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
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 ...
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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?
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
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### 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 ...
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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
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### 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
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
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
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
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
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 ...
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 ...
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}...
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