Intereting Posts

What is the derivative of this?
What is continuity, geometrically?
$A \subseteq \mathbb R^n $ s.t. for every continuous function $f : A \to \mathbb R$ , $f(A)$ is closed in $\mathbb R$ , is $A$ closed $\mathbb R^n$?
Show that $A=\{ \frac{m}{2^n}:m\in \mathbb {Z},n\in \mathbb {N} \} $ is dense in $\mathbb {R}$
Probablility of a dart landing closer to the center than the edge of a square dartboard?
An extrasensory perception strategy :-)
Lipschitz and uniform continuity
Regularity of elliptic PDE with coefficients in some Sobolev space
Without AC, it is consistent that there is a function with domain $\mathbb{R}$ whose range has cardinality strictly larger than that of $\mathbb{R}$?
Probability/Combinatorics Problem. A closet containing n pairs of shoes.
Topology exercises
Conditional expectation of product of Bernoulli random variables
Positive integer multiples of an irrational mod 1 are dense
Proof on p. $16 \;$ of Lang's Algebraic Number Theory
Evaluating $\int^1_0 \frac{\operatorname{Li}_3(x)}{1-x} \log(x)\, \mathrm dx$

We can build $\frac{1}{33}$ like this, $.030303$ $\cdots$ ($03$ repeats).

$.0303$ $\cdots$ tends to $\frac{1}{33}$.

So,I was wondering this:

In the decimal representation, if we start writing the $10$ numerals in such a way that the decimal portion never ends and never repeats; then am I getting closer and closer to some irrational number?

- Constructive proof that algebraic numbers form a field
- Is there a rational number between any two irrationals?
- Does every sequence of rationals, whose sum is irrational, have a subsequence whose sum is rational
- Relationship between degrees of continued fractions
- Why must $a$ and $b$ both be coprime when proving that the square root of two is irrational?
- Proof that the irrational numbers are uncountable

- Different types of transcendental numbers based on continued-fraction representation
- Uncountable set of irrational numbers closed under addition and multiplication?
- Fractions in binary?
- binary representation of a real number
- Proving that for each prime number $p$, the number $\sqrt{p}$ is irrational
- Teaching irrational numbers?
- Can someone explain these strange properties of $10, 11, 12$ and $13$?
- What is a real number (also rational, decimal, integer, natural, cardinal, ordinal…)?
- How to find the radix (base) of a number given its representation in another radix (base)?
- Enough Dedekind cuts to define all irrationals?

The decimal expansion of a rational number is always repeating (we can view a finite decimal as a repetition of $0$’s)

If $q$ is rational we may write it as an irreducible fraction $\dfrac{a}{b}$ where $a,b\in\mathbb{Z}$. Consider the Euclidean division of $a$ by $b:$

At each step, there are only finitely many possible remainders $r\;\;(0\leq r< b)$. Hence, at some point, we must hit a remainder which has previously appeared in the algorithm: the decimals cycle from there **i.e.** we have a repeating pattern.

Since no rational number can be non-repeating, a non-repeating decimal must be irrational.

Closer and closer? If you mean by *closer and closer* as in approximating an irrational, yes. The idea behind *closeness* is that there is some destination behind where you are going whilst writing the number, but in an irrational number there is no destination, you simply keep writing. If it would be possible, with perfect information, to keep writing out the decimal expansion of an irrational number, making sure there is absolutely no repetitiveness or patterns, then you would be getting *arbitrarily close* to the irrational number, (in terms of $\epsilon$ close). An irrational number has a non-terminating, non-repeating decimal expansion. So there is no real idea of closeness here, unless you are talking about distance, ($\epsilon$ close), in which case yes you are getting closer.

- Is there a general formula for $\sin( {p \over q} \pi)$?
- Prove $a^ab^bc^c\ge (abc)^{\frac{a+b+c}3}$ for positive numbers.
- How to verify this limit?
- Fibonacci Generating Function of a Complex Variable
- Describe all the complex numbers $z$ for which $(iz − 1 )/(z − i)$ is real.
- I can't remember a fallacious proof involving integrals and trigonometric identities.
- Big Bang Theory Reference to Formal Logic
- Set of Ideals of a Polynomial Ring
- Quadrature formula on triangle
- Maps between Eilenberg–MacLane spaces
- Is closure of convex subset of $X$ is again a convex subset of $X$?
- Characteristics in the theory of PDE's – What's Going On?
- Does a median always exist for a random variable
- Prove that $\pi$ is a transcendental number
- Show uncountable set of real numbers has a point of accumulation