I have been thinking about a continuing decimal, i.e. 0.707007000700007...
obviously, the decimals composite parts are 0.7+0.007+0.000007+...
My question is this, is this decimal rational or irrational? And a reason why?
I first started thinking of geometrical progressions and summing to infinity, but I found that my common ratio, r, had to change to create the decimal, so I personally think it's irrational as the repeating block is changing?
i.e. 0.7 + 0.7(10^-2) + 0.7(10^-2)^2 + ... +
Does anyone have a neat proof of this or even better, think that I am wrong?
Kind regards, tom
That's incorrect though. That would work if our repeating block was constant wouldn't it? Because the fraction is recurring i.e. the sequence goes to infinity?
Therefore the number, with your reasoning would be:
S = a/(1-r)
S=1/(1-10^-2)
S= 1.01010101
Which has a constant repeating block, where as my decimal is, with removing 0.7 as a factor, would be 0.101001000100001000001... etc
Anymore ideas? Or am I missing something?
It was my fault, I don't think I described my question in a mathematically concise way...
So, because of this, we would state that the decimal is, in fact, IRRATIONAL? Because it cant be expressed in the following way:
a/(10^l - 1), where a,l are some positive integers? Because, for example, if our fraction had a denominator of 9999, this would imply a repeating block of 4 digits long, but my decimals repeating block varies, and thus has a varying magnitude of l?
Your number is 7 times the number 0.101001000100001000001... = , where the exponents are the triangular numbers (Triangular number - Wikipedia, the free encyclopedia). The number is irrational because all rational numbers either have a terminating decimal representation, or one that consists of a group of continually repeating digits. The reason I know this is because of how the long division algorithm works (Long division - Wikipedia, the free encyclopedia).
An arbitrary rational number is given by the quotient of two integers which we shall represent by m and n. Without loss of generality, we may assume that both m and n are positive and that m < n. In each step of the algorithm, we are left with a unit digit divisor and a remainder r such that 0 ≤ r < n. Since there are at most n possibilities for r, the digit sequence of divisors must either terminate (r = 0) or repeat in a cycle with size < n.
Since your number does not have a constantly repeating, fixed length cycle of digits in its decimal representation it is irrational.
I suspect that it is also transcendental; however, considering how long it took mathematicians to prove that e and are transcendental, I doubt that proving it would be easy.