as the dual of the previous one.
If we add 1 to y = ...999, the result is zero: y+1 = 0. Therefore, y = −1.
If we have x = 0.999.... 10x = x+9 x=1, then we have analogously:
y = ...999 y/10 = y + 0.9 y = −1.
It is seen that x and y are opposites of each other: their sum is zero: x+y = 0.
The paradox occurs that an infinitely large number (...999) is equal to −1, and that 0.999... and ...999 are opposites of each other: 0.999.... = −...999
This result can be generalized for any sequence of digits that repeat indefinitely. For example,
x = 0.341341341....
103x = x+341
x = 341/(103−1)
y = ...341341341 y/103 = y + 0.341
y = −341/(103−1)
The general formulas for a sequence m of length n are:
x = m/(10n−1)
y = −m/(10n−1)
All numbers composed of infinite sequences of digits fall between −1 and +1.
Numbers of the type x+y are always zero, regardless of the sequence of digits. For example, ...341341341.341341341... = 0
Explanation of paradoxes
The numbers 1/3 and 1/9 are inexpressible in an exact and finite way in the base 10 numbering system because 3 and 9 are not submultiples of 10, with 0.333... and 0.111... are the maximum possible approximations, respectively. Instead, in base 3 these numbers are expressed in an exact and finite way: 1/3 = 0.1(3 and 1/9 = 0.01.
0.999... is not a concrete number, but a borderline number between finite and infinite. It is infinitely close to 1 and is expressed in infinite decimal places. It is a number that we can qualify as imaginary because it implies infinity. The number 0.999... and all those with infinite decimal places can be called "imaginary".
The expression 1 - 0.999... is an infinitesimal, a number infinitely close to zero, but it is not zero. An infinitesimal is a positive real number less than any positive real number. This number does not exist, it is imaginary. Since 1 - 0.999... is an infinitesimal, 1 - 0.999.... > 0 y 1 > 0.999...
The expression 1 = 0.999... is an expression that implies consciousness because it connects the two modes of consciousness, it unites the opposites: the finite and the infinite, the quantitative and the qualitative, the operative and the descriptive, the real and the imaginary, the rational and the intuitive.
Therefore, 0.999... is not 1. And ...999 is not −1. In the case of 0.999... it is the sum of infinitely many summands whose limit is 1. The equality 0.999.... = 1 is a convention based on the notion of limit: the value of an infinite series whose limit is a finite value is the value of its limit.
Cantor's Representation
Cantor [1869] invented a system that allows any rational number to be represented exactly by a finite sequence. With this system fractions can be represented without any loss of accuracy [Wayner, 1988].
In Cantor's system a factorial basis is used: each place value is equal to the factorial of that place:
For example,
301.102(F = 3*3! + 0*2! + 1*1! + 1/2! + 1/3! + 2/4!
where the subscript F indicates "factorial basis".
It can be shown that a number anan−1. ..a1.a1a2. .. an(F based on factorials meets property 0 ≤ ai ≤ |i|, i.e., the number having the maximum digits is
...5 4 3 2 1.1 2 3 4 5...
These properties allow analogies to be drawn between the factorial-based system and the traditional base-10 exponential system. For example:
Exponential
Factorial
999 = 1000 − 1
321(F = 1000(F − 1
0.999 = 1 − 0.001
0.123(F = 1 − 0.001(F
Examples of integer representation
Number
Representation
1
1(F
2
10(F
3
11(F
4
20(F
5
21(F
6
100(F
7
101(F
8
110(F
9
111(F
10
120(F
Examples of representation of fractions less than 1
Fraction
Representation
1/2 = 1/2!
0.1(F
1/3 = 2/3!
0.02(F
2/3 = 1/2! + 1/3!
0.11(F
1/4 = 1/3! + 2/4!
0.012(F
3/4 = 1/2! + 1/3! + 2/4!
0.112(F
1/5 = 1/3! + 1/4! + 2/5!
0.0104(F
The number e
As a curiosity it is worth mentioning that the number e is represented as
10.111...(F = 10.1(F since
e = ∑(1/i!) (summation between 0 and infinity)
The factorials as "anti-primes"
Just as prime numbers refer to themselves and represent mathematical archetypes, factorials are the polar opposite of primes − we can call them "anti-primes"− they are the ultimate surface manifestations because they contain as factors a series of consecutive numbers starting from 1, which are all their divisors. The factorial of a number n contains (arithmetically speaking) the numbers 1 to n, that is, n! can be divided into equal groups of 1 (ngroups), 2 (n/2 groups), 3 (n/3 groups), ..., n (1 group).
Therefore, representing numbers with Cantor's system has two fundamental advantages:
It is founded or supported by the numbers of highest level of manifestation.
Rational numbers are represented finitely and without loss of precision. There are no cases where infinite decimal places appear, as with the traditional representation.
Bibliography
Cantor, George. Über Die Einfachen Zahlensysteme. Zeitschrift für Mathmatik und Physik, vol, 14, 1869, pp. 121-128.
Dauben, Joseph Warren. George Cantor. His Mathematics and Philosophy of the Infinite. Princeton University Press, 1990.
Gazale, Midhat. Number. From Ahmes to Cantor. Princeton University Press, 2000.
Klein, Benjamin G.; Bivens, Irl C. Proof without words. Mathematics Magazine, 61 (4), October 1988.
Strogatz, Steven. El placer de la X. Taurus, 2013.
Wayner, Peter. Error-Free Fractions. BYTE, Junio 1988, pp. 289-298.
