Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am following a proof from a book, but it seems to have a typo or use some strange notation I have never seen, and I have already spent a lot of time trying to decipher this, but cannot find the solution.

It is basically showing one how to construct general real numbers from Dedekind's definition by expressing those as a finite or infinite n-ary function.

So, it is trying to prove that any real number $x = a_{0}\cdot a_{1}a_{2}a_{3}.... = a_{0}+\frac{a_{1}}{n} + \frac{a_{2}}{n^2} + ...$ with $ 0\leq a_{k}\leq n-1 $ for $k=1,2,3,...$.

It starts by pointing out that $x$ being irrational, it can be defined as the Dedekind cut $x=D_{a}|D_{b}$. Then it considers the sets

$C_{k}=\left \{ \frac{m}{n^k} \right \}$ , $m=...,-2,-1,0,1,2...$

such that $C_{0}\subset C_{1}\subset C_{2} \subset ... \subset C_{k} \subset C_{k+1}$. In each of these there is a largest number $m_{k}$ such that $m_{k}/n^k$ belongs to $D_{a}$ while $m_{k}+1/n^k$ belongs to $D_{b}$ so that

$ \frac{m_{k}}{n^k}\leq x \leq \frac{m_{k}+1}{n^k} $

After this it says that since $C_{k} \subset C_{k+1}$, it follows that

$m_{k}|n \leq m_{k+1} \leq m_{k} n+(n+1)$

and in the author's words "that is, $m_{k+1}=m_{k}n+a_{k+1}$, where $0 \leq a_{k+1}\leq n -1$".

So it says that this completes the informal proof, since by choosing k=0,1,2,..., this leads to an algorithm generating n-ary fraction.

I get lost after the part that says "since $C_{k} \subset C_{k+1}$", as in the equation that follows this the author uses the cut symbol |, for $m_{k}$, and at the last inequality there is no mathematical operator between $m_{k}$ and n + (n+1) and I assume a product is not involved since there are no parenthesis. Same thing happens in the equation below between and $m_{k}$ and $n$, where in the book there is even a visible space between the two numbers.

Any hints on what happens from the point I got lost on will be extremely helpful.

The book I am reading is "Linear Algebra for Quantum Theory" by Per-Olov Lowdin.

share|cite|improve this question
up vote 2 down vote accepted

The cut symbol is a typo. This is supposed to be $ m_k n \le m_{k+1} \le m_k n + n - 1$.

share|cite|improve this answer
Thanks. It all makes sense now. – Raphael R. Mar 25 '11 at 18:01

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.