One time I was pondering about expressing irrational numbers as infinite series of fractions.
I came to the conclusion that, If we allow an infinite number of fractions, we can express every positive real number as an infinite series of fractions, by taking the decimal expansion of the number and converting each decimal to fractions.
For example,
$\frac{10}{4} = \frac{2}{1} + \frac{1}{2} + \frac{0}{1} + \frac{0}{1} + \cdots$
Or for an irrational example,
$\pi = \frac{3}{1} + \frac{1}{10} + \frac{1}{25} \cdots$
After that, I started pondering about expressing positive reals as infinite sums of fractions where none of the fractions were like - I thought for 2 seconds more and realized we could just scale each fraction so its denominator is different from all others.
But here’s a problem:
Can we express every positive real number $n$ as a sum of an infinite number of fractions, where none of the fractions are equivalent to $n$ or any other fraction in the series, nor like?
It’s always a good first step to check integers:
$2 = \frac{3}{3} + \frac{4}{4} + \frac{0}{5} + \cdots$
No, that won’t work as $\frac{3}{3}$ is equivalent to $\frac{4}{4}$.
The good thing is I already verified two statements:
- 1: Yes, we can express every positive real number $n$ as an infinite sum of fractions, using decimal expansions and converting them to fractions.
- 2: Yes, we can express every positive real number $n$ as an infinite sum of unlike fractions, which we can achieve by taking the argument from statement 1 and scaling each fraction so that all denominators are different.
Now, I really don’t know how I should try going at the third statement:
- 3: Can we express every positive real number $n$ as a sum of an infinite number of fractions, where none of the fractions are equivalent to $n$ or any other fraction in the series, nor like?
What I would usually do is try to check the following checklist:
- Is it true for integers?
- Is it true for rationals?
- Is it true for irrationals/transcendentals?
- If all above are ticked, can you confirm it works for all positive reals?
We can actually check the first condition for integers:
If $n$ is an integer, $n = (n-1) + 0.999999\cdots$. Then we can expand it as follows:
$n = \frac{n-1}{1} + \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \cdots$
Thanks to @peterwhy for this idea.
Help is appreciated for the other $3$ checkboxes!
Note: Two fractions are equivalent if their simplest/reduced forms are equal, and two fractions are like if they have the same denominator.
Note 2: Fractions can be proper ($\frac{a}{b}$ where $a < b$) or improper ($\frac{a}{b}$ where $a>b$).
Yes, in lots of ways. The simplest is probably the greedy algorithm for unit fractions: given a positive real number $x$ and some already chosen unit fractions $\frac1{m_1},\dots,\frac1{m_k}$ with $\frac1{m_1}+\cdots+\frac1{m_k}<x$, choose $m_{k+1}$ to be the smallest positive integer not in $\{m_1,\dots,m_k\}$ such that $\frac1{m_1}+\cdots+\frac1{m_{k+1}}<x$. Then it's straightfoward to show that $\sum_{j=1}^\infty \frac1{m_j} = x$.