Linked Questions
21 questions linked to/from Proof that every repeating decimal is rational
7
votes
7
answers
4k
views
Is the number 0.2343434343434.. rational? [duplicate]
Consider the following number:
$$x=0.23434343434\dots$$
My question is whether this number is rational or irrational, and how can I make sure that a specific number is rational if it was written in ...
- 111
3
votes
3
answers
7k
views
How write a periodic number as a fraction? [duplicate]
What I call as a periodic number is for exemple
$$0.\underbrace{13}_{period}131313...$$ or $$42.\underbrace{465768}_{period}465768465768.$$
So how can we put theses numbers like a integer ...
- 11.9k
363
votes
31
answers
62k
views
Is it true that $0.999999999\ldots=1$?
I'm told by smart people that
$$0.999999999\ldots=1$$
and I believe them, but is there a proof that explains why this is?
40
votes
4
answers
34k
views
How can I prove that all rational numbers are either terminating decimal or repeating decimal numerals?
I am trying to figure out how to prove that all rational numbers are either terminating decimal or repeating decimal numerals, but I am having a great difficulty in doing so. Any help will be greatly ...
- 793
32
votes
2
answers
3k
views
Why is the decimal representation of $\frac17$ "cyclical"?
$\frac17 = 0.(142857)$...
with the digits in the parentheses repeating.
I understand that the reason it's a repeating fraction is because $7$ and $10$ are coprime. But this...cyclical nature is ...
- 14.7k
6
votes
7
answers
26k
views
Why is a repeating decimal a rational number?
$$\frac{1}{3}=.33\bar{3}$$
is a rational number, but the $3$ keeps on repeating indefinitely (infinitely?). How is this a ratio if it shows this continuous pattern instead of being a finite ratio?
...
- 5,063
37
votes
1
answer
12k
views
Is there a proof that $\pi \times e$ is irrational?
A little reading suggests:
It is known that either $\pi + e$ or $\pi \times e$ is transcendental (or possibly both), but no proof is known that one of those two numbers in particular is ...
- 2,561
6
votes
4
answers
3k
views
Seven line segments, with lengths no greater than 10 inches, and no shorter than 1 inch, are given. [closed]
Seven line segments, with lengths no greater than 10 inches, and no shorter than 1 inch, are given. Show that one can choose three of them to represent the sides of a triangle.
- 1,559
10
votes
4
answers
4k
views
Rational numbers and periodic decimal representation [duplicate]
I'm trying to prove that a number is rational if and only if it has an eventually periodic decimal expansion. One part is simple; without loss of generality we consider $q=0.\overline{d_1\dots d_k},$ ...
- 101
5
votes
4
answers
7k
views
Prove that a number is rational if and only if from some point on its decimal expansion becomes periodic. [duplicate]
Q: "Prove that a number is rational if and only if from some point on its decimal expansion becomes periodic"
Please help!! I am relatively new to algebra and I find these questions very abstract.
...
- 101
11
votes
3
answers
23k
views
Detecting that a fraction is a repeating decimal
Given any fraction where both the numerator (N) and denominator (D) are both positive and are both whole numbers.
Without manually dividing N by D, is it possible to pre-determine if the resulting ...
- 225
13
votes
3
answers
6k
views
How do you calculate how many decimal places there are before the repeating digits, given a fraction that expands to a repeating decimal?
If you have a fraction such as $$\frac{7}{26}=0.269230\overline{769230}$$ where there are a number of digits prior to the repeating section, how can you tell how many digits there will be given just ...
- 133
6
votes
4
answers
678
views
Are there any natural proofs of irrationality using the decimal characterization?
Mathematicians typically define rational number to mean quotient of two integers. It is not hard to show that a number is rational by that definition if and only if its decimal expansion terminates ...
3
votes
3
answers
3k
views
Is there an alternative proof for periodic expansion of decimal fraction?
I'm currently reading Elementary Number Theory and Its Applications by Kenneth H. Rosen. In chapter 12 - Decimal Fraction, he provided a proof about the period length of the base $b$ is $\mathrm{ord}...
- 12.2k
4
votes
2
answers
4k
views
Hint for: Prove any terminating decimal can be represented as a rational number
I am currently working on a problem from Hardy and have been stuck trying to figure out what to do. I was wondering if someone could provide me with a hint that may help jump-start my thought process.
...
- 2,676