Decimal numbers can be either rational numbers or they can be irrational numbers such as pi, sqrt(2), e, phi, etc. In the world of positive integers, a question arises. Can decimal values be represented using only positive integers? The answer is yes. Rational numbers are simply a ratio of positive integers, and irrationals can be represented using integers in continued fractions or series.
Depending on the base used, the rational numbers either have terminating or repeating decimal representations. For example, in base 10, we have:
1/5 = 0.2 which terminates since 5 is a factor of the base 10.
1/8 = 0.125 which again terminates since 8=23 and 2 is a factor of the base 10.
1/7 = 0.142857... which has a period of 6 decimal digits and repeats.
Repeated decimals can be converted to fractional equivalents using the following algorithm :
Let f = 1/7 = 0.142857... for example (in base 10)
Its has a decimal period of 6, so multiply f by 106 :
Therefore : 1000000 x f = 142857.142857...
f = 0.142857...
Subtracting : 999999 f = 142857
Therefore f = 142857/999999 (which of course reduces to 1/7).
For those interested in the nature of the repeating decimal, 1/7 is now
replaced with a fraction of positive integers which retains the decimal digits
of the cycle. These digits (in base 10) reflect the nature of 1/7 in base 10.
So all rational decimal numbers can be expressed as the ratio of positive integers in either reduced or unreduced form.
In addition, fractions (and therefore rationals), represent partitions. For example, 100/5 really means take 100, and split it into 5 parts. Therefore this is equivalent to partitioning 100 into 20+20+20+20+20. In partition theory, this is only one of many partitions of 100 into 5 parts. Similarly, 100/21 is the same as 21+21+21+21+16. The value 100/21=4.76190... is really saying that we have 5 parts and one of them (the remainder 16=21*.76190...) is smaller than 21. Partitions have the advantage that they can represent splits with more than 2 parts that are different. For example, it is not possible to represent 42+21+16+14+7 as a fraction, but this is also a valid partitioning of 100 into 5 parts. Partitions cover all such splits. Therefore fractions, or the division operation, are really very specific partitions. So rationals can be replaced with partitions of integers. Partitioning is an important operation in the realm of positive integers, and I address this subject further when I discuss inverse operators on another page.
Irrationals can be also be expressed as continued fractions or infinite series using only positive integers.
The following are continued fractions for phi, sqrt(2), pi, and e.
phi = 1 + 1/(1 + 1/(1 + 1/(1 + ... )))
sqrt(2) = 1 + 1/(2 + 1/(2 + 1/(2 + ... )))
4/pi = 1 + 12/(2 + 32/(2 + 52/ (2 + 72/(2 + ... )))
e = 2 + 1/(1 + 1/2 + 1/(1 + 1/(1 + 1/(4 + 1/(1 + 1/(1 + 1/(6 + 1/(1 + ... )))))))
The following are a few of the many possible infinite series that use integers to represent the irrationals pi and e for example.
Pi = 1 + 1/2 + 1/3 + 1/4 - 1/5 + 1/6 + 1/7 + 1/8 + 1/9 - 1/10 + ...
Here the sign of the term is determined as follows:
Given 1/n is the term, if n = prime == 1(mod 4) then use minus sign.
If n = prime == 3(mod 4) then use plus sign.
If n = composite, then the sign is the product of the prime factors of n
as determined above.
The nature of pi is more clearly revealed in this continued fraction
than it is in the decimal value 3.1415926... which is base 10 dependent.
Pi2/6 = 1 + 1/22 + 1/32 + 1/42
+ ... + 1/n2 + ...
This is a specific case of the famous Riemann-Zeta series. And there are
many other series representations of pi.
e = 1 + 1/1! + 1/2! + 1/3! + 1/4! + ... + 1/n! + ...
The idea here is that decimal values are redundant since an integer representation exists for each decimal value. I fully realize the usefulness of decimal values in calculations, however the point is that in the theory of numbers, decimals are not necessary. In otherwords, there are equivalent integer expressions that reflect the same value. In fact, the fractional version reflects the internal structure of the number better than its decimal equivalent. In the decimal value, only the digits exist to reflect its nature, whereas the fractional values (using either partitions or continued fractions) clearly reflect an orderly and complex pattern.
Enjoy