# Tagged Questions

Questions about real numbers not expressible as the quotient of two integers. For questions on determining whether a number is irrational, use the (rationality-testing) tag instead.

204 views

### Expansion of $(1+\sqrt{2})^n$

I was asked to show that $\forall n\in \mathbb N$ there exist a p $\in \mathbb N$* such that $$(1+\sqrt{2})^n = \sqrt{p} + \sqrt{p-1}$$ I used induction but it wasn't fruitful,so I tried to use the ...
27 views

### Can a change of basis modify irrationality/transcendance?

Fix a real number $x$. We can consider its binary expansion, for instance $x = (0.01101001100101101001011\ldots)_2$. Now we consider the real number $y = (0.01101001100101101001011\ldots)_{10}$ : we ...
253 views
+50

152 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 ...
106k views

### Does Pi contain all possible number combinations?

I came across the following image, which states: $\pi$ Pi Pi is an infinite, nonrepeating (sic) decimal - meaning that every possible number combination exists somewhere in pi. Converted ...
867 views

### Understanding non-solvable algebraic numbers

Background We know from Galois theory that the zeros of a polynomial with rational coefficients whose Galois group is solvable can be expressed in a formula that involves rational powers of the ...
116 views

### Any real number has at most two decimal representations [closed]

Here, Tao says that any real number has at most two decimal representations. Is this really true? I always thought $\pi$ has only one decimal representation.
55 views

### Is $\pi e$ irrational? [duplicate]

During our ongoing research, we need to prove that $\pi e<\lceil \pi e\rceil$. Is $\pi e$ irrational? How to prove it? Thanks- mike
70 views

### Notation for representing ANY number?

i'm working on a mathematics/number-manipulation program, and i was wondering if you could practically have a representation that could holds the value of any number. This would need to include ...
177 views

### Addition, subtraction, multiplication and division of irrational numbers, correct to $n$ decimal places

Suppose we want to do one of the four basic arithmetic operations on two irrational numbers, and we want some confidence that our answer is correct to $n$ significant figures/decimal places. Doing ...
57 views

### Can an irrational number be expressed as a sum of other irrational numbers, at least one of which is not an integral multiple of the required number?

For example, $\pi = Ae + B\sqrt 2+ \cdots$ ($A,B,\ldots\in\mathbb R$) (Equations like "$\pi = 3\pi - 2\pi$" are not allowed.)
96 views

54 views

### Difference between rationalizing factor and conjugate surd

I have some confusion regarding rationalizing factor and conjugate surd. For binomial surds for example $2+\sqrt{3}$ is conjugate of $2-\sqrt{3}$ and it is also rationalizing factor of $2+\sqrt{3}$. ...
4k views

### Sum of two periodic functions is periodic?

I have following paragraph taken from the Stanford's study material. Question: Is the sum of two periodic functions periodic? Answer: I guess the answer is no if you are Mathematician, yes ...
77 views

### Is there any $\alpha$ for which $e^{\alpha}$ is an integer?

Is there any $\alpha$ which gives $e^{\alpha}$ an integer. $\alpha=0$ is the trivial one. But is there any other than $0$?
30 views

### Show that $θ_0 < Arg (z^α) < θ_0+\epsilon$ for infinitely many values of $z^α$, where $−π < θ_0 < π$ and $\epsilon > 0$

For $z \neq 0$ and $α$ irrational, show that $θ_0 < Arg (z^α) < θ_0+\epsilon$ for infinitely many values of $z^α$, where $−π < θ_0 < π$ and $\epsilon > 0$. I am trying to solve this ...