Understanding Repeating Decimals: Standard Notation & Behaviour

[ebaywannabe]

This is a simple question and I'd appreciate a simple answer. ;-)

So, 2/3 is .6666... Is it a repeating decimal like this only in base 10 decimal system? Is this just an artifact of using base 10? It's still rational, even though you can't write it out? What is the standard notation for this type of thing? 10/11 is another example. Is this just due to the base 10? If so, are there repeaters that repeat independently of the number system you are using? Or is this kind of "weird" behavior number system specific?

Thanks.

 

[robert Ihnot]

What's 2/3 look like in base 3? Hint: What does 2 look like and what does 6 look like in base 3?

 

[D H]

If you count 0 as a repeating decimal (e.g., 1/2=0.5000...), then every rational number expressed as a decimal fraction is a repeating decimal. If you want to be picky, things like 1/2 are called terminal decimals. Now, what about other bases? Being picky again, the word "decimal" implies base 10. The generic term is "place-value notation". Every rational number has either a terminating or repeating place-value notation in any integer base. For example 1/2 in base three is 0.111... A number that does not have a terminating or repeating place-value notation in a particular integer base (e.g., pi in base 10) does not have a terminating or repeating place-value notation in any integer base. The only numbers that behave this way are the irrationals, and all of the irrationals behave this way.

 

[Xevarion]

So, here's an interesting question: what proportion of rational numbers in [0, 1] are repeating and what proportion are terminal in base $b$? Define the proportion of repeating rational numbers to be
$$\lim_{N\to \infty} \frac{\#\left\{\frac{p}{q} : q \le N, (p, q) = 1, \frac{p}{q} \text{ is repeating}\right\}}{\#\left\{\frac{p}{q} : q \le N, (p, q) = 1\right\}}.$$

 

[Office_Shredder]

Note that anything that terminates has two representations, e.g. 1/2=.5=.4999999... one of which isn't just trivial zeroes over and over again

 

[Xevarion]

Perhaps we should be clear about our definitions then: a place-value representation is repeating iff there is a unique place-value representation, and that representation has nonzero terms after the $$n^{th}$$ place for any $$n$$. A representation is terminating iff there is some place-value representation which only has finitely many nonzero terms.

 

[CRGreathouse]

The proportion is zero. Almost all denominators will have a prime factor not dividing the base.

 

[CRGreathouse]

$$\lim_{N\to \infty} \frac{\#\left\{\frac{p}{q} : q \le N, (p, q) = 1, \frac{p}{q} \text{ is repeating}\right\}}{\#\left\{\frac{p}{q} : q \le N, (p, q) = 1\right\}}.$$

The denominator is http://www.research.att.com/~njas/sequences/A002088 (N) ~ $3N^2/\pi^2$. The numerator varies a lot, but I'm sure it's $o(N^2)$.

 

[Hurkyl]

So, 2/3 is .6666... Is it a repeating decimal like this only in base 10 decimal system? Is this just an artifact of using base 10? It's still rational, even though you can't write it out?

What do you mean "you can't write it out?" You wrote it out in this very passage: '2/3'.

 

[mathwonk]

look at the proof that a rational number is repeating. it has nothing to do with base 10.

 

[HallsofIvy]

Perhaps we should be clear about our definitions then: a place-value representation is repeating iff there is a unique place-value representation, and that representation has nonzero terms after the $$n^{th}$$ place for any $$n$$.

Where does that say anything about "repeating"? Wouldn't the decimal representation of $\pi$ fit that?

 

[Xevarion]

haha I guess I wasn't sufficiently specific there. Yeah I meant to exclude irrationals, so my definition should also specify that after some $n$, the sequence of digits is periodic.

Another interesting question: how long can the period be, as a function of the size of the denominator? What is the average relationship?

 

[CRGreathouse]

haha I guess I wasn't sufficiently specific there. Yeah I meant to exclude irrationals, so my definition should also specify that after some $n$, the sequence of digits is periodic.

My answer was only considering rational numbers. The proportion of rational numbers that are terminating in base b (b fixed) is zero. _The proportion of real nunmbers that are terminating in base b is also zero, of course.)

 

[CRGreathouse]

Another interesting question: how long can the period be, as a function of the size of the denominator? What is the average relationship?

The length of the period of a/b (if a/b is in lowest terms) divides $\varphi(b)$, so the length of the period of a/b is between 1 and b - 1. I'm not aware of any results on the average length of periods.

Also, you can divide out any factors of the base in the denominator. In base 10, factors of 2 and 5 don't change the period.

 

[sathish mat]

may i know please the meaning of squaring the circle

 

[sathish mat]

what is the meaning of locally exact in complex

 

[D H]

sathish,

First, welcome to Physics Forums.

Now a word of admonition: Please don't hijack existing threads to ask a question that is quite unrelated to the topic at hand. The thing to do is to start a new thread with your questions. These sound like homework questions, so I would suggest posting them in our homework section.

 

[mathwonk]

thank you and may i ask the proof of the poincare conjecture?

 

[robert Ihnot]

Originally Posted by Xevarion
Another interesting question: how long can the period be, as a function of the size of the denominator? What is the average relationship?

97 has a period of 96 decimals. Looking into the matter, 257 has a period of 256 = 2^8. These are called "full reptend primes."

William Shanks found pi to 707 decimals, but was only correct to 527 of them; also worked out tables of the length of all reciprocals of primes to 20,00. Proceeds of the Royal Society, London, #22, 1874.

C R Greathouse: The length of the period of a/b (if a/b is in lowest terms) divides , so the length of the period of a/b is between 1 and b - 1.

The length of the period is the smallest power n such that 10^n==1 Mod b. So the length of the period is b-1 when 10 is a primitive root of b. Otherwise it is a divisor of b-1.

For example, Sloane's Table A046146, http://mathworld.wolfram.com/PrimitiveRoot.html lists 3 as the smallest primitive root of 7. BUT 3^6 Mod 7 is the same as 10^6 Mod 7, so the length of the period is 6. Similarly we can see that if 3 is the smallest primitive root of 17, we have 3==3+17==20 Mod 17. Looking at (20)^8==(2^8)(10)^8==(1)*10^8==-1. So that 10^16 is the smallest power of 10 congruent to 1. Thus the period of 17 is 16.