Number if rational iff it has periodic decimal expansion

In summary, the conversation discusses the proof that a number is rational if and only if its decimal expansion becomes periodic at some point. The method of long division is used to show that rational numbers have repeating decimal expansions. The other direction is also proven, but there is some confusion about notation in the proof. The notation for the periodic part of the decimal expansion is explained as each ai representing a non-repeating digit and each ci representing a digit in the repeating part. The conversation ends by acknowledging the helpfulness of the forum as a resource.
  • #1
Logik
31
0
My teacher gave us as excercices this:
Prove that a number is rational if and only if from some point on its decimal expansion becomes periodic.

I'm pretty certain you have to prove it by contradiction, but I don't get how to represent to periodic decimal expension in a proof?

Any hint is welcome, thanks in advance.
 
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  • #2
First, to show that rational => repeating decimal expansion, remember that long division is how you get from a rational number to its decimal expansion. If r=a/b, then there are only b possible remainders at each step. What happens when the same remainder comes up a second time (after you've gone through all the digits in a)? I'll let you work on the other direction.
 
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  • #3
Ok well I guess my intuition was bad. I made a few search on books.google.com and found a couple of proofs. One is pretty easy and I actually though of that solutions before but didn't know how to generalise it. My only problem now is that I'm not familiar with one notation in the proofs.

I don't understand the reprensetation of the period, so in this image the first line where it says x= a_n ...

http://img131.imageshack.us/img131/3996/proof2or5.jpg [Broken]
 
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  • #4
each ai is a non repeating digit of the number; each ci is a digit of the repeating part of the number. So, for example

1.1234343434...
a0 = 1
a-1 = 1
a-2 = 2

c1 = 3
c2 = 4

m=2
 
  • #5
Thanks... that was kinda easy but I guess you just need to know it. Btw good job, this forum is a great ressource :P
 

1. What does it mean for a number to have a periodic decimal expansion?

A number has a periodic decimal expansion if its decimal representation contains a repeating pattern of digits. For example, the number 1/3 has a periodic decimal expansion of 0.333..., where the digit 3 repeats infinitely.

2. How can I determine if a number has a periodic decimal expansion?

To determine if a number has a periodic decimal expansion, you can divide the numerator by the denominator using long division. If the division results in a repeating pattern of digits, then the number has a periodic decimal expansion.

3. Is every rational number guaranteed to have a periodic decimal expansion?

Yes, every rational number has a periodic decimal expansion. This is because rational numbers can be expressed as fractions with a finite number of digits in the numerator and denominator, which will result in a repeating pattern when divided.

4. Can irrational numbers have a periodic decimal expansion?

No, irrational numbers cannot have a periodic decimal expansion. Unlike rational numbers, they cannot be expressed as fractions and their decimal representations do not contain a repeating pattern of digits.

5. What significance does a periodic decimal expansion have in mathematics?

The concept of a periodic decimal expansion is important in understanding the properties of rational numbers and the relationship between fractions and decimals. It also helps in calculating and approximating irrational numbers. Additionally, it has applications in various fields such as number theory, geometry, and physics.

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